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English-French Pages 326 [340] Year 1982
Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
962 Category Theory Applications to Algebra, Logic and Topology Proceedings of the International Conference Held at Gummersbach, July 6-10, 1981
Edited by K.H. Kamps, D. Pumplen, and W. Tholen
Springer-Verlag Berlin Heidelberg New York 1982
Editors
Klaus Heiner Kamps Dieter Pumpl0n Walter Tholen Fachbereich Mathematik und Informatik Fernuniversit~t - Gesamthochschule L0tzowstr. 125, 5800 Hagen Federal Republic of Germany
AMS Subject Classifications (1980): 18-06, 03D, 05C, 06D, 08A, 13C, 13E, 16A, 18A, 18B, 18C, 18D, 18F, 18G, 20L, 26E, 46A, 46B, 46G, 46M, 54B, 54D, 54E, 55F, 55N, 55P, 57M ISBN 3-540-11961-2 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-38?-11961-2 Springer-Verlag NewYork Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. © by Springer-Verlag Berlin Heidelberg 1982 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
During
the last stages
this volume
the editors
death of our colleague Her personality remembered
of the p r e p a r a t i o n learnt Graciela
of the tragic Salicrup.
and her work will
by all of us.
of
always
be
PREFACE
The I n t e r n a t i o n a l to Algebra, 1981;
C o n f e r e n c e on C a t e g o r y T h e o r y - A p p l i c a t i o n s
Logic and T o p o l o g y - was h e l d in G u m m e r s b a c h , J u l y
it was a t t e n d e d by 93 m a t h e m a t i c i a n s
6-10,
from 19 d i f f e r e n t coun-
tries. Financial
support
for this c o n f e r e n c e was p r o v i d e d by a grant of
the D e u t s c h e F o r s c h u n g s g e m e i n s c h a f t a d d i t i o n a l means of the M i n i s t e r Landes N o r d r h e i n - W e s t f a l e n . their sincere thanks
(grant no.
4851/140/80)
fur W i s s e n s c h a f t
and by
und F o r s c h u n g des
The o r g a n i z e r s w o u l d like to e x p r e s s
for this f i n a n c i a l
assistance,
without which
this c o n f e r e n c e w o u l d not have been possible. The c o n f e r e n c e had been d i v i d e d into three sections: c a t e g o r y theory,
c a t e g o r y theory and logic,
c a t e g o r y theory to analysis,
General
and a p p l i c a t i o n s
t o p o l o g y and c o m p u t e r science.
very m u c h a p p r e c i a t e d by the o r g a n i z e r s
of It was
that John Gray agreed to
be c h a i r m a n of this c o n f e r e n c e
and special thanks are due to him
for his e s s e n t i a l
to its success.
contribution
also very g r a t e f u l
to Horst H e r r l i c h
the section on a p p l i c a t i o n s logy and c o m p u t e r
The o r g a n i z e r s w o u l d
ning of the c o n f e r e n c e
of c a t e g o r y theory to analysis,
like to express Prof.
Peters,
its p r e p a r a t i o n e s s e n t i a l
D u r i n g the c o n f e r e n c e
and e f f e c t i v e help was given and this help has
b e e n g r a t e f u l l y a c k n o w l e d g e d by the o r g a n i z e r s . from the u n i v e r s i t y a d m i n i s t r a t i o n
Thanks
E s p e c i a l l y Mr.
should be m e n t i o n e d
for
for this conference.
are due to the F a c h b e r e i c h M a t h e m a t i k und I n f o r m a t i k of
the F e r n u n i v e r s i t ~ t Many c o l l e a g u e s its preparation. Mrs.
for his ope-
and for the w e l c o m e he e x t e n d e d to the par-
by the a d m i n i s t r a t i o n of the F e r n u n i v e r s i t ~ t ,
Bl0mel
topo-
t h e i r thanks to the Rektor
Dr. Dr. b . c . O .
t i c i p a n t s on b e h a l f of the F e r n u n i v e r s i t ~ t .
his e n g a g e m e n t
are
science.
of the F e r n u n i v e r s i t ~ t ,
and d u r i n g
The o r g a n i z e r s
for his help as c h a i r m a n for
for s u p p o r t i n g this c o n f e r e n c e
in every respect.
a d v i s e d and a s s i s t e d us d u r i n g the c o n f e r e n c e We w o u l d
I. M U l l e r and Mrs.
like e s p e c i a l l y to
and
thank the s e c r e t a r i e s
K. T o p p for their m o s t e f f i c i e n t work.
VJ
Last, Dr.
b u t by no m e a n s
G. Greve,
T. MUller,
Dr. W.
least, Sydow,
all m e m b e r s
of the F e r n u n i v e r s i t ~ t forts
that
there w e r e
ference
and they
ference
feel
This of this
this
our s i n c e r e
Klaus
Heiner
to e x p r e s s
D. BrUmmer,
of the F a c h b e r e i c h for their
Dr.
und
It is due
difficulties
to m a k e
our t h a n k s
B. H o f f m a n n
Mathematik
engagement.
no o r g a n i z a t i o n a l
did t h e i r best
of S p r i n g e r
conference.
series.
Dr.
like
to
and Dr.
Informatik
to t h e i r
during
the p a r t i c i p a n t s
ef-
the con-
of the con-
at ease.
volume
ger L e c t u r e
we w o u l d
Notes All
Lecture
We w o u l d
like
in M a t h e m a t i c s contributions
thanks
Kamps
Notes
Dieter
volume
referees
PumplUn
the p r o c e e d i n g s
the e d i t o r s
for a c c e p t i n g
to this
go to all the
constitutes
to t ha n k
of the S p r i n -
the p r o c e e d i n g s have
been
for
refereed
for their work.
Walter
Tholen
and
PARTICIPANTS
M. A d e l m a n C. A n g h e l H. B a r g e n d a M. B a r r J.M. Beck H.L. B e n t l e y G.J. B i r d R. B S r g e r D. B o u r n H. B r a n d e n b u r g R.D. B r a n d t R. B r o w n C. C a s s i d y Y. Diers G. D u b r u l e A. Duma J.W. D u s k i n R. D y c k h o f f A. Frei P. F r e y d A. F r ~ l i c h e r J.W. Gray C. G r e i t h e r G. G r e v e R. G u i t a r t R. H a r t i n g M. H ~ b e r t H. H e r r l i c h P.J. H i g g i n s M. H ~ p p n e r B. H o f f m a n n R.-E. H o f f m a n n M. H u ~ e k J. Isbell B. Jay P.T. J o h n s t o n e K.H. K a m p s G.M. K e l l y H. K l e i s l i A. K o c k J. L a m b e k H. L i n d n e r F.E.J. L i n t o n H. L o r d R.B. LGs chow J. M a c D o n a l d S. M a c L a n e
L. M ~ r k i G Maury A MSbus T MGller C J. M u l v e y A Mysior R Nakagawa G Naud~ L.D. Nel S.B. N i e f i e l d A. O b t u ~ o w i c z B. P a r e i g i s J. P e n o n M. P f e n d e r A.M. Pitts H.-E. P o r s t T. P o r t e r A. P u l t r D. P u m p l G n R. R e i t e r G. R i c h t e r R. R o s e b r u g h J. R o s i c k ~ G. S a l i c r u p B.M. S c h e i n D. S c h u m a c h e r F. Schwarz Z. S e m a d e n i T. S p i r c u G.E. S t r e c k e r R. S t r e e t T. S w i r s z c z W. S y d o w M. T h i ~ b a u d T. T h o d e W. T h o l e n V.V. T o p e n t c h a r o v V. T r n k o v ~ K. U l b r i c h R.F.C. W a l t e r s H. W e b e r p a l s S. W e c k R. W i e g a n d t A. W i w e g e r R.J. W o o d O. Zurth
AUTHORS'
H.L.
Bentley
ADDRESSES
D e p a r t m e n t of M a t h e m a t i c s U n i v e r s i t y of T o l e d o Toledo, Ohio 43605 U.S.A.
R. Betti
Istituto Matematico U n i v e r s i t ~ di M i l a n o Via Saldini 50, M i l a n o Italy
F. B o r c e u x
U n i v e r s i t ~ C a t h o l i q u e de Louvain 1348-Louvain-La-Neuve Belgium
D. B o u r n
U n i v e r s i t ~ de P i c a r d i e U.E.R. de M a t h ~ m a t i q u e s 33, rue St Leu 80039 A m i e n s France
H. B r a n d e n b u r g
I n s t i t u t fur M a t h e m a t i k I Freie U n i v e r s i t ~ t B e r l i n A r n i m a l l e e 2-6 10OO B e r l i n 33 Fed. Rep. of G e r m a n y
R. B r o w n
School of M a t h e m a t i c s and Computer Science U n i v e r s i t y C o l l e g e of N o r t h Wales Bangor, G w y n e d d LL57 2UW U.K.
Y. Diers
D ~ p a r t e m e n t de ~ ""a t h e"m a t l"q u e s U.E.R. des S c i e n c e s U n i v e r s i t ~ de V a l e n c i e n n e s 59326 V a l e n c i e n n e s France
A. Frei
Mathematics Department U n i v e r s i t y of B r i t i s h C o l u m b i a V a n c o u v e r , B.C. C a n a d a V6T IY4
A. F r 6 1 i c h e r
S e c t i o n de M a t h ~ m a t i q u e s U n i v e r s i t ~ de G e n ~ v e 2-4, rue du Li~vre 1 2 1 1 G e n ~ v e 24 Switzerland
IX
J.W. Gray
Department of Mathematics University of Illinois Urbana, Ill. 61801 U.S.A.
G. Greve
Fachbereich Mathematik und Informatik Fernuniversit~t 5800 Hagen Fed. Rep. of Germany
P.J. Higgins
Department of Mathematics University of Durham Science Laboratories South Road Durham DHI 3LE U.K.
R.-E. Hoffmann
Fachbereich Mathematik Universit~t Bremen 2800 Bremen 33 Fed. Rep. of Germany
M. H~ppner
Fachbereich MathematikInformatik Universit~t-GesamthochschulePaderborn 4790 Paderborn Fed. ReD. of Germany
M. Husek
Matematick~ Ustav University Karlova Sokolovsk~ 83 18600 Praha Czechoslovakia
S. Kaijser
Uppsala University Uppsala Sweden
J. Lambek
Department of Mathematics McGill University 805 Sherbrooke St. West Montreal, PQ Canada H3A 2K6
J. MacDonald
Mathematics Department University of British Columbia Vancouver, B.C. Canada V6T IY4
L. M~rki
Mathematical Institute Hungarian Academy of Sciences Re~itanoda u. 13-15 1053 Budapest Hungary
A. M e l t o n
D e p a r t m e n t of C o m p u t e r S c i e n c e W i c h i t a State U n i v e r s i t y W i c h i t a , K a n s a s 67208 U.S.A.
A. M y s i o r
I n s t i t u t e of M a t h e m a t i c s U n i v e r s i t y of G d a n s k 80952 G d a n s k Poland
L.D. Nel
D e p a r t m e n t of M a t h e m a t i c s Carleton University Ottawa, O n t a r i o C a n a d a KIS 5B6
S.B. N i e f i e l d
Union C o l l e g e S c h e n e c t a d y , N.Y. U.S.A.
J.W.
Pelletier
12308
F a c u l t y of Arts York U n i v e r s i t y 4700 Keele Street Downsview, O n t a r i o C a n a d a M3J IP3
M. P f e n d e r
M A 7-I Technische Universit~t Berlin Str. des 17. Juni 135 1OOO B e r l i n Fed. Rep. of G e r m a n y
H.-E.
Fachbereich Mathemaik Universit~t Bremen 2800 B r e m e n 33 Fed. Rep. of G e r m a n y
Porst
T. P o r t e r
School of M a t h e m a t i c s and C o m p u t e r Science U n i v e r s i t y C o l l e g e of North Wales Bangor, G w y n e d d LL57 2UW U.K.
A. P u l t r
Matematick~ Ustav University Karlova S o k o l o v s k ~ 83 18600 P r a h a Czechoslovakia
R. Reiter
Fachbereich Mathematik Technische Universit~t Berlin Str. des 17. Juni 135 10OO B e r l i n Fed. Rep. of G e r m a n y
Xl
G. R i c h t e r
F a k u l t ~ t fHr M a t h e m a t i k Universit~t Bielefeld U n i v e r s i t ~ t s s t r . 25 4800 B i e l e f e l d I Fed. Rep. of G e r m a n y
M. S a r t o r i u s
Fachbereich Mathematik Technische Universit~t Berlin Str. des 17. Juni 135 1000 B e r l i n Fed. Rep. of G e r m a n y
T. S p i r c u
National Institute for S c i e n t i f i c and T e c h n i c a l Creation D e p a r t m e n t of M a t h e m a t i c s Bdul P~cii 220 79622 B u c h a r e s t Romania
A. Stone
Mathematics Department UC Davis Davis, C a l i f o r n i a U.S.A.
G.E.
D e p a r t m e n t of M a t h e m a t i c s K a n s a s State U n i v e r s i t y M a n h a t t a n , Kansas 66506 U.S.A.
Strecker
R. Street
S c h o o l of M a t h e m a t i c s and Physics Macquarie University N o r t h Ryde, N.S.W. 2113 Australia
W. S y d o w
F a c h b e r e i c h M a t h e m a t i k und Informatik Fernuniversit~t 5800 H a g e n Fed. Rep. of G e r m a n y
J. T a y l o r
D e p a r t m e n t of M a t h e m a t i c s U n i v e r s i t y of D u r h a m Science Laboratories South Road D u r h a m DHI 3LE U.K.
W. T h o l e n
F a c h b e r e i c h M a t h e m a t i k und Informatik Fernuniversit~t 5800 H a g e n Fed. Rep. of G e r m a n y
Xil
V. Trnkov~
Matematick9 Ustav University Karlova Sokolovsk~ 83 18600 Praha Czechoslovakia
R.F.C. Walters
Department of Pure Mathematics University of Sydney N.S.W. 2006 Australia
R. Wiegandt
Mathematical Institute Hungarian Academy of Sciences Re~itanoda u° 13-15 1053 Budapest Hungary
A. Wiweger
Institute of Mathematics Polish Academy of Sciences ~niadeckich 8 00-950 Warszawa Poland
CONTENTS
H.L.
Bentley A note
R. Betti The
on the h o m o l o g y
and R.F.C. symmetry
of r e g u l a r
nearness
spaces
Walters
of the C a u c h y - c o m p l e t i o n
of a c a t e g o r y
F. B o r c e u x On a l g e b r a i c
localizations
13
D. B o u r n A canonical to c o h e r e n t
H. B r a n d e n b u r g A remark
R. B r o w n
Y.
on c a r t e s i a n
An a p p l i c a t i o n 23
closedness
33
Higgins
complexes
and n o n - a b e l i a n
extensions
39
Diers Un c r i t ~ r e de r e p r ~ s e n t a b i l i t ~ de f a i s c e a u x
A.
limits.
and M. H u ~ e k
and P.J.
Crossed
a c t i o n on i n d e x e d homotopy
par
sections
continues 51
Frei Kan e x t e n s i o n s
and s y s t e m s
of i m p r i m i t i v i t y
62
A. F r ~ l i c h e r Smooth
J.W.
structures
69
Gray Enriched
algebras,
spectra
and h o m o t o p y
limits
82
G. Greve General construction topological, uniform
of m o n o i d a l c l o s e d s t r u c t u r e s and n e a r n e s s spaces
in 100
XlV
P.J.
H i g g i n s and J. T a y l o r The f u n d a m e n t a l g r o u p o i d and the h o m o t o p y c r o s s e d c o m p l e x of an o r b i t space
115
R.-E. H o f f m a n n Minimal
topological
completion
of ~ B a n I -->
~Vec
123
M. H ~ p p n e r On the freeness of W h i t e h e a d - d i a g r a m s
133
M. Hu~ek Applications
of c a t e g o r y t h e o r y to u n i f o r m structures
138
S. K a i j s e r and W. P e l l e t i e r A categorical
framework
for i n t e r p o l a t i o n t h e o r y
145
J. L a m b e k 153
T o p o s e s are m o n a d i c over c a t e g o r i e s
J. M a c D o n a l d and A. Stone Essentially monadic adjunctions
167
J. M a c D o n a l d and W. T h o l e n Decomposition factors
of m o r p h i s m s
into i n f i n i t e l y m a n y 175
L. M&rki and R. W i e g a n d t 190
R e m a r k s on r a d i c a l s in c a t e g o r i e s
A. M e l t o n and G.E.
Strecker
On the s t r u c t u r e of f a c t o r i z a t i o n
structures
197
A. M y s i o r A remark on s c a t t e r e d spaces
209
L.D. Nel B o r n o l o g i c a l L 1 - f u n c t o r s as Kan e x t e n s i o n s Riesz-like representations
and 213
XV
S.B. Niefield Exactness and projectivity
221
M. Pfender, R. Reiter, and M. Sartorius Constructive arithmetics
228
H.-E. Porst Adjoint diagonals for topological completions
237
T. Porter Internal categories and crossed modules
249
A. Pultr Subdirect irreducibility and congruences.
256
G. Richter Algebraic categories of topological spaces
263
T. Spircu Extensions of a theorem of P. Gabriel
272
R. Street Characterization of bicategories of stacks
282
W. Sydow On hom-functors and tensor products of topological vector spaces
292
V. Trnkov~ Unnatural isomorphisms of products in a category
302
A. Wiweger Categories of kits, coloured graphs, and games
312
A Note on the Homology of Regular Nearness Spaces H. L. Bentley Abstract:
I t is shown that the homology and cohomology groups of a regular near-
ness space can be defined by means of a v a r i a t i o n on the ~ech method, which uses nerves of uniform covers:
the v a r i a t i o n involves associating with each uniform
cover, not the nerve, but a complex, called the vein, defined by means of nearness In a recent paper, the author showed that the ~ech homology and cohomology groups (: Vietoris homology and Alexander cohomology groups) of merotopic and nearness spaces s a t i s f y , in a variant form, a l l the axioms of Eilenberg-Steenrod. For d e f i n i t i o n s of these groups and f o r h i s t o r i c a l information, the reader is referred to that paper [ I ] .
We are interested here in regular nearness spaces
(for the d e f i n i t i o n , see Herrlich [5]) and in the p o s s i b i l i t y of using what i s , formally, a d i f f e r e n t d e f i n i t i o n of the homology and cohomology groups, but a d e f i n i t i o n which we prove gives r i s e to the usual ~ech groups. By a pair (X, Y) of nearness spaces we mean a nearness space X together with a nearness subspace Y of X. where ~ I
A uniform cover of (X, Y) is a pair 0~= ( C ~ I , O ~ 2)
is a uniform cover of X, ~2C-011 , and ~ 2 L ) { X - Y} is a uniform
cover of Y.
~
The nerve K(01) of a uniform cover ~ =
( C)~I , (~I 2) of (X, Y) is a pair of
s i m p l i c i a l complexes K ( ~ I ) = ( K I ( O I ) , K 2 ( ~ ) ) . elements of ~ I ; ~-~C~ ~
•
a simplex of K I ( ~ I )
The vertices of K I ( ( ~ ) are the
is a f i n i t e subset C~of ~ I such that
The vertices of K2(OI ) are the elements of ~ 2 ;
K2((~) is a f i n i t e subset ( ~ o f
OI 2 such that Y ~ ~ C ~
a simplex of
~ ~o
Recall that a collection C)jL of subsets of a nearness space X is said to be near in X i f f o r each uniform cover ~ of X there exists C e ~ G e C~ ,
C F'~G #
~ .
such that for a l l
Recall also that i f Y is a nearness subspace of X then
a collection ( ~ o f subsets of Y is near in Y i f and only i f ~ is near in X. Now we are ready to make our main d e f i n i t i o n ; i t is a v a r i a t i o n on the d e f i n i t i o n of the nerve.
The vein J(01 ) of a uniform cover C~: ( (-~I' 012) of (X, Y) is a pair of simplicial complexes J( ~ ) = (Jl ( C)I ), J2 ( 0 ] ) ) . the elements of ~ l ;
The vertices of Jl ( 0 1 ) are
a simplex of Jl ( ~ ) is a finite subset (~of 011 such that
C~ is near in X. The vertices of J2 ( 01 ) are the elements of (#~2; a simplex of J2 (
) is a finite subset ~ o f
C)I2 such that
If ~ = ( (Y~I' 01 2) and ~ =
( ~l'
C~A {Y} is near in Y.
J~2 ) are uniform covers of the pair
(X, Y) of nearness spaces then we say that (#~ is a refinement of ~_~ i f refinement of ~Fl and 012 is a refinement of ~2"
('~l is a
Under this relation of re-
finement, the set of all uniform covers of a pair of nearness spaces becomes a directed set. Thus, there is a spectrum of complexes K(OI )
~. K( ~LF )
J( Ol )
"~J(~)
and of complexes
for ~ a refinement of ~J.
From these spectra there arise two spectra of homology
groups and two of cohomology groups. From now on, let G be a fixed abelian group. G will be the coefficient group of our homology and cohomology theories but explicit denotation of G will be suppressed. The direct spectrum of cohomology groups oC~
: Hn(K( ~J ))
~.Hn(K( 01 )) V
has for its limit group the n-dimensional Cech cohomology group of (X, Y) which we will denote by ~n(x, Y).
/~
The inverse spectrum of homology groups
: Hn(K(OI ))
!>Hn(K(~T ))
has for its limit group the n-dimensional ~ech homology group of (X, Y) which we will denote by ~n(X, Y). The direct spectrum of cohomology groups ~
: Hn(j( ~
))
> Hn(j( C)] ))
has for its l i m i t group the n-dimensional vascular cohomology group, of (X, Y) which we will denote by Hn(x, Y).
The inverse spectrum of homology groups
3
OI ~Zy
: Hn(J( 01 ))
-Hn(J(~
))
has for its limit group the n-dimensional vascular homology group of (X, Y) which we will denote by Hn(X, Y). We are now ready for the statement of our main result. Theorem.
I f (X, Y) is a pair of regular nearness spaces then the ~ech and
vascular homology, and cohomology, groups coincide, i.e. Hn(X, Y) = ~n(X, Y)
and
~n(x, Y) = ~n(x, Y) for a l l n. Proof: dual.
We give a proof only for the homology groups; the proof for cohomology is With each collection~u% of subsets of X, we associate the collection ~*
=
{E C X I for some D e ~
Of course, as usual we are using the notation uniform cover of X. write
E < D to mean that {D, X - E} is a
For each uniform cover ~ = ( ~ I '
~ * = ( (~ I * '
012) of (X, Y) we w i l l
~ 2 * ) ; note that because X is regular then ~ *
uniform cover of (X, Y). note that
, E < D},
(To show that
( CY~ 2 LJ {X - Y})*
refines
is again a
C~2" U{X - Y} is a uniform cover of X, (~2" ~ { x - Y}.)
For each uniform cover ~ of X, there exists a s i m p l i c i a l map
gc~ : J ( C ~ * ) which, on vertices E e ~ l * ' g~
satisfies
>K( L~ ) g~(E) e ~ l
and E < g~(E).
Of course,
is not determined by this condition but any two such simplicial maps have to
be contiguous and so, at the homology level, a unique homomorphism f~
= (g~),
:
H n ( J ( O * ) ) - - - ~ Hn(K( ~
))
is determined, which depends only on ~ and not on the p a r t i c u l a r choice of g ~ . Before going on, i t should be noted that the fact that g ~
is a s i m p l i c i a l map
arises from the fact that ~enever ~ is a f i n i t e subset of ~ I * ' and only i f the form
~{g~(E)
I Ee ~ }
# B.
~ is near i f
Also, since the set of a l l covers of
L~* is a cofinal subset of the set of a l l uniform covers of (X, Y), i t
follows that the fc~ form a homomorphism of the inverse spectrum.
For each uniform cover L~ of (X, Y), K ( ( ~ )
is a subcomplex of J( ~I ) so we
have the homomorphism k~ : Hn(K( ~ ))
>Hn(J( ~
))
induced by the inclusion map. Turning our attention now to the l i m i t groups, we have the projection homomorphisms u ~ : Hn(X, Y)
Hn(J( C~ ))
V~ : Hn(X, Y)
Hn(K((~I)),
and as well as the l i m i t homomorphisms f~: Hn(X, Y)
v X > Hn( , Y)
and k:
Hn(X, Y)
Hn(X, Y).
Consider the following diagram: Hn(X, Y)
Hn(J(~*))
f~
t~o~, > H n ( d ( ( ~ I ) )
kc~
----a.
Hn(K(651")) .
~c~
k~
k
> Hn(K( 01 ))
\
>
Hn(X, Y)
I t is clear that each of the inner triangles is commutative, because each homomorphism is induced either by a projection of refinements or by an inclusion map. To show that f o k = I, let x e vHn( X, Y) and compute as follows:
v
O)
f
k
x
=
=
x
f ~ k 6 ~ , VC~m x X
=
V ~ X.
Consequently, f k x = x. An equally pleasant computation shows that
k Ofoo = 1
and the proof
of the theorem is complete. For regular nearness spaces, the above theorem provides an a l t e r n a t i v e method v
of computing the Cech groups:
one can compute by means of the vascular theory.
I f X is a regular nearness space and Y is a dense nearness subspace of X and i f is a c o l l e c t i o n of subsets of Y which s a t i s f i e s is near in Y.
(3{clxA i A e ~
} ~ ~
then
This observation, together with the knowledge t h a t , in the above
s i t u a t i o n , the homology groups of X are the same as those of Y, indicates t h a t , instead of passing to the extension X and using the Cech theory, one could stay i n side Y and use the vascular theory. Of course, not every nearness space Y is a subspace of a topological nearness space X so, even i f X is the completion of Y, there may e x i s t c o l l e c t i o n s C~ of subsets of Y such that
dl {clxA I A e C21 } : ~ .
In such a case, i t might also be
advantageous to use the vascular theory. We w i l l now present an example using thevascular homology groups Hn(X, Y). Consider the Euclidean plane as a nearness subspace (= uniform subspace) of i t s Alexandrov one-point compactificaton.
Let X be the nearness subspace (= uniform
subspace) induced on the subset 1 1 { ( I , y)[ - I ~ y ~ I } l . ) { ( x , ~) [ 1 < x} t_J { ( x , - T) I 1 ~ x} , The completion of X is a c i r c l e S1 on a 2-sphere S2.
Thus, by the fact proved
in [ 2 ] , the homology of X is the same as that of SI . The point here though is that the homology of X can be computed without going outside X.
The d e t a i l s are as follows.
The set of a l l f i n i t e uniform covers of X is a cofinal subset of the set of a l l uniform covers of X. So, consider an a r b i t r a r y f i n i t e uniform cover ~ :~Y = {A e ~
of X.
Let
I A is unbounded}
and l e t ~ > 0 be such that {GC X I diam G < E }
refines ~ .
Let x+ be the supremum of the set { l } L W { x e R I f o r s ~ e y > 0 and f o r some A e 0 ) - ~ ,
(x, y ) e A}
and l e t x- be the supremum of the set {l}U{x Let ~
e R I for some y < 0 and f o r some A e ~ - ~ ,
be a set of i n t e r v a l s on X of diameter at most E such that
{(l,y)
I -I F(A) F(f) F
satisfying obvious coherences,
for
compositions
and
')
2-morphisms. These cohe-
rences are such" that a lax cone is exactly an object of the total category of the following diagram : ~B(X,FA) A with
t~ 'A,A'
[~(A,A'),B(X,FA') ] -- ~ ~ ~ [A(A,A') x~(A' ,A"),B(X,FA") ] ~--~-~,A,A' ,A"
t 0((TA)Ae A) = F(f) . TA ,
[qO(OA,A ')] A,A' ,A''(f'g) = F(g).eA, A,(f)
t I((~A)AeA ) = TA, ,
((n(@A,A,)~A,AtA,,(f,g)
= @A,A,,(~.f)
[J (@A,A ') ] A = @A,A(IA )'
[(qI(@A,A ')] A,A',A ''(f'g) = @At, A"(g)
Actually, this diagramm is determined by the right K~n extension of /~ / - : & - - - ÷ A
2 , where
l~/~
~ J_~ ~(A,~) ~(A,A')x~(A', A ~---A,A' wbere
d0CA f A' g ~3
=
A' g
d I(A f A' ~ ~) = A g÷f
~ F
along
is the internal category in Cat : ~ ~o ~)(** [ I A(A,A') x•(A' ,A") xA(A", ~) ~ A,A' ,A"
28
i(A ~f ~) =(A = A +f ~) P0(A+f A' ~ A" h ~) = (A f A' h~g c0 P I ( A I'+-A
aA"h-~) h ~)
m ( A f A' ~ A "
Whence t h e l a x l i m i t s We s h a l l d e n o t e
= (A'
~A" ~ ~)
A g-~f A" h ~)
=
are the
Y2 ® ~ / - l i m i t s .
Y2 ® A / -
by
Remark. T h i s new d e s c r i p t i o n
L(A) .
of lax limit leads to a generalization,
[6 ] , o f t h e B o u s f i e l d - K a n homotopy l i m i t s &/~
is an internal
simplicial
category in
: if
~
S and N e r ( A / ~ )
is a simplicial
studied in category,
is a cosimplicial
then
s p a c e . The
i n d e x a t i o n f o r t h e s e homotopy l i m i t s i s t h e p r o f u n c t o r
H(A) = A - - - * ! to generalize
, defined by the
H@A) (~) = D i a g ( N e r ( ~ / ~ ) )
replacement
scheme o f
[71
. This indexation allows us
to simplicial
categories.
The monads and t h e s e m i a d s . Let
~
be the 2-catego~
2-cells
~ : ~ + t ~.
with a single object
and
k t
~ : t2 ÷ t
=~.tk=t,
~.~
Then it is clear that a monad on a category such that its value at
~
limit of this 2-functor
is
C
Let
D
t
=~.
t~
is a 2-functor
C . It is well known too
:
from
[13] , [4]
~
to Cat
that the lax
is the category of algebras of the monad and that there
is a cotriple on this category of algebras, of
~ , g e n e r a t e d by a 1 - c e l l , and two
satisfying the well-known relations
that is an action of
D c°
(the dual
for the 2-cells). ~
be the sub 2-category of
semiad a 2-functor from
~
and a natural transformation Then the lax limit of are the pairs
a
D
generated by the 2-cell
to Cat, that is a category ~ : T2 ~ T
C
such that
(c, h • c ÷ T c )
such that
are the morphisms
The universal lax
cone
is
U(c, b) = c , and the 2-cell
b . ~(c) = b . T b
f: c ÷ c'
given by the
such that
forgetful
B : T . U + U
T
~ . ~ T = ~ . t ~.
semiad is the category of algebras
(c, b) ÷ (c', b')
~ . Let us call a
with an endo-functor
whose objects
and whose morphisms
f . b = b'
functor
given by
CT
. T f .
U : CT + C
6(c, b) = b : T c + c .
There is no longer an adjunction between C and C T, but it is clear too, that we have a functor
F : C ÷ CT
with
F(c) = (T(c), ~(c))
, such that
U . F = T . Fur-
ther more there is a natural transformation
q " F . U + I CT
Indeed
defines a natural transformation. Let
dl
q(c, b) = b : (T(c), ~(c)) + ( %
b)
be the 2-category with only one object
a 2-cell Proposition.
q : t-4~
~ , generated by a 1-morphism
. We can sum up this result in the following
There is a canonical action of
t
and
:
d7 on the category of algebras of a
29
semiad. III. The c~nonical action on G-indexed limits. These two last examples raise the question: is this fact general, is there always an action on the category of algebras! Let
~
be a
~g-category, A --~-~ ]I an
indexation. The profunctor ~ can be viewed as a functor Proposition. The
t-indexed limit of
nical action on each Proof.
~
A ÷ V •
is a ~-mono~d and this mono~d has a cano-
~-indexed limit,
~-lim t = U ~, ~]] = Nat(~, @) has an obvious structure of
F : ~ +~g t-lim ~
be a M-functor. So
~-lim F = ~ ,
acts on the ~-indexed limit of
~-functor and
L
M-mono~d. Let
and it is clear that
F . More generally, let
the t-indexed limit of
and the canonical action
F]] = Nat(t, F)
F :~ + B
be a
F . So we have ~(B,L) =Nat(t, ~(B,F -))
~-lim @ x ~(-, L) ÷ B(-, L)
which is equivalent by
the Yoneda le~raa to a morphism : t-lim ~ ÷ B(L, L) . It is easy to see that it is a morphism of V-monoids. Examples. The monad case. We have seen that the indexation limit of
y : ~op ÷ Cat~
tegory of the monad
k L~_~ I
is the lax
so that we can exhibit it as a monad on the Kleisli ca-
D(-, ~)
on
k(~, ~) . A simple but tedious computation of
its category of algebras shows us that this category is exactly mono~d structure is that of
~co(~, ~) and the
~co . The canonical action on the category of algebras
of a monad is the usual comonad. In the same way, we can study the semiad case. We must calculate the category of algebras of a semiad on the lax colimit of the semiad ~(-, ~) which is just action is
J/ (~, ~)
the
and the monoid structure is that of
on
~(~, ~) ,
J/ . The canonical
one described by the former proposition. More details will be
given in the proof of the next proposition. Remark. I choose this proof for sake of simplicity and quickness. But it is a very general result that (as in
the
case
of
V-functors [9] ) right Kan extensions
of profunctors are equipped with an action of the codensity monad (which always exists since we deal with profunctors) so that they can be factorized through the Kleisli category
[ ]6 ] of that codensity monad of profunctors, which in our case
has only one object and so is a mono~d. This general result is used in
[ 5 ] to
show in a very simple and categorical way that Kan extensions are shape invariant, so we could say that
~-lim t
is the "shape" mono~d of ~ .
IV. An appT,icatio~ to coherent homotopy. Following Dwyer-Kan gory
[10] , the standard resolution
C (a single object
t 2 = t)
•
F~ C
of the idempotent cate-
with a single non trivial morphism
is a simplicial category.
t
such that
30
But a category
is a particular simplicial set (via its nerve), so a 2-category is
a particular simplicial category. Now forthis category C , no composite of non identity maps is an identity, so F~ C see that this 2-category is just
is actually a 2-category and it is not hard to ~
.
We are now going to study the consequences of the higher homotopy coherences involved in the data of a simplicial functor from
B
to a simplicial category
which I keep on calling a semiad. In the special case
B
B = Top , I shall speak of
a coherent homotopy idempotent. The 2-pro-functor
L~)
simplicial profunctor. joint
K
: B ---+ |
indexing
lax limits can be considered as a
The simplicial embedding Cat L
preserving products, preserves
X
of
F
F : ~] ~ B
[14] ), being
the case of a 2-functor
ends and along
in a 2-category
B
F : [4 + IB
F
L(~), is still the 2-mono~d
we studied previously the action of
Y
F
has an L~H)-indexed limit L,
such that, if
c~(~)
a(~) . v = F(t) , and a 2-cell between
and the constant 2-functor on
iH(~, -)
then there
is the canonical projection v . ~(~) ~
given by
9(~) =~(t,-)
and
L(~) O
-)
~(t,-)j
:
~
"
~(~, -)
since we can verify that : @(t)
. @(~) I[-I(~, - )
= IH(U, - )
and
O ( t 2) = @(t)
. (9(t) ~ I ( t ,
. E(t, -)
- ) IH(!J, - )
= tI(p,
-)
= M(]~ . t 1~, - )
. I~(1~ t ,
-)
= tt(p
. ]] t ,
Whence a natural transformation : L(I0 ÷ H(~, -)
such that
~ T(~) = ~](t, -)
is the
between
: I](~,-) ÷~(~,-)
JH(~, -) = IH(ll,
IL .
on the li-
Y : B °p ÷ Cat~ . On the other hand, we have a lax cone
@(t)
In
a simplicial
L . Firstly let us consider the 2-enriched situation. We saw that
lax colimit of
~.
JJ. We have the B
Proof. The proof will be given by a careful study of the action of mit
. Thus
a simplicial functor.
v : F(~) ÷ L
L ÷ F(~) , we have
L(~)
L(l~)-indexed limits (homotopy limits in
L~H)-lim
Proposition. If the simplicial semiad exists a map
is also the
is the simplicial
following result about this action in the general situation of category and
L(~) -indexed
L(~)
considered as a simplicial functor. Therefore the sim-
plicial mono~d acting on the simplicial the sense of Gray
X
considered as a simplicial profunctor along
the lax limit of a 2-functor L(~)-indexed limit
has a simplicial ad-
exponentiations,
limits. So the right Kan extension of a 2-profunctor right Kan extension of
S
-)
31 if
• is the universal lax cone associated to the lax colimit of
there is a 2-1ax cone
6 between ~(~)O
and
Y . Furthermore,
T, given by
s i n c e t h e second members o f t h e f o l l o w i n g e q u a l i t i e s ~(~).T(N)o(t) are equal,
=
T(t).T(~)H(~,--)
,
b e c a u s e o f t h e c o h e r e n c e o f t h e l a x cone
~. Whence a 2 - n a t u r a l
transformation
L = Nat(LOt),F)
. The u n i v e r s a l
=
T(t).T(t)
S . Let L be t h e L (H) - i n d e x e d l i m i t o f F , t h a t L (N) - i n d e x e d cone o a s s o c i a t e d
We have a map o f s i m p l i c i a l Nat(~,F) :
and we v e r i f y
~(~).~
= F(t)
to the 2-cell
.
o(~) = Nat (T (~) , F) : Nat(L(tt) ,F) ---, N a t ( I ~ ( ~ , - ) ,F) = F(~) , and so on.
[q(t,-)
T with respect
d: T ( ~ ) a ~ I L ( H )
Now l e t us b e g i n w i t h a semiad F:14 ~ is
~(t).~(~)~(t,-)
s e t s w: F ( ~ ) - ~ L
F(e) = N a t ( I t ( ~ , - ) , F )
to o(t)
L
i s g i v e n by
= Nat(T(t),F)
, that is: , Nat(L(V),F) = L
, since
~(~) .~ = N a t ( T ( ~ ) , F ) , N a t ( ~ , F )
= Nat(T(~)~,F)
~ Nat(H(t,-),F)
= F(t).
F u r t h e r m o r e we have a 2 - c e l l b e t w e e n ~ . o ( ~ ) and 1 L g i v e n by N a t ( d , F ) and so F "splits"
at
L .
More g e n e r a l l y
l e t F : [ t * A be a semiad i n a s i m p l i c i a l
i n d e x e d l i m i t o f F and ¢ t h e u n i v e r s a l commutative d i a g r a m w i t h n a t u r a l
isomorphisms:
N a t ( T ( ~ ) , / A ( X , F - ) ) : N a t ( L 0 t )/A(X,F-)) A(X,o~) F u r t b e r m o r e we g e t a n a t u r a l A(X,F(~)) m
c a t e g o r y A. Let L be t h e L ( ~ ; -
L O t ) - i n d e x e d c o n e . Thus we have t h e f o l l o w i n g
* NatOH(~,-),IA(X,F-))
: /A(X,L)
, A(X,F(~))
f i n X) t r a n s f o r m a t i o n :
Nat(H(~,-)),/A(X,F-))
mat(~JA(X,F-))
Nat(L(H),N(X,F-))
~-~ N(X,L)
and so by t h e Yoneda len~na a morphism
v: F(~)--,I,
such that this natural
t i o n i s j u s t /A(-,v)
that o(~).v
is F(t), since
. Then i t
And now t h e 2 - ( n a t u r a l )
is clear
~.r(~)
cell
dJ atd
A(X,L) ~ Nat(L(i~) , A ( X , F - ) )
N a t ( r (~) ~,N(X,F-) )
A(X,L) ~ Nat(L(tl) ,N(X,F-})
transorma= ~(t,-).
32 determines, by Yoneda ! , a 2 - c e l l i n
N(L, L)
between
~ . ~(x)
and
1L •
Corollary. The homotopyidempotent ( i . e . idempotent i n the homotopy category H0-Top) a s s o c i a t e d to a coherent homotopy idempotent, s p l i t s . References. I. C. Auderset, Adjonctions et monades au niveau des 2-categories, Cahiers de Top. et G6om. Diff., XV (1974), 3-20. 2 J. B~nabou,
les distributeurs, Inst. Math. Pures et Appl. Univ. Louvain la Neuve
Rapport n ° 33 (1973). 3
F. Borceux and G.M.KelIy, A notion of limit for enriched categories, Bull. of
the AustralianMath. 4
Soc., 12 (1975) 45-72.
D. Bourn, Natural anadeses and catadeses, Cahiers de Top. et G~om. Diff. XIV
(1974) 371-480. 5
D. Bourn and J.M. Cordier, Distributeurs et th~orie de la forme, Cahiers de Top.
et G~om. Diff., XXI (1980) 161-189. 6
D. Bourn and J.M. Cordier, Une formulation g~n~rale des limites homotopiques,
Notes, Univ. Amiens (1980). 7 A.K. Bousfield and D.M. Kan, Homotopylimits, completions and localizations, Springer Lecture Notes in Math., 304 (1972). 8
J.M. Cordier, Sur la notion de diagran~e homotopiquement coherent , Proceedings
3~me colloque sur les categories Amiens 1980 (~ para~tre). 9
E.J. Dubuc, Kan extensions in enriched category, Springer Lecture Notes in
Math., 106 (1969). 10
W.G. Dwyer and D.M. Kan, Simplicial localizations of categories, Journal of
P.A. Algebra, 17 (1980) 267-284 . 11
J. Dydak, A simple proof that pointed FANR-spaces are regular fundamental re-
tracts of ANR's, Bull. Acad. Polon. Sci. Math., 25(1977) 55-62. 12
P. Freyd and A. Heller
(in preparation).
13
J.W. Gray, Formal category theory, Springer Lecture Notes in Math., 391 (1974).
14
J.W. Gray, Closed categories, lax limits, homotopylimits, Journal of P.A. Alge-
bra, 19 (1980) 127-158.
15
F.W. Lawvere, Teoria d e l l e c a t e g o r i e sopra un topos di base, Mimeographed notes,
Perugia (1973).
Topology,13,(1974),
16
G.B. Segal, Categories and cohomology t h e o r i e s ,
17
R. S t r e e t , Limits indexed by category valued 2 - f u n c t o r s , Journal of P.A. Alge-
293-312.
bra, 8 (1976), 149-181. 18 M. Thiebaud, S e l f - d u a l s t r u c t u r e semantics and a l g e b r a i c c a t e g o r i e s , Dalhousie Univ., Halifax, N.S., (1971).
A REMARK ON C A R T E S I A N
H. B r a n d e n b u r g
I. A c a t e g o r y
A with
if for every A - o b j e c t For b a c k g r o u n d for c e r t a i n
and M.
finite p r o d u c t s
X the functor
concerning
aspects
cartesian
of a l g e b r a i c
c l o s e d n e s s and its
to
cited
there.
the c a t e g o r i e s
they c o n t a i n (for UNIF or UNIF,
spaces
[12]).
gorical
Aspects
tively
of T o p o l o g y
cartesian
respect
in TOP or UNIF.
(see r e m a r k
general
problem
category
of TOP or UNIF w h i c h
problem
is included
gorical
Topology
tain the
THEOREM
there
following
theorem,
point discrete space, As a c o n s e q u e n c e subcategory str o n g l y
of TOP m u s t
easily
and nega-
in the m o r e
reflective
closed.
recent
Note
survey
that
article
subthis
on Cate-
For the case of TOP we obbe proved
subcategory
in section
of TO__~P contains
I every c a r t e s i a n
consist
of c o n n e c t e d
space
of c a r d i n a l i t y
continuous
of X form an example contains
subspaces
here
car-
space w h i c h
can be a n s w e r e d
interested
w h i c h will
of T h e o r e m
ty is the only n o n - c o n s t a n t
TOP w h i c h
of usual
have
i.e.
2:
the two-
then it is not cartesian closed.
rigid H a u s d o r f f
all powers
TO_~P or UNIF
a non-trivial
11).
F.
on Cate-
subcategories,
is c a r t e s i a n
[7], P r o b l e m
If a reflective
I.
we are exists
in TOP are
fact,
Conference
whether
his q u e s t i o n
in H. H e r r l i c h ' s
(see
by this
a non-indiscrete
formation
Since
(c) below),
whether
Motivated
that
subcategories
that their p r o d u c t s
epireflective
to the
closed
1980 O t t a w a
containing
it is known
are c o r e f l e c t i v e
and Analysis)
closed
subcategories
are c l o s e d with products
(at the
importance and topolo-
and the l i t e r a t u r e
closed,
cartesian
products.
asked
[3].
TOP of t o p o l o g i c a l spaces and
are not c a r t e s i a n
from the usual
has r e c e n t l y
closed
[15],
All these c a t e g o r i e s
Schwarz
non-trivial
analysis,
[14],
and they have the d i s a d v a n t a g e
different
tesian
[5],
some nice n o n - t r i v i a l
see
cartesian closed
is called
topology,
algebra we refer Although
[2],
Hu~ek
X x - has a right adjoint
gical
UNIF of u n i f o r m
CLOSEDNESS
show that all c a t e g o r i e s
spaces. ~2
mapping
of a r e f l e c t i v e
only c o n n e c t e d
spaces
[4].
_AX o b t a i n e d
closed r e f l e c t i v e If X is a
(i.e.
the
identi-
from X into X) , then subcategory However,
~X of
one can
in this w a y are not
34
cartesian
closed.
cartesian
closed
We c o n j e c t u r e
rem shows
the v a l i d i t y
reflective
that
there exists
subcategory
of TOP.
of the c o r r e s p o n d i n g
no n o n - t r i v i a l
Our
second theo-
statement
for u n i f o r m
spaces. THEOREM
If a reflective subcategory
2.
indiscrete
space,
2. T h r o u g h o u t
this note all
and i s o m o r p h i s m - c l o s e d .
information
about
is i n t e r e s t i n g
PROPOSITION.
functor
I we will
Since A c o n t a i n s closed,
exists
T on the
a topology
(ii)
• is admissible,
Hence
where
Y is A - p r o p e r ,
i.e.
set C(Y,Z)
of c o n t i n u o u s
the e v a l u a t i o n
map
for every A - o b j e c t
is continuous,
topology
from
e:Y ×
(C(Y,Z),T)
~ Z
X and
for every contin-
w h e r e ~(f) (x) (y)=f(x,y).
exist
on C(Y,Z)
on C(Y,Z).
to A.
In order
X in A and a c o n t i n u o u s (C(Y,Z),~)
two spaces Y,Z
in A such
is not A-proper.
with
the usual
To this
topology
and
space Y × Yo' w h e r e Yo is the s u b s p a c e
e , both Y and Z b e l o n g
~
mappings
there
e(y,g)=g(y).
space of i r r a t i o n a l s
let Z be the p r o d u c t
~(f):X
that
of A - o b j e c t s
properties:
Y U {o} of the reals w i t h the usual metric.
topology
[8].
the countable
it is easy to v e r i f y
for each pair Y,Z
to show that there
admissible
end let Y be the
to
proposition
f:X × Y ~ Z the m a p p i n g
(C(Y,Z),T)
it suffices
that every
in TOP and contains
then
i.e.
is continuous,
~(f) :X ~
following
Then A is not cartesian closed.
the f o l l o w i n g
uous m a p p i n g
For addi-
we refer
of TOP which is closed with
a singleton,
if A is c a r t e s i a n
(i)
use the
Let A be a subcategory
respect to countable products
Y into Z w i t h
subcategories
to be full
C is reflec-
for itself.
infinite discrete space ~. Proof:
are a s s u m e d
A of a c a t e g o r y
has a left adjoint.
reflective
In the p r o o f of T h e o r e m which
subcategories
A subcategory
tive in ~ if the e m b e d d i n g tional
of UNIF contains a non-
then it is not c a r t e s i a n closed.
Being
homeomorphic
Let T be an a r b i t r a r y to p r o v e
mapping
that there
admissible
exists
a space
f:X × Y ~ Z such that
is not c o n t i n u o u s
we
start w i t h an a r b i t r a r y
to
35
continuous
mapping
g:Y ~ Z satisfying
for e a c h
y 6 Y, w h e r e
pl:Z
Consider
an a r b i t r a r y
point
hood.
By the
borhoods (yo,g)
continuity
V and V'
6 V'
cl V is not
tinct
points
X be the
Yo
~ Yo are
in Y w h i c h
of the
where
compact,
of Y
no c o m p a c t
map
that
that
exists
{Ynln
consisting
e w e can cl V c V'
a sequence
6 ~}
neighborfind
for
i=I,2}. of dis-
in Y.
non-negative
neigh-
and
(yn)n6~
is c l o s e d
of all
= 0
the p r o j e c t i o n s .
W = {z 6 Z I IPi(Z) I < I
there
in cl V such
subspace
has
evaluation
of Yo and a U 6 T s u c h
x U c e-l[W],
Since
Ip1(g(y)) I < I and p2(g(y))
~ Y and p2:Z
Now
let
numbers.
O
Being
homeomorphic
subspace retract
to Y, t h e
of X x y the of X × Y
continuous
(see
mapping
space
X belongs
space A =
({0}
[11],
II,
§26
f r o m A into
× Y)
has
a continuous
[
Z defined
that
an o p e n
neighborhood
there
exists
hand we conclude
from and
diction
shows
that
6 ~})
is a
Consequently
the
is c o n t i n u o u s .
an m 6 ~
that
Y=Yn
Then
there
~(f) [B] c U.
satisfying
e(Ym,~(f) (x))
is not
and
x y ~ Z.
e ( Y m , ~ ( f ) (x))
~(f)
2).
if x > 0
B of 0 in X such
=
(pl(g(ym)),mx)
f:X
(C(Y,Z),T)
an x 6 B and
(X x {Ynln
by
(pl(g(yn)),nx)
~(f) :X ~
As a c l o s e d
if x = O
extension
Suppose
U
Corollary
S g(Y)
(x,y)
to A.
mx
> I.
exists
Moreover On the o t h e r
= ~(f) (x) (y m) = f ( x , y m) 6 W that m x
continuous,
< I.
which
This
completes
contrathe
proof. Proof
of T h e o r e m
taining
the
countable
1:
L e t A be a r e f l e c t i v e
two-point
discrete
discrete
space
tue of t h e p r o p o s i t i o n .
space
e, t h e n A Hence
subcategory
{0,1}.
is not
we assume
of T O P
If A c o n t a i n s
cartesian that
closed
e does
con-
the by v i r -
not b e l o n g
to
A. Suppose is g i v e n
t h a t A is c a r t e s i a n by r e × r e : e x e
reflection
of e.
the ~ - r e f l e c t i o n and
that
of exe
Then
~ re × re, w h e r e
To v e r i f y
l_~In£er~ x {n}
closed.
this
always = re x
fact
has
the
I Jn6e{n}
one
the A - r e f l e c t i o n re:e only
form
~ re d e n o t e s has to n o t e
r(exe)
= re x r~
=
of e × e the Athat
l~In6ere x {n},
by t h e c a r t e s i a n
36
closedness Now
of A
consider
, where
the
I_~i d e n o t e s
continuous
f(n,m)
mapping
tinuous that
re-re[e]
fact,
n = m
and
m • 1
if
n • m
and
m = I
{O,1}
To prove Xo
% @ and
re = re[e]
(since
the
second
£ re-re[e]
such
h : r e ~ re d e f i n e d
~
observation {0,1}
[
re(1)
Applying
essentially
of r e x r e
we can show that
{x}×re
6 rear
if
g(x,x)
= 0
the
same argument
We conclude (a) T h e o r e m taining
- by
te space.
to our assumption. exists
if
n % I n
this
section
I implies
an mapping
that
In fact,
[9], T h e o r e m
1.1
rex{re(n)}
I
g(x,y)
g(x,x)
A cannot
every
it h a s
that
is i m p o s s i b l e .
subspaces
the argument
that
with
space X which
=
, which
to the
if
Repeating
Consequently
a finite
sian closed. which
[e].
in p a r t i c u l a r
a contradiction!
In
is a h o m e o m o r p h i s m
there
= I
of r e x r e w e o b t a i n
[e],
con-
We claim
Then the continuous
g(x,x)
I
spaces
that
= O.
if
g'x're'n~'~ ~ ,; = If 0
x,y
~ re
, contrary
assume
a unique = f"
every x 6 re-re[e].
h o r e = r e = i d r e o r e a n d h • idre
each x 6 rear
exists g o r xre
by f x
for
re:e
e 6 A
assertion
there that
= I for
that
that
that g(Xo,Xo)
h(x)
satisfies
such
g(x,x)
imply
i.e.
by
otherwise
that
would
6 A),
defined
if
g:rexre
in A.
~ {0,1}
I
to t h e p r e c e d i n g
mapping
f:e×e
= ~ I 0
According
the coproduct
again
for t h e
sub-
= 0 for e a c h p a i r
= 0 for
each x 6 rear
be cartesian
closed.
[e]-
[]
some remarks. reflective
is n o t
to c o n t a i n
- always
subcategory
indiscrete
cannot
the reflective
contains
of TOP
con-
be cartehull
the two-point
o f X, discre-
37
(b) If a r e f l e c t i v e s u b c a t e g o r y A of TOP contains a n o n - i n d i s c r e t e space X w h i c h is not TI, then it is not c a r t e s i a n closed.
In case
that X is not T this follows from the fact that A m u s t contain the o b i r e f l e c t i v e hull of X (e.g. see [10]) and hence a n o n - i n d i s c r e t e finite space.
If X is a To-Space,
spaces is c o n t a i n e d in A
then the c a t e g o r y of sober
([10], T h e o r e m
1.3).
In particular,
the
t w o - p o i n t d i s c r e t e space belongs to A. (c) No
non-trivial epireflective
sian closed,
s u b c a t e g o r y of TOP can be carte-
since it has to contain the t w o - p o i n t d i s c r e t e space.
This answers S c h w a r z ' s q u e s t i o n m e n t i o n e d However,
in the introduction.
a simpler proof of this fact results
from the o b s e r v a t i o n
that there exist z e r o - d i m e n s i o n a l T 1 - s p a c e s X,Y,Z and a c o e q u a l i z e r q:Y ~ Z such that i d x X q : X x y ~ XxZ is not a q u o t i e n t m a p p i n g in TOP (e.g. see
[I], Exa/nple 4.3.4).
E s s e n t i a l l y the same a r g ~ e n t
that there is no n o n - t r i v i a l e p i r e f l e c t i v e tegory of p s e u d o t o p o l o g i c a l
spaces
shows
s u b c a t e g o r y of the ca-
[13].
(d) Every e p i r e f l e c t i v e s u b c a t e g o r y of an e p i r e f l e c t i v e subcategory of TOP is r e f l e c t i v e in TOP. a/nple, to e p i r e f l e c t i v e
Hence T h e o r e m
I applies,
for ex-
s u b c a t e g o r i e s of the c a t e g o r i e s of Haus-
dorff spaces or c o m p l e t e l y regular spaces. 3. In order to prove T h e o r e m 2 let A be a r e f l e c t i v e s u b c a t e g o r y of UNIF c o n t a i n i n g a n o n - i n d i s c r e t e space X.
If A is c a r t e s i a n
closed,
l_~ldenotes the copro-
then X~xl In6 {n} = l_~In6 X~×{n}, w h e r e
duct in _A" jection,
For each n 6 ~ let in :X~x{n} ~ X~xi---in6~{n} be the in-
and let Pn:Xex{n}
Pn is u n i f o r m l y continuous, mapping
f:X~xi_~in6~{n}
~ X be defined by
((xi),n) ~ x n.
Since
there exists a u n i f o r m l y c o n t i n u o u s
~ X s a t i s f y i n g foi n = Pn for each n 6 ~.
M o r e o v e r there are two points x,y 6 X and a u n i f o r m cover U of X such that no element of U c o n t a i n s both x and y. c o n t i n u i t y of f there exist u n i f o r m covers
By the u n i f o r m
V of X and W of
{~in£~{n}
such that V×W refines f-1(U). It follows that V refines every -I Pn (U), c o n t r a d i c t i n g the fact that the subspace {x,y} e of X e is not t o p o l o g i c a l l y discrete.
Hence A cannot be c a r t e s i a n closed
w h i c h c o m p l e t e s the proof of T h e o r e m 2. It is w o r t h m e n t i o n i n g that our proof of T h e o r e m 2 m a k e s no use of star-refineraents of u n i f o r m covers,
hence T h e o r e m 2 is v a l i d
38
also for the category N E A R of nearness follows that no non-trivial reflective
subcategory
tion applies,
[6].
Moreover
subcategory
of NEAR is cartesian closed.
for example,
to the category
spaces
epireflective
This observa-
to the category of proximity
of contiguity
spaces
it
of an epispaces or
[6].
REFERENCES
Elements of Modern Topology, McGraw-Hill,
[I]
R. Brown, (1968).
[2]
E.J. Dubuc and H. Porta, Convenient categories of topological algebras and their duality theory, J. pure appl. Algebra I (1971)
New York
281-316.
[3]
S. Eilenberg and G.M. Kelly, Closed categories, in: Proc. of the Conference on Categorical Algebra, La Jolla 1965, ed. by S. Eilenberg et.al., Springer-Verlag, Berlin-New York (1966).
[4]
H. Herrlich, On the concept of reflections in general topology, in: C o n t r i b u t i o n s to E x t e n s i o n Theory of T o p o l o g i c a l Structures , (Proc. Sympos., Berlin, 1967), 105-114, Deutscher Verlag d. Wissensch., Berlin (1969).
[5]
H. Herrlich, Cartesian closed topological categories, C o l l o q u i u m Univ. Cape Town 9 (1974) 1-16.
[6]
H. Herrlich, A concept of nearness, (1974) 191-212.
[7]
H. Herrlich, Categorical Topology 1971-1981, in: General Topology and its Relations to M o d e r n A n a l y s i s and Algebra V (Proc. of the Fifth Prague T o p o l o g i c a l Symposium, Prague 1981), H e l d e r m a n n Verlag, Berlin, (to appear).
[8]
H. Herrlich and G. Strecker, Category Theory, H e l d e r m a n n Verlag, Berlin, (1979).
[9]
R.-E. Arch.
Hoffmann, der Math.
33
(1979)
[11]
K. Kuratowski, (1966).
[12]
M.D.
ed.,
n~chterner Rdume, Manus-;
Topology, Vol. I, Academic Press, New York,
Rice and G.J. Tashian, Cartesian closed coreflective subcategories of uniform spaces, (preprint). F. Schwarz, Cartesian closednes8, exponentiality, and final hulls in pseudotopological spaces, (preprint). U. Seip, Kompakt erzeugte Vektorrdume und Analysis, Springer Lecture Notes
[15]
4
258-262.
R.-E. Hoffmann, Charakterisierung cripta Math. 15 (1975) 185-191.
[14]
sec.
Appl.
Reflective hulls of finite topological spaces,
[10]
[13]
General Topol.
Math.
in Math.
273
N.E. Steenrod, A convenient M i c h i g a n Math. J. 14 (1967)
(1972).
category of topological spaces, 133-152.
CROSSED COMPLEXESAND NON-ABELIAN EXTENSIONS Ronald Brown School of Mathematics and Computer Science, University College of North Wales.
Philip J. Higgins Oepartment of Mathematics, Science Laboratories, Durham University.
and
Durham, U.K.
Bangor, Gwynedd, U.K.
Introduction
Crossed complexes may
be thought of as chain complexes with operators from a
group (or groupoid) but with non-abelian features in dimensions one and two.
We start
by surveying briefly their use. The definition of crossed complex is motivated by the standard example, the
otopy crossed conrplex ~ Here
~]~
of a filtered space
is the fundamental groupoid
hom-
~ : X 0 c X] c ... c X n c Xn+ ! c ... c X.
~I(X|, X0)
of
homotopy classes
rel i
of maps
(I, i) ÷ (X;, X O) , with the usual groupoid structure induced by composition of paths. For
n a 2 , ~n_X is the family of relative homotopy groups
p E X0 . map
For
n ~ 2 , there is an action of
~]~
on
~n(Xn, Xn_ ] , p)
6 : ~n~ + ~n_]~ ; there are also the initial and final maps
The rules which are satisfied by all such crossed complex (§]). complexes.
~
for all
~n~ , and there is a boundary 60
6 ] : ~|X + X 0
are taken as the defining rules for a
In particular, the rule
66 = 0
shows the analogy with chain
Of course the individual rules are connnonly used in homotopy theory, with-
out necessarily considering the total structure. By a
reduced crossed complex
C
we mean one in which
have been considered for some 35 years. [2].
CO
is a point.
These
They were called "group systems" by Blakers
He writes that he follows a suggestion of Eilenberg in using these group systems
to apply the homotopy addition lemma in his investigation of the relationship between the homology and homotopy groups of pairs.
His proofs involve a functor from reduced
crossed complexes to simplicial sets; the values of this functor have been shown recently by Ashley [I] to be
simplicial T-complexes, and Ashley has proved the hard theo-
rem that this functor gives an equivalence T-complexes.
N
between crossed complexes and simplicial
This equivalence generalises the well known equivalence of chain com-
plexes and simplicial abelian groups, due to Dold and Kan [27; Theorem 22.4], and the functor
N
generalises also the nerve of a groupoid, which we use in §3.
Reduced crossed complexes satisfying in each dimension a freeness condition were called "homotopy systems" by Whitehead [3], 32], and his main example was is the filtration of a CW-complex
K
by its skeletons.
~
where
The paper [3]] gives inter-
esting relations between homotopy systems and chain complexes with operators: we shall generalise these results to crossed complexes in [10]. the papers [30, 31, 32] is
reallsability.
In §]7 of [32] Whitehead sketches a proof
of a theorem announced in §7 of [3]], that if of finite dimensional homotopy systems, and
An overall consideration in
~ : C ÷ C' is a homotopy equivalence C
is realisable as
~
for some
40
CW-complex K + K' .
K , then
C'
is also realisable as
~'
and
~
is realisable by a map
The approach to simple homotopy theory in this section of [32] seems to have
Deen ignored and indeed its predecessor [31] is not widely read. Huebschmann, Holt and others (cf. [20, 17] and the historical note [26]) have shown how crossed complexes may be used to give an interpretation of all the cohomology groups
Hn(G; A)
of a group
G
with coefficients in a G-module
A .
Lue has explain-
ed in [24] how related ideas had been developed earlier for varieties of algebras, rather than just for groups.
However, the tie-up with classical cohomology was not
made explicit (cf. p.172 of [24]). We have given in [6, 7] a colimit theorem for the homotopy crossed complex of a union of filtered spaces.
This theorem includes the usual Seifert-van Kampen theorem
on the fundamental groupoid of a union of spaces; it also includes the Brouwer degree theorem
(~n Sn = ~ ) ,
the relative Hurewicz theorem, and a subtle theorem of J.H.C.
Whitehead on free crossed modules [31; §16].
The proof of the colimit theorem in [7]
involves in an essential way two other categories equivalent to crossed complexes, namely m-groupoids and cubical T-complexes [6, 8].
With simplicial T-complexes [l],
~-groupoids [9] and poly-T-complexes [22], there are now five categories known to be equivalent to crossed complexes, the proofs in each case being highly non-trivial. The papers [16, 18] give other work on crossed complexes. One of our aims here is to show how the homotopy addition lemma (which plays a key rSle in the work of Blakers [2] and of the authors [6, 7]) is also important in the cohomology of a group G . We do this by showing that the standard crossed resolution CG , which is constructed algebraically in [20] and applied further in [2]], in fact arises as
~BG , the homotopy crossed complex of the classifying space of
The boundary maps in
CG
G .
are determined by the homotopy addition lenmla.
Our further aim is an exposition of the Schreier theory of non-abelian extensions. Much
has been written on non-abelian extensions and cohomology,
(cf. [5, 12, 13
23] and the further references there), but it is notable that, while tnereare accounts in several books on group theory, texts on homological algebra remain largely silent on the subject, presumably because there is no known exposition using chain complexes, on which expositions of the abelian case are rightly based.
Here we show that the non-
abelian features of crossed complexes allow an exposition closer to the abelian case, involving morphisms and homotopies.
We strengthen the theory, by presenting an equiv-
alence of groupoids which on components induces the usual one-one correspondence of sets.
We also generalise the theory, to extensions of groupoids rather than just
groups, and to "free" equivalences of extensions. l.
Crossed Complexes We recall from [6] the definition of the category (here denoted
complexes. A crossed complex
C
(over a groupoid) is a sequence
XC) of crossed
41 60 "'" ---+ Cn satisfying (].I)
the following
C1
"'6> Cn-I
J> "'" ---÷ C2 ~
C! ~ 61
C0
axioms:
is a groupoid with
CO
as its set of vertices
and
60 , 6 !
as its initial
and final maps. We write
Cl(p, q)
for the group (1.2)
For
n e 2 , Cn
disconnected (1.3)
for the set of arrows
is a family of groups
groupoid
The groupoid
C1
over C 0) operates
(x,a)~-+ x a
Here if
Cn(p) ~ Cn(q)
if
We use additive
p
For
p
to
q
notation
n e 3
(p,q E C O )
and
C](p)
q
and
a £ C1(p,q)
Cn(P)(n
C! , where
, then
e 2)
for all their identity is a morphism
CI
Cn(P)
Cn(n e 2)
lie in the same component
for all groups
0
(equivalently, C n i s a
the groups
on the right of each
n ~ 2 , 6 : C n ÷ Cn_ l
the action of a =-a+x+a.
{Cn(P)}PEC0
and for
x E Cn(p) and
and we use the same symbol (l .4)
from
C](p,p). totally
are abelian.
by an action denoted
x a c Cn(q)
.
(Thus
of the groupoid
CI .)
and for the groupoid
CI ,
elements.
of groupoids
acts on the groups
C|(p)
over
CO
and preserves
by conjugation:
x (1.5)
= 0 : C n ÷ Cn_ 2
~
for
n _> 3 (and
~0~ = 616 : C2 ÷ CO
as follows
from
(1.4)). (1.6)
If C2
c e C 2 , then
as conjugation
6c
by x
In the case when We observe
operates
trivially
CO
= -c + x + c
~roupoid
as a
, or, simply, C](p)
n ~ 3
and operates
on
.
C
C2(p)
two as defining
over
for
(x, c e C2(P))
is a single point, we call
that the above laws make each
(Cl, CO)
Cn
c , that is 6c
we take the laws up to dimension
is a module
on
a
a crossed module
C2
over
CI(p)
;
crossed module over the
as a
crossed Cl-mOdule.
, and we take the laws
reduced crossed complex.
Let
n ~ 3 .
Then
(1.1) - (1.3) as defining
Cn(p)
Cn
as
module over the groupoid (Cl, c O ) , or, simply, as a Cl-module. A morphism f : C + D of crossed complexes is a family of morphisms of groupoids fn : Cn ÷ On , compatible of
C 1 , D!
on
with
Cn , D n .
the boundary
We denote by
maps
XC
Cn ÷ Cn-!
the resulting
' Dn ÷ Dn-l category
and the actions
of crossed
com-
plexes. By restriction (over groupoids). identity fl
of structure, Let
(as happens
Suppose union of the
erators
f : C ÷ D
throughout
are the identity,
we call
be a morphism
§5) we write f
Cl(P)
, p ~ CO .
[x] £ C 2 with
and of crossed
of crossed modules.
f
C , a set
Then we say
6[x] = hx
of modules,
as a pair
If
(f1' f2 ) "
f0 If
modules
is the f0
and
morphism of crossed Cl-mOdules.
a
given a crossed module
and for any other crossed for all
we have categories
.for all
Cl-module
C'
C
X
and function
is
the free crossed cl-module on gen-
h
from
x E X , if such elements and elements
x ~ X , there is a unique morphism
f : C ÷ C'
[x]' c C 2' of crossed
X
to the
[x] are given,
with
6[x]'
= hx
Cl-mOdules
such
42
that
f[x] = [x]'
for all
uced case [31]. [11]),
C2
is constructed,
x a E C2(q) +yb+x
for all
This d e f i n i t i o n becomes
to the group case, given
C1 , X
x ~ X , a ¢ Cl(60Xx ' q), and
q ~ C 0 , wita the usual relations
w h e r e these make sense, and witll C1
is
free on generators
free crossed C1-module on these generators be a crossed complex.
[15] of the groupoid
C1
A crossed complex C2
C
A crossed complex
C
If
C
is exact and
(or, equivalently,
G C
~i C
Px ' x E X.
is the quotient
subgroupoid
6C 2 .
The
(for some
(on some graph
% : X 2 + CI) , and for
XI),
n e 3 , Cn
Xn).
exact if for
is
is a groupoid,
n e 2 =
Im
then
C
(6
G .
Cn+ 1 --+ C n)
:
•
together with an isomorphism
with a quotient m o r p h i s m
crossed resolution of
called a
equal to the zero at
is a free groupoid
(6 : C n --+ Cn_l)
Ker
6 : C2 ÷ C I
n e 3 , the induced structure of #iC-module.
CI
is a free crossed Cl-mOdule (on some
6[x]
with
, x E X , if it is a
totally disconnected
C n , for
free if
is
C
[x] e C2(Px)
with
-x a
~ x a) = - a + % x + a
fundamental groupoid
Its
by the normal,
rules for a crossed complex give
is a free ~iC-module
Ix] = x 0
can be regarded as a crossed Cl-mOdule
Such a Cl-module
C
is given in
% , as t h e g r o u p o i d w i t h g e n e r a t o r s
A module over
Let
the usual one in the red-
(an exposition of w h i c h
and
a=yb-a+%x+a
trivial.
x ~ X .
Analogously
C1 + G
w h o s e kernel
is
Wl C ÷ G
6C2) is
free crossed resolution if also
It is a
C
is
free. Let
G
follows. groupoid w
be a groupoid.
Let on
X
generating
C2
÷ C I) is the G-resolution
w
is
G
~ : C1 ÷ G .
be a function to the union of the
closure of the image of Let
G
X , with quotient m o r p h i s m
: R ÷ C1
G .)
A free crossed resolution of
be any subgraph of
Ker ~ .
CI(p)
may be constructed
Let
R
(CI, CO)
be any set and let
(X; R, w)
determined by
w
.
is a Then
G-module of identities for the presentation (cf. [II]). + Cn ÷
... ÷ C 3 ÷ K
of
K
by G-modules;
: C 2 + C 1 to give a free crossed resolution
of
as
be the free
, p E C O , such that the normal
(The triple
be the free crossed Cl-mOdule
G
and let
G .
presentation of K = Ker(6
: C2
Choose any free
this may be spliced (Such a construction
into for groups
is used in [20, 21].) As explained
in the introduction,
homotopy crossed complex defined by the skeletons (This is due to Whitehead case follows. 71~ is
w~
7n X = 0
for
~ l ( X , x o) •
p
example of a crossed complex
space
~ .
X ; then
~
Let
, and the homology of Hn(Xp)
(cf. [32; Footnote n e 2) , then
w~
in dimension ~
(i.e.
is exact,
is the
b e the filtered space
from w h i c h
the more general
two is given in [II].)
Ker 6/Im 6) is for
, p E X 0 , where
41]).
~
is a free crossed complex.
[31; §16] in the reduced case,
A simple proof of freeness
~I(X, X0)
based at
key
of a CW-complex
phic to the family of groups X
a
of a filtered
Xp
In particular,
n e 2
is the universal if
X
Further isomor-
cover of
is aspherical
(i.e.
and so it is a free crossed r e s o l u t i o n
of
43
2.
The homotopy addition lemma This is a basic,
it expresses
but not so easy to prove,
the idea that "the boundary
Its formulation
involves
all the structural
and so for completeness Let and let n > I
An An
from
then determine vI
to
n-simplex
have its filtration
~I(A l, A~)
of the homotopy
with ordered
set of vertices
by skeletons
A rn "
vI
Then
o , say.
Intuitively,
of its faces".
crossed
complex,
is also written
a .
determines
~n(A n, A nn-l' Vn)
The face maps
3io ~ ~n_l ~n , and the map
respectively,
{Vo,Vl,...,Vn} , is for
The unique arrow of 3 i : A n-I + A n
u : A 1 ÷ A n , which
sends
v0 ,
uo ¢ Zl ~n .
(The homotopy addition le,~na) The elements
1.
Proposition
to
elements
Vn_ l , v n
theory.
is the composite
elements
cyclic group with generator v0
in homotopy
we state it here.
be the standard
an infinite
lemma
of a simplex
a may be chosen so that
the boundary : ~n(A n, A nn - l '
Vn) ___+ ~ n - l ( A ~ - I ' Ann-2' Vn)
is given by -81o + 800 + 320
if
n = 2 ,
300 - (830)u° - 31o + 820
if
n = 3 ,
n-I E (-l)lSi O + (-l)n(3no)U° i=O
if
n > 4
For a proof of this result, homotopy
3.
[29].
lemma is given for m-groupoids
A corresponding
cubical
form of the
as Lemma 7.1 of [6].
The standard crossed r e s o l u t i o n Let
of
addition
see for example
o
G
be a groupoid.
G , in which
of n-tuples
NnG
of elements
ui
The geometric
realisation
The simplicial
structure
homotopy write
crossed
CG
There
on
complex
2.
Let
G
of
G
X = INGI
~
for this crossed
Proposition
is a well-known
is the set of composable
NG
such that
simplicial
set
(ul,...,Un)
u i + ui+ !
is defined
is known as the classifying
induces
a structure
(for the skeletal complex and call
be a groupoid.
NG
elements
on
for
space BG
of CW-complex
filtration
[28], of
X)
on
the nerve
G n , i.e. 1N
i < n .
of
G .
X , and so the
is defined.
We
it the standard crossed resolution of
Then
is a free crossed resolution of
CG
G .
G
and has the following structure. (i)
CoG = G O ; CIG
is the free groupoid on the sub-graph
vertices and all the non-identity arrows of
G .
The basis element of
CIG
u E G*
ation is extended to
G
(ii)
C2G
corresponding to by setting
is written
(u,v) ¢ N2G*
consisting of all the [u] , and this not-
[0p] = 0p
is the free crossed CiG-module on generators ~[u,v]
for all
G*
: -[u + v| + [u] + Iv|
(the composab~e pairs of
G*)
.
[u,v]
E C2G(~Iv)
with
44
(iii) For
is the free G-module on generators
n >- 3 , CnG
for all (u l,...,un) ~ NnG* . We also let [Ul,...,u n] ~ CnG some
ui : 0 .
(iv)
6 : C3G ÷ C2G
be the identity at
~lun
[Ul, .... u n] E CnG(~lUn )
if
(Ul,...,Un) ~ NnG
and
i8 given by
6[u,v,w] = [v,w] - [u,v] [w] - [u + v,w] + [u,v + w] ,
for al~
(u,v,w) E N3G .
For
(v)
n >- 4 , 6 : CnG ÷ Cn_IG
is giVen by n-I i Z (-1) [Ul,...,u i + ui+i, .... u n] i=l
~[Ul, .... u n] = [u 2 ..... u n] +
+ (-l)n[ul, . . . ,Un_l ]
[u n]
.
D
This proposition follows from the homotopy addition lemma, the standard description of the face operators in if
G
NG , and the fact that
is a group, then Proposition 2 shows
homogeneous) crossed resolution of how
CG
G
CG
BG
as defined in §9 of [20].
6 : C3G ÷ C2G
We have now shown
should be noted; the values of this
are in a family of (generally) non-abelian groups. a crossed complex
A
Note that
arises geometrically.
The curious formula for
functor
is aspherical [281.
to be the same as the standard (in-
A(CG)
abelianises
C
a chain complex
AC
is the bar resolution of C2
There is a funetor assigning to
with operators from G
TIC
[I0]; for this
(cf. [25]), for the group case).
However,
and so loses information.
The 3-simplices of
NG
may be pictured as 3
u
+
0
(cf. p.12 of [25]). n > I .
Now
NG
v
~
u
I
is a T-complex in which every n-simplex is thin for
Every T-complex has a groupoid structure in dimension I, and the above picture
illustrates the 3-simplex used to prove associativity [I] of this groupoid. suggests the link between
4.
~ : C3G ÷ C2G
This
and associativity in extension theory.
Homotopies The notion of homotopy has a similar importance for crossed complexes to that for
chain complexes.
However, because of the more complicated structure of crossed com-
plexes, there are several possible conventions for the definition of homotopy, and there are also two levels of generality (corresponding to free and based homotopy in
45
the topological
case).
Our definition
lemma in the algebra of m-groupoids m-groupoids, Let
If If
(4.2)
be morphisms
and
Cop e Dl(fp,gp)
x c Cn(q)
: CI ÷ D2
of crossed complexes.
e n : C n ÷ Dn+l(n >- 0)
p ~ C O , then
n ~ 2 81
from the cubical homotopy addition
a topic w h i c h we hope to develop elsewhere.
f , g : C ÷ D
is a family of functions (4.1)
follows
[6], applied to a natural notion of homotopy for
, then
.
A ~omotopy
0
: f : g
with the following properties.
If
x e Cl(p,q)
, then
Ol x E D2(g q) .
0nX e Dn+l(g q) .
is a derivation
over
gl
, that is if
x + y
is defined
in
Cl
then O1(x + y) = (01x)gY + ely (4.3)
For
n e 2 , e n : C n ÷ Dn+ l
a ~ Cl(P, q) , x ~ Cn(p)
,
, y c Cn(q)
If
x e C1(p, q)
then
(4.5)
If
n ~ 2 , and
x E Cn(q)
gy = gl y .
where
ga = gla .
gx = -e0p + fx + 00q - (6elx)
(A similar definition, [23].
A homotopy p ~ CO
but with different
For further comments,
0 : f = g
conventions,
eq = e0q . is given in the reduced case
see Remark 4 at the end of the paper.)
which are used by H u e b s c h m a n n
if
00p
is an identity for all
(It is these homotopies, [20].)
For emphasis,
with different
the more general
called ~re~ homotopies.
e : f = g , e' : g = h
is defined by then
conventions,
f0 = go ) "
kinds of homotopy are sometimes If
where
is said to be tel C O
(so that in consequence
.
then
gx = (fx) eq - 8n_l~X - ~0nX ,
by Whitehead
gl , that is, if
, then
en(xa + y) = (enx)ga + 8ny (4.4)
where
is an operator m o r p h i s m over
are (free) homotopies,
their composite
~ = e + 8'
~0 p = Cop + 8~p , p ~ C O , and if n ~ I and x ~ C1(p, q) or x ~ C (q), e' n q . It is easily checked that # is a homotopy f = h .
~n x = e~x + (0nX)
In the next section we will be considering which are the identity on
CO = D O .
only crossed complex morphisms
C + D
Therefore we w r i t e
(C, D)f
and
for the groupoids w h i c h have such morphisms vely the free, and the
rel C O , homotopies.
are thus the respective
sets of homotopy
(C, D) as objects,
and w h o s e arrows
The sets of components
classes of morphisms
over
are respecti-
of these groupoids C O = D O , and they
are w r i t t e n respectively [C, D]f
5.
[C, D]
.
Non-abelian extensions Throughout
A
and
this section,
is totally disconnected
~t~nsion
of
A
by
G
G
(i.e.
and A
A
will be groupoids
is a family
is a pair A ~ E ~ G
A(p)
such that
GO = A 0
, p e A 0 , of groups).
and An
46
of morphisms
of groupoids
(5.2)
p
is a quotient m o r p h i s m of groupoids.
(5.3)
i
maps
p
A
i
p
are the identity on objects.
isomorphieally
onto
is a quotient m o r p h i s m means
for more details action of
E
see [15].)
on
A
~
A
A free equivalence
a crossed
is an isomorphism;
i
(large)
free equivalences
A .
~ E
if ~
Extf(G,
A) of
E
induces an
This can be extended
trivially
is (with the quotient m o r p h i s m
A
by
P
> G
G
is a commutative
diagram
G that
~
also is an isomorphism.
is the identity.
Here
Act A
is an isomorphism
on
A
and A
and the equivalences
Under our assumption Act A
of
this implies
the extensions
For any groupoid
q
in
E/Ker p ÷ G ;
Such
W e can thus form two
groupoids
both having objects
of
conjugation
which
i' ~ E' ~
equivalence
is an
an isomorphism
G .
A ~
induces
E-module.
of such extensions A
such that
p
: ... ÷ 0 ÷ 0 ÷ A ÷ E
a crossed resolution of
a free equivalence
Ker p . that
For such an extension,
making
to a crossed complex p)
and
the following properties.
E0 = GO
(That
and
satisfying
(5.1)
A
there is a groupoid
A(p) ÷ A(q) that
A
A)
,
G , but having arrows respectively
the
of extensions.
has the same objects
determine
Ext(g,
by
Act A
as
of actions on the vertex groups
A , and an arrow in
of groups.
Act A
There is a conjugation map
is totally disconnected,
from
p
to
~ : A ÷ Act A.
this map and the action of
a crossed complex ---+ 0 ---+ ... ---+ 0 ----+ A--~+~ Act A
w h i c h we w r i t e
XA .
olution
G , then the action of
~
of
(o, 1) : E ÷ X A isomorphism further
If
(where
A i
E
R> G
o : E ÷ Act A)
(~, ~) : X A ÷ X A
where
is an extension with associated E
on
.
A free equivalence
~ : Act A ÷ Act A
, Extf(G,
e : (CG, X A) The m o r p h i s m
e
since this result
details.
is given by
is the restriction
our m o r p h i s m
H, N, G
of
ef .
of standard
a $~ = ~((~-1a)8) ;
, .
W e give the proof only for
read our
for the group case, and
but with differences
G, A, E ; his factor set
k : C2G ÷ A ; his a u t o m o r p h i s m
a ~-+ a u
ef .
theory, w e do not give full
are given in [14],§|5.1
rather than free equivalence,
: for Hall's
A)
* Ext(G, A)
is a reformulation
[Some of the calculations
for equivalence follows
induces a m o r p h i s m
as in (*) induces an
There are canonical equivalences of groupoid8 ef : (CG, xA)f
Also,
by conjugation
o'n = ~o : E ÷ Act A .
T h e o r e m 3.
Proof.
A
crossed res-
of
N
for
in notation as (u,v) e N u ~ G
becomes
becomes
our
47
morphism
h : CIG ÷ Act A ; his choice
morphism
£ : CIG ÷ E ; his f u n c t i o n
A morphism
CG ÷ x A
+ Act A , k : C2G ÷ A
over
u ~-+ ~
GO = A0
such that
of coset r e p r e s e n t a t i v e s becomes our
~ : H ÷ N
k
becomes our d e r i v a t i o n
~ : CIG ÷ A . ]
is d e t e r m i n e d by a pair of m o r p h i s m s
is an operator m o r p h i s m over
h : CIG
h , and such that
the equations h~ = ~k , k~ = 0 hold.
(These equations are equivalent to the first two equations
of [14], and indeed
k~ = 0
in T h e o r e m 15.1.1
is, b y P r o p o s i t i o n 2, equivalent to the "factor set" con-
dition k[u + v,w] + k[u,v] h[w] = k [ u , v + w] + k[v,w] for all by
G
(u,v,w) ~ N3G*
.)
G i v e n such a m o r p h i s m
is defined by setting
set of pairs
(u,a)
E0 = GO
such that
and for
u E G(p,q)
CG ÷ X A , a n e x t e n s i o n
p, q e G O , letting
, a e A(q)
v e G(q,r)
, b ~ A(r)
.
The v e r i f i c a t i o n that
reader (cf. p.220 of [14]).
We write
Suppose now given two morphisms (h,k)
, (h',k')
as above.
B = 80 ' ~ = @I "
Then
Let
~
E
E(p,q)
of
A
be the
, with addition
(u,a) + (v,b) = (u + v, k[u,v] + a h[v] + b) for
,
E
,
is a groupoid is left to the
E = e(h,k) CG ÷ x A
over
G O , w h i c h w e w r i t e as pairs
@ : (h,k) = (h',k')
is a d e r i v a t i o n over
h'
b e a (free) homotopy, and if
and w r i t e
u e G(p,q) , v ~ G(q,r) ,
w e have h'[v] = -Bq + h[v] + Br - ~ [ v ] k'[u,v] = k[u,v] Br - a~[u,v]
,
.
A s t r a i g h t f o r w a r d c a l c u l a t i o n shows that k'[u,v] + ~ [ u , v ]
= -~[u + v] + k'[u,v] + (~[u]) h'[v] + ~[v]
(and this verifies that our d e f i n i t i o n of e q u i v a l e n c e agrees w i t h that on p.22! of [14]).
Define ef(8)
: e(h,k) ---+ e(h',k') (u,a) ~
Then by
ef(8)
(u, ~[u] + a Bq)
, u ~ G(p,q)
, a E A(q)
is an i s o m o r p h i s m of groupoids which, w i t h the a u t o m o r p h i s m
a ~-+ a Bq , a ~ A(q)
, defines a free equivalence of extensions.
. A ÷ A
Conversely,
given any
free e q u i v a l e n c e
arises in the above w a y if ÷ A
A
~ e(h,k)
> G
A
~ e(h',k')
> G
B : G O ÷ Act A
ined b y
n(u,0) = (u,~'u)
e(h,k)
complex
E
.
Let
h'
B(q) = ~]A(q)
, and : G ÷ A
~ : C|G
the f u n c t i o n
~'
def-
of
is equivalent to
.
Finally, w e show that any e x t e n s i o n some
is d e f i n e d by
is defined b y extending to a d e r i v a t i o n over
~ : CIG ÷ G
A _~i E -P+ G
A
by
G
b e the quotient m o r p h i s m and consider the crossed
obtained by trivial extension of the crossed E - m o d u l e
A .
Consider
48
the d i a g r a m C3G l I + 0
6 ~ C2G i Ik + ~ A
6 .> CIG i I~ + ~ E
A ~ A c t The crossed complex groupoid. (h,k)
CG
is free, while
So the identity on
is a m o r p h i s m
is defined by
G
CG ÷ X A
over
GO
A
is exact,
has a lift
G I I= % G
(~,k)
and both have
: CG -> E .
and an equivalence
G
Let
as fundamental
h = o~ .
of extensions
Then
e(h,k) -> E
(u,a) ~-+ ~[u] + ia .
Thus the crossed complex approach in non-abelian resolution
E
~
is successful
extension theory are so-to-speak
(a kind of universal
example)
because
compressed
some of the difficulties
into the standard crossed
and in particular
into the formula for
: C3G ÷ C2G • By standard homotopy Corollary
Let
4.
C
arguments,
we obtain from T h e o r e m 3;
be any free crossed resolution of the groupoid
G .
Then there
are equivalences of groupoids
Corollary
Let
Let
5.
e~ : (C, xA)f
> Extf(G, A),
e'
~ ext(G,
: (c, ×A)
N i--~ F -P-+ G
F-module
N .
G
Let
C
C ÷ ~
induces
n : F ÷ Act A
is injective. of groupoids
a set normally generating
Then a m o r p h i s m such that
h(r)
such that
C! = F
and
N = 6(C2)
5 is when P ÷ XA
A
. is centreless,
i.e. w h e n
is determined by a m o r p h i s m
is a conjugation
of
A
for each
r
in
N .
isation of Dedecker's work on non-abelian
"
.
also enable one to give a crossed complex version of a general-
be as above and suppose given a crossed ~0 = GO)
G
,
, (~, X A) --+ (C, X A)
special case of Corollary
The above methods
is free.
isomorphisms
(~, xA)f--+ (C, xA)f
: A ÷ Act A
A)
~ Ext(G, A)
be a free crossed resolution of
An interesting
F
obtained by trivial extension of the
~f(G,
e" : (~, xA)
Then the projection
D
Then there are equivalences of groupoids e~ : (~, xA)f ~
Proof.
.
be an extension of groupoids such that
P denote the crossed resolution of
crossed
A)
A H-~tension of
A
by
G
ether with a m o r p h i s m of crossed modules
cohomology ~-module
and extensions
A
is an extension
(where A-~
~ E ~
[12].
Let
G , A
is a groupoid with G
as above tog-
49
i
A
~E
I In fact if, by extending exes
~
and
of crossed
xHA
trivially,
respectively,
we regard
a function
then the above diagram
a co~ugation $
from
HO
xHA ÷ x~A
alence of
H-extensions
and a conjugation
in which
(~,~)
and
: x~A ÷ x~A
a groupoid
' q ~ ~0
(~,~)
such that
Ext,(G,
' such that Define a free equiV-
.
: ~ ÷ ~'
o'n = ~
generalisation
e is similar
on components
over the identity
•
: (CG, ×~A)
G , A , ~
~ ~(G,
A)
,
~ Ext'(G,
A)
.
(Dedecker's
result
is the bijection
results
that given a morphism
~ * xHA
to Oedecker's
A theory of extensions
internal
category
includes
the above equivalence
of groupoids,
to define similar
for the group case. ~ ÷ x~A
(where
e
If
X
is a CW-complex,
the aohomology of
ants of function
spaces
Remark 4. such that
A homotopy ft(Xn)
are discrete,
c Zn+ 1
for
complexes.
C
maps
fo' f|
We will
induces
do not include
and also extensions
are not used.
complex,
it seems reasonable
simply as
[~,
C] .
(A
to Postnikov
invari-
to have applications
invariant.
: ~ ~ ~
is a homotopy
prove elsewhere
a homotopy
the non-abelian
is a homotopy
theory,
in
cohomology.
n ~ 0 .
Consequently,
of
[19] he rel-
is given in [23], using
in E2] and applied
It would be interesting
of filtered
then such a homotopy
above for CW-complexes,
is a crossed
is developed
in [3].)
ft
C
complexes
X with coefficients in
theory of such a non-abelian
to Dedecker
Dedecker's
The results
and crossed and
On p.309 of is as in Coroll-
to be the coequaliser
of T-algebras
This generalises
of groupoids.
nor free equivalences,
~
2-cocycles.
and cohomology
for T-algebras.
idea for chain complexes
homotopy
crossed
objects
induced
are groups.)
has proved related
(for groups)
ates such morphisms
3.
G ,
Such free equivalences
ary 5), one can define an extension A ÷ E ÷ G by taking E __+ two ma~9 A--+ F m A (the semi direct-product). In a letter
Remark
on
of Theorem 3.
to that of Theorem 3. when
Huebschmann
[20] he shows
2.
for which there is
The (strict) equivalences are those
A) .
ef : (CG, x~A)f
Remark
(o,l) : ~ ÷ xHA
There are equivalences of groupoids
Theorem 6.
1.
compl-
is the identity.
We have the following
Remark
H(q)
(~,~)
~(a) = a Bq , a ~ A(q)
to be an isomorphism
(~,~)
form under composition
e
as crossed
is a morphism
to be an isomorphism
to the union of the
~(x) = -Bp + x + Bq , x e N(p,q)
by
these crossed modules
complexes.
Define
The proof
I
~fo = ~f|
cohomology
that if
X0
ft : X ÷ Z and
of morphisms
suggested
Z0
of
in Remark
3
50
R E F E R E N C E S O.
H. ANDO*, A note on the Eilenberg-MaeLane invariant, Tohoku
I.
N.K. A q~iLEY, Crossed oompl~e8 and T-~omplexe8, Ph.D. Thesis, University of Wales, (197fl).
Math. J. 9 (1957), 96-104.
2.
A.L. BLAKEKS, So~e relations between homology and homotopy groups, Ann. of Math., (49) 2 (1948), 428-46].
3.
R. ~ROWN, Cohomology with chains as coefficients, Proc. Lond. Math. Soc., (3) 14 (1964), 545-565.
4.
R. BROWN, On K~nneth suspensions, Proc. Camb. Phil. Soc., (1964), 60, 713-720.
5.
R. BROWN, Groupoids as coefficients, Proc. Loud. Math. Soc., (3) 25 (1072), 413-426.
6.
R. B~OWN and P.J. HIGGINS, The algebra of cubes, J. Pure Appl. A1 E. 21 (1981), 233-260.
7.
R. BI~OWN and P.J. HIGGINS, Colimit theorems for relative homotopy groups, J. Pure Appl. AlE. 22 ( 1 9 8 1 ) , 11-41.
8.
R. BROWN and P.J. HIGGINS~ The equivalence of w-groupoids and cubical T-complexes, Can. Top. G~om. uiff., (3e Coll. sur les cat4gories, d~di4 a Charles Ehresmann), 22 (1981), 349-370.
9.
R. BROWN and P.J. HIGGINS, The equivalence of crossed complexes and ~-groupoids, CaLl. Top. G~om. Diff. , (3e Coll. sur les categories, d4di~ a Charles Ehresmann), 22 ( 1981), 371-386.
10.
R. BROWN and P.J. HIGGINS, On the relation between crossed complexes and chain complexes with operators, (in preparation).
1|.
K, BROWN and J. HUEBSCHMANN, Identities among relations, in L~Dimenaiona~ and T.L. Thiekstun, Lond. Math. Soc. Lecture Note Series 48 (1982).
12.
P. OEOECKER, Les foncteurs Ext~ , H 2 4891-4894.
13.
P, OEDECKER and A. FREI, Gdn4ralisation de la suite exacte de cohomologie non ab~lienne, C.R. Acad. Sci. Paris, 263 (1966), 203-206.
14.
M. HALL, JR., Tile gheo~d of groups., MacMillan (]959).
]5.
P.J. HIGGINS, Ca~egorv~es c ~
16.
P.J. HIGGINS and J. TAYLOR, The fundamental groupoid and homotopy crossed complex of an orbit space, (these proceedings).
17.
O.F. HOLT, An interpretation of the cohomology group~
]8.
J. HOWIE, Pullback functors and crossed complexes, Cah. Top. G~om. Diff., 20 (1979), 281-295.
et
H2
TapoLogy, Ed. R. Brown
non ab~liens, C.R. Acad. Sci. Paris, 258 (1964),
~l~oupo~ds, van Nostrand Math. Studies, 32 (1971).
Un(G, M) , J. Alg., 60 (1979), 307-318.
19.
J. HUEflSCHMANN, Letter to P. Dedeeker, (4th June, 1977).
20.
J. NUEBSCHMANN, Crossed N-fold extensions of groups and cohomelogy, Comm. Math. Helv., 55 (1980), 302-314.
2].
J. HUEBSCtD4ANN, Automerphisms of group extensions and differentials in the Lyndon-Hoehsehild-Serre spectral sequence, J. Algebra, 72 (1981), 296-334.
22.
D.W. JONES, Po~l-T-comp~exe8,
23.
R. LAVEND~OMME and' J.R. ROISIN, Cohomologie non-ab~lienne de structures alg4briques, J. Algebra, 67 ( 1 9 8 0 ) , 385-414.
24.
A.S-T. LUE, Cobomology o f groups r e l a t i v e
25.
S. MAGLANE, Topology and l o g i c as a s o u r c e o f a l g e b r a ,
26.
S. MACLANE, H i s t o r i c a l
27.
J.P. MAY, S~npl~cia~ objects in a~gebralc topologyj van Nos~rand Math. Studies II (1967).
Ph.D. Thesis, University of Wales, (in preparation).
to a v a r i e t y ,
n o t e , J . A l g e b r a , 60 ( 1 9 7 9 ) ,
J . A l g e b r a , 69 ( 1 9 8 ] ) ,
155-174.
B u l l . Amer. Math. S o c . , 82 ( 1 9 7 6 ) ,
1-40.
319-320.
28.
G. SEGAL, Classifying spaces and spectral sequences, Publ. Math. I.H.E.S., 34 (1968), 105-112.
29.
G.W. WHITEHEAD, Elemen~a of /~omo~op~ ~heoz,~, Graduate texts in Maths. No. 61, Springer, Berlin~eidelberg-New York, (1978).
30.
J.H.C. WHITEHEAD, Combinatorial hometopy I, Bull. Amer. Math. Soc., (55) 3 (1949), 213-245.
31.
J.H.C. W~ilTEHEAD, Combinatorial hometopy II, Bull. Amer. Math. Soc., 55 (1949), 453-496.
32.
J.R.C. WHITEHEAD, Simple homotopy type, Amer. J. Math., 72 (1950), 1-57.
* NO~.
Reference [0] continues work of [2J.
UN CRITERE DE REPRESENTABILITE PAR SECTIONS CONTINUES DE FAISCEAUX Yves DIERS D~partement de Math~matiques, U.E.R. des Sciences Universit~ de Valenciennes, 59326 VALENCIENNES O. Introduction. Etant donn~ un foncteur tions chaque objet
B
de
B
d'un faisceau ~ valeurs dans
~
et fibres dans
globales d~fini sur la catggorie dans
~. La cat~gorie
~
flexive dans la cat~gorie
U : $ ÷ ~, on d~termine dans quelles condi-
est isomorphe g l'objet des sections globales continues
~ais~A
~, universel pour le foncteur sections
des faisceaux g valeurs dans
~
et fibres
peut alors ~tre plong~e d'une fa~on pleinement fiddle cor~~ais~A
si bien que chaque objet de
~
peut s'identifier
son faisceau repr~sentant. On utilise la construction universelle des spectres, topologies sepctrales et faisceaux structuraux donn~e dans [6] et on est rameng ~ d~terminer dans quelle condition le morphisme canonique de chaque objet de
~
vers l'objet des sections globales con-
tinues de son faisceau structural, est un isomorphisme. On montre qu'une condition n~cessaire et suffisante est que le foncteur
U
soit cog~n~rateur finiment r~gulier.
Cette notion, plus forte que celle de foncteur cog~n~rateur prDpre [7] et plus faible qne celle de foncteur codense [12], est obtenue ~ partir des notions de famille monomorphique stricte on effective
E8] on r~guligre
cog~n~ratrice par monomorphismes stricts
~]
[5] de morphismes, de famille d'objets
et de morphismes de presentation finie
relative [7~, et est d~crite de plusieurs fa~ons diffgrentes. Dans certaines conditions, un foncteur est cog~n~rateur finiment r~gulier si et seulement si il est cog~n~rateur. Ainsi si et si
~
est une sous-cat~gorie de
~
B
est une cat~gorie arithm~tique
[5] et [15],
ferm~e pour les ultraproduits et dont les mor-
phismes sont exactement les monomorphismes de
~
dont le but est dans
$, alors
est une sous-cat~gorie cog~n~ratrice finiment r~guli~re si et seulement si une sous-cat~gorie cog~n~ratrice, d'un produit d'objets de
c'est-g-dire si tout objet de
~
~
est
est sous-objet
&. II s'en suit un th~orgme de representations par sections
continues de faisceaux qui contient t o u s l e s
thgor~mes de reprgsentations qui utili-
sent habituellement des versions g~n~ralis~es du th~or~me chinois sur les syst~mes de congruences. En appliquant les r~sultats ~ des foncteurs
U
oubli de structure ad~quats entre ca-
tegories d'ensembles munis de structures alg~briques, on obtient d'une part, de tr~s nombreux th~or~mes connus de representation par sections continues de faisceaux dont quelques uns sont d~taill~s ici, et d'autre part, des nouveaut~s parmi lesquelles la representation des anneaux commutatifs r~guliers formellement r~els par des faisceaux de corps ordonn~s, celle des groupes ab~liens sans torsion par des faisceaux de groupes abgliens totalement ordonn~s, celle des espaces veetoriels r~els par des faisceaux d'espaces vectoriels euclidiens ou par des faisceaux d'espaces vectoriels norm~s, celle des ensembles par des faisceaux d'ordinaux finis. Une originalit~ de ces derni~res
52
representations
est que les faisceaux repr~sentsntsont
ces topologiques
non "spectraux"
au sens de Hochster
en g~n~ral pour bases des espa[9] car non To-s~pargs
et ~ven-
tuellem~nt non quasi-compacts. On utilise les notations et les rgsultats de [5], ]. Foncteurs
cog~n~rateurs
de presentation IB
finie
~
tout morphisme
B/~
U-injective
des objets de de
B
si tout morphismes
core appel~es
U : /A ÷ ~. Un morphisme
au-dessous U
de
B. ll est dit
de
finie
[7]
U-injectif
si
se factorise ~ travers lui. Plus ggngrale-
de morphismes
g : B -~ UA
de
B
Les families monomorphiques
families monomorphiques
f : B -> C
s'il est un objet de presentation
~
vers
(fi : B -> Ci)ic I
l'un de ses membres.
dams
finie relative
g : B -~ UA
ment, une famille
On consid~re une cat~gorie localement
[7] et un foncteur
est dit de presentation
dams la catggorie
finiment rgguliers.
[6].
de m~me source de vers
U
IB, est dite
se factorise g travers
r~guli~res
de morphismes
strietes ou effectives
dans
de
~, en-
[81, sont ~tudi~es
[5].
|.0. D~finition. famille
Le foncteur
U-injective
U : ~A ÷ ~
de morphismes
est coggn~rateur
de prgsentation
finiment r~gulier si toute
finie relative de
~
est monomor-
phique r~guli~re. Rappelons
qu'un foncteur
au sens de Grothendieck tion
HO~B(f,UA)
U : /A ÷ ~
est dit coggn~rateur
[|4] si tout morphisme
: HO~B(C,UA)
-~ HO~B(B,UA)
est n~cessairement
isomorphique.
|.]. Proposition.
Si le foneteur
U
f : B ÷ C
propre de
~
[7] ou cog~n~rateur tel que l'applica-
soit bijeetive pour tout objet
est cog~ngrateur
A
de
/A,
il est cogg-
finiment rggulier,
ngrateur propre doric coggn~rateu_r. Preuve
: Soit
f : B ÷ C
un morphisme de
bijective pour tout objet finie,
la categoric
objets de
B/B
de presentation monomorphique
B/~
A
de
Soit
tel que l'application ~
l'est aussi et le morphisme finie au-dessus de lui
finie relative.
Chaque morphisme
rggulier.
Le morphisme
m,n : C ~ D
r~guliers.
deux morphismes
g : D + UA
f
f
est colimite filtrante des i.e. f = lim f. i-~+ i
Ii reste g montrer que mf = nf
gmf = gnf
donc
et soit gm = gn
f
k : D + K puisque
; il se factorise done ~ travers
k ; ce qui implique que
est bijective
et donc que
et par suite
k
est monomorphique
dams les representations
jamais d'adjoint ~ gauche, mais ils ont ngcessairement donn~ par les fibres des faisceaux repr~sentants. a un multiadjoint
g gauche et pour chaque objet
une famille universelle
de morphismes
de
avec
f. :B+C. l i
est ~pimorphique.
est bijective
Les foncteurs qui interviennent
soit
f. : B ÷ C. est U-injectif donc i I est alors monomorphique r~gulier comme colimi-
v~rifiant
vgrifie
HomB(f,UA)
~tant localement de presentation
de prgsentatiOn
te filtrante de monomorphismes
Tout morphisme
~
~. La cat~gorie
B
vers
leur COnoyau~
Hom~(f,UA) HO~B(k,UA)
m = n.
par faisceaux ne poss~dent
un multiadjoint
~ gauche
[4]
On suppose donc que le foncteur B
de U.
U
~, on note (~i :B÷UAi)icSpecu(~
53 1.2. Proposition. Si le foncteur
a un multiadjoint ~ gauche, il est cog~-
U :~ ÷ 8
n~rateur finiment r~gulier si et seulement s i i l de
~
existe une classe
de morphismes
telle que
(I) tout morphisme diagonalement universel de morphismes de
~
de source
B
B
vers
U
est colimite filtrante de
et
(2) toute famille U-injective de morphismes de
~ , est monomorphique r~guli~re.
Preuve : La condition n~cessaire est satisfaite en prenant pour phismes de presentation finie relative de 0
~
de morphismes de
universelle
B
0
la classe des mor-
~. R~ciproquement supposons qu'une classe
satisfasse (I) et (2). Pour chaque objet
B
de
~, la famille
(Ni : B ÷ UA.) est monomorphique. En effet, si T est un objet de pr~i ~ et m,n : T ~ B sont deux morphismes vgrifiant D.m = N.n i i i c Specu(B), alors d'apr~s (|), pour chaque i, il existe un morphisme
sentation finie de pour tout d. : B ÷ D. i i
de
~
au-dessus de
N. i
tel que
d.m = d.n. La famille l i
est U-injective, donc monomorphique r~guli~re d'apr~s (2). Par suite tat est aussi vrai pour un objet quelconque d'objets de presentation finie de
(Ni). Soit
morphisme
fk(i) : B + Ck(i)
d.1 : B ~ D.I
de
~
)
m = n. Le =~sul-
~, puisque celui-ci est colimite (Ni)
~tant monomorphiques,
route
est aussi monomorphique puisque plus fine une famille U-injective de morphismes
~. Chaque morphisme
~. se factorise ~ travers un i k(i) c K. D'apr~s (I), il existe un morphisme
avec
au-dessus de
(d i : B + Di)icSpecu(B)
~
(fk : B ÷ Ck)kc K
de presentation finie relative de
de
~. Les families
famille U-injective de morphismes de [5] qu'une famille
T
(di)icSpecu( B
N.,I qui se factorise ~ travers
fk(i)" La famille
ainsi obtenue est U-injective donc monomorphique r~guli~re.
elle est moins fine que la famille
(fk)kcK
et m~me r~guli~rement moins fine [5]
puisque route image directe de la famille Ii s'en suit que la famille Lorsque les objets de
A
(fk)kc K
(d.) est U-injective donc monomorphique. i est monomorphique r~guli~re (prop. 2.1 [5]).
sont des ensembles munis d'une structure alg~brique d~finis-
sable par une th~orie logique du premier ordre, la cat~gorie
$
est ~ ultraproduits.
Nous allons montrer que, dans ce cas, il suffit de consid~rer les families finies de morphismes. 1.3. Proposition. Si le foncteur ultraproduits d'objets de (I)
U
U :A ÷ ~
a un multiadjoint g gauche et rel~ve les
& ([6] 4.1), les assertions suivantes sont gquiValentes
:
est cog~n~rateur finiment r~gulier,
(2) toute famille finie U-injective de morphismes de (3) il existe une classe
D
~, est monomorphique r~guli~re,
de morphismes de presentation finie relative de
~
telle
que a) tout morphisme diagonalement universel de de morphismes de
0
B
vers
b) toute famille finie U-injective de morphismes de Preuve :
U
est colimite filtrante
et
(I) => (2):Soit
(fi : B ~ Ci)ic[],n]
est monomorphique rgguli~re
une famille finie U-injective de
54
morphismes de
~. La cat~gorie
(B/~) n
existe une petite cat~gorie filtrante
~tant localement de presentation finie, il ~
et un diagramme
d'objets de presentation finie de
(B/B) n
que les morphismes
sont de presentation
tout
fik : B + Cik
((fik:B + Cik)icE1,n~kcK
dont la colimite est
(fi). C'est-g-dire
finie relative et que pour
i £ El,n],
fi = ~ fik" Pour ehaque k ¢ ~, Is famille (fik:B ÷ Cik)ie[l,n] keK est U-injective done monomorphique rfiguli~re. Notons (fijk : Cik + Cijk' f~jk: Cjk ÷ Cij k)
la somme amalgam~e de
(fik : C + Cik, fjk : C + Cjk)
et
n
n
Pik :
R Cik + Cik la projection canonique. Le morphisme (fik) : B + ~ Cik i=I i=l est noyau des deux morphismes (f~jkPik)(i,j)e[l,n]2 et (f~jkPjk)(i,j)e[1,n]2 de f,. (f~j:C. " i ÷ C.., xj xj :C.j -~ n Cij) la somme amalgam~e de (fi : B ÷ Ci, f. : B -+ Cj) et Pi : ~ C. ÷ C. la J i=l i i projection d'indice i, alors par passage ~ la colimite filtrante suivant ~, le morN phisme (fi) : B ~i=IH C.x est noyau des deux morphismes (f~:pi)J (i,j)cE1,n]2 et source
n i~l Cik
(f'.'.p.) 2 lj j (i,j)e[l,n] la famille
(~Tn) (i,j)=(l,l)Cijk.
et de but
n ~ C. i=l i
de source
(fi)ieEl,n ]
est monomorphique
morphismes
de
r~guli~re. ~
la classe des morphismes de presentation
~.
(3) => (1) :avec la proposition de
(n,n) ~ C .. Cela implique que (i,j)=(],l) ij
et de but
(2) => (3) : est satisfait en prenant pour finie relative de
Si l'on note
~
].2, il suffit de montrer que toute famille U-injective
est monomorphique
r~guli~re.
Soit
(fk : B + Ck)kc K
une telle
famille. Supposons qu'il n'existe aucune sous-famille finie U-injective de Pour chaque partie finie que
K de K, notons D(K o) o se factorise g travers l'un des morphismes
Hi
D(Ko) # Specu(B) ,
D(~) = ~
et
D(KoO
(fk)kcK .
l'ensemble des i ¢ Specu(B) fk
avec
K I) = D(Ko ) 0 D(K l)
tel
k e Ko. Les relations montrent que les parties
compl~mentaires finies de
des parties D(K ) dans Specu(B) quand K parcourt les parties o ~ o K, forment une base de filtre sur Specu(B ). Soit F un ultrafiltre plus
fin. Ii existe un objet (UAi)icSpecu(B)
AF
suivant
de
$
tel que
torise ~ travers un morphisme
fk
avec
colimite filtrante et que le morphisme existe
I ¢ F
fk" L'inclusion
tel que le morphisme I C D({k})
r~sulte que la famille
k c K. Puisque fk
D({k})
de
NF : B + UA F
UA F = I ~
se fac-
icl~ UA.I est une
finie relative,
il
H UA i se factorise ~ travers iel D({k}) e F, ce qui est en contradiction dans
Specu(B)
appartient ~
poss~de une sous-famille finie U-injective
est monomorphique
r~guli~re. La famille
est done monomorphique
injectives. La sous-famille
d~fini
est de presentation
implique alors
(fk)kcK
(fk)keKo
(H i : B * UAi)icSpecu(B)
soit l'ultraproduit
(Ni)icl : B ÷
avec le fair que le compl~mentaire de
La sous-famille
UA F
F. Le morphisme canoniquement
(fk)kcK
F. Ii en (fk)kcK . o
de m~me que toutes les families U-
est r~guligrement plus fine que la famille o
55
(fk)keK
(2 [5])
puisque toute image directe de
(fk)k~K
est U-injective donc mono-
morphi~ue. De la proposition 2.1 [5], il r6sulte que la f~mille
(fk)keK
est monomor-
phique rgguli~re. ].4. Proposition. Si le foncteur
U
est codense []2], il est cog~ngrateur finiment
r~gulier. Preuve : Si &
U
est codense, tout objet
B
de
B
est limite de t o u s l e s objets de
au-dessous de lui, ce qui implique que la famille de t o u s l e s morphismes de
vers
U
B
est monomorphique r~guli~re (prop. 5.4 F5]). Toute famille U-injective de
morphismes de source
B
est plus fine que la famille pr6c6dente donc est monomorphi-
que ; elle 8st m~me rgguli~rement plus fine puisque toute image directe d'une famille U-injective est U-injective donc monomorphique (prop• 2.1
; elle est donc monomorphique r6guligre
~]).
1.5. Proposition.
Le foncteur
existe un foncteur
V : K ÷ ~
U :~ ÷ B
est cog6n~rateur finiment r~gulier s'il
tel que le foncteur
UV
soit cog~n~rateur finiment
r~gulier. Preuve : Toute famille U-injective de morphismes de presentation finie relative de m~me source de
~
est UV-injective donc monomorphique r~guli~re.
2. Le crit~re de repr6sentabilit~. 2.0. Th~or~me. Soit
U :~ ÷ ~
un foncteur tel que : I)
ment de presentation finie, 2) U diagonalement universel d'un objet
~
est une cat~gorie locale-
admet un multiadjoint ~ gauche, 3) tout morphisme B
pr6sentation finie relative de source
de B
~
vers
U
est colimite de morphismes de
diagonalement universels pour
U, 4)
U est
cog~n~rateur finiment r~gulier. Alors tout objet dans
~
B
de
et fibres dans
~
d~termine un faisceau
FB
Spe_cu(B)
~ valeurs
$, dont l'objet des sections globales est isomorphe ~
qui est universel pour le foncteur sections globales que le foncteur
de base
Bet
F : ~ais ~A ÷ ~ ; c'est-g-dire
F admet un adjoint g gauche pleinement fiddle. Si les conditions
I), 2), 3) sont satisfaites, la condition 4) est en fait n~cessaire et suffisante pour obtenir la conclusion. Preuve : Les conditions ]), 2), 3) sont les conditions d'applications du th6or~me 3.1 de E6] dont on utilise ici les notations et les r~sultats (cf. 3.0, 3.], 3.3, 3.4, 3.5). Soit
B
un object de
a) la famille universelle foncteur
U
est monomorphique puisque le
est coggn~rateur. Cela implique que les morphismes de
universels pour f, g : C ~ D
~. (H i : B + UAi)ieSpecu(B)
U
sont ~pimorphiques. En effet si
sont deux morphismes v~rifiant
on a ~3f6 = ~jg6
ce qui implique
(nj• : D ÷ UAj)jaSpecu(D)
~jf
=rljg
6 : B + C
~
est l'un d'eux et si
f6 = g6, alors pour tout donc
f = g
diagonalement
j e Specu(D),
car la famille
est monomorphique.
b) Montrons que le foncteur
D : ~(B) ÷ D(Specu(B)) °p
est une 6quivalence de cat6-
56
gories. II est surjectif sur les objets d'apr~s la construction de est fiddle puisque les morphismes de morphisme entre deux objets de ~(B) de
tels que
A'(B)
~(B). Soit
D(6) C D(6'). Notons
(6,6'). La relation
6tant 6pimorphiques, 6 : B ÷ C, 6' : B ~ C'
D(6|6) = D(6) N D(@') = D(6)
gonalement universel de presentation finie relative
@|
est U-injectif.
6
est plein.
(dk : (C,@) ÷ (Ck,@k))keK
D(@) = keK ~J D(6k)" La famille relative est U-injective.
6 : B ÷ C
PB(Specu(B))
dans
~(B), est un faisceau. On en d~duit
canoniques
B ÷ FB(Specu(B))
F
F : ~ais ~A ÷ ~
FB(D(6)) ÷ Fc(Specu(C))
F : ~ais ~
FB(Specu(B))
on d~duit
est pleinement
fiddle
est un isomorphisme.
que l'application
Specu(~ ) : Specu(C) ÷ Specu(B)
~.j : A.1 ÷ A.j
ouvert (prop. 3.3.6 [6]) la fibre de
l'isomorphisme
le morphisme canoniquement ~ ~. Alors
Soit
et
j. Le morphisme
Fc(Specu(@)) ~
A'(B)
est un faisceau sur
est monomorphique
au point
= B
et du
(U~j) ~i = ~.6. Puisj i
fibre de
est la fibre de F6
en
i
(F~)D(@)
r~guli~re.
Or on a
(6k : B ÷ Ck)ke K
1.2, il
de morphismes
ke~K D(6 k) = Specu(B) , doric, puisque (FB(Specu(B)) ÷
r~guli~re. Compte tenu des isomorphismes
e), la famille pr~c~dente est isomorphe g la famille
3. Un crit~re de reprgsentabilit~
FC
est doric
: FB(D(6)) ÷
~'(B). D'apr~s la proposition
Specu(B), la famille de morphismes
est monomorphique
:
est cog~n~rateur finiment r~gulier. Tout morphisme
est colimite filtrante de morphismes de
FB
(F6)D(6)
est un plongement hom~omorphique
suffit donc de montrer que toute famille U-injective de
d~fi-
j e Specu(C ). Posons
U~. : UA. ~ UA.. On en d~duit que le morphisme i i j est un isomorphisme. U
On
F~ : F B ÷ Fc(Specu(~)) ~
l'isomorphisme d6fini par
(F@)i : (FB)i ÷ (Fc(Specu ( 6 )))i
f) Montrons que le foncteur
@ : B ÷ C e &'(B),
est un isomorphisme.
Specu(B). Montrons que le morphisme
i = (Specu(6))(j)
FB(Specu(B))
et
I), 2), 3) que le foncteur adjoint ~ gauche
(Specu(6),F 6) : (Specu(B),F B) ÷ (Specu(C),F C)
FB(D(~k)))ke K
F B ~ PB
PB(D(6)) = C
~tant des isomorphismes,
FB(D(~)) ÷ Fc(Specu(C))
est un morphisme de faisceaux sur
Ni
Cela exprime pr~ci-
dgfini par
soit pleinement fiddle. Montrons d'abord que pour
ni par le foncteur adjoint ~ gauche ~
Fc(Specu(C))
telle que
~12])
le morphisme canoniquement d6fini
en
r~guligre.
PB : V(Specu(B)°P ÷ ~
e) Supposons maintenant avec les conditions
note
vers
de morphismes de presentation finie
Elle est donc monomorphique
que le foncteur adjoint ~ gauche au foncteur
au foncteur
~(B)
6'
= PB(D(IB)) = B.
d) Les morphismes
(Th. 1, p.88,
C'est donc
est un morphisme de
une famille de morphismes de
(dk : C ÷ Ck)ke K
s~ment que le pr~faisceau structural pour
D
la somme amalgam~e
implique que le morphisme dia-
6~1@~
dans la cat~gorie ~(B). Ainsi le foncteur
et il
deux objets de
(61 : C ÷ CI, 61 : C' ÷ C|)
un isomorphisme d'aprgs 4). Ce qui implique que
c) Soit
N(Specu(B))
il y a au plus un
(6k).
special pour les categories arithm~tiques.
Le th6or~me suivant contient les th~orgmes de repr~sentabilit~ de faisceaux qui utilisent habituellement
par sections continues
une version g~n~ralis6e du th~or~me chinois
57
sur les syst~mes de congruences. arithm~tiques, 3.0. Th~orgme. trice de
6
exactement Alors
A
So it
6
une cat~gorie arithm~tique
les monomorphismes
de
FB
de base
teur sections globales
: Soit
valences
R
on note
B
B
de
B
g
et
un objet de
vers
~
et
~. Notons
phismes diagonalement
B
de
L'ensemble ~
nalement universels pour
de
~
d~ter-
$, dont
F
admet un
R
~. Pour chaque
(DR : B + B/R)RcSpecu(B )
En effet, si
g : B ÷ X
la relation d'~quivalence de
h : B/R + X
B
est dans
est un m~rphis-
sur
tel que
B
~
~, donc
B/R
multir~flexive ~R : B + B/R
les morphismes
$. Ils sont aussi de presentation B
par un
de
o0
~. Notons R
est une
et o0
~R
est
est fermg pour les colimites finies dans
~tant monomorphiques,
de
g
est
II est imm~diat qu'une telle fac-
de la forme
A'(B)
engen-
h D R = g ; puis-
engendr~e par un nombre fini d'~Igments
universel
R e Specu(B) ,
vers
~
de
A'(B)
sont diago-
finie relative.
~tant colimite
Tout mor-
filtrante de morphismes
A'(B), la condition 3) du th~or~me 2.0 est satisfaite.
II reste g montrer que 6. Pour chaque objet phique puisque de morphismes
~ de
$ B
est une sous-cat~gorie de
¢og~nfiratrice dans ~
est monomorphique.
B, les morphismes
De la proposition
de morphismes
4. Applications. de representation foncteurs
Le th~or~me 2.0
de
A'(B)
7.11
[5], il r~sulte que
est monomorphique
r~guli~re.
1.3 en prenant comme morphismes
permet de retrouver de tr~s nombreux
oubli de structure ad~quats.
de classe
th~or~mes
II suffit de l'appliquer
connus g des
De nombreux exemples de foncteurs
I), 2), 3) du th~or~me
sont satisfaites
[6]. II reste au lecteur ~ d~terminer dans quels cas, le foncteur
rateur flniment r~gulier.
est monomor-
A'(B).
par sections continues de faisceaux.
U :$ + 6
pour lesquels les hypotheses
dans
de
finiment r~guli~re de
(NR : B + B/R)
~. Par suite toute famille U-injective
Le r~sultat d~coule alors de la proposition de source
coggn~ratrice
6, la famille universelle
toute famille finie U-injective
U
B
et fibres dans
que le foncteur
La famille
est donc une sous-cat~gorie
le morphisme
de
B
; on obtient ainsi une factorisation de
sur
de
et tout objet
l'ensemble des relations d'~qui-
(qR : B ÷ B/R)RcSpecu(B).
relation d'~quivalence quotient.
sont
~.
est dans
l'unique monomorphisme
l'ensemble des morphismes
B/~. Les morphismes
B/R
quotient.
$, on note
R e Specu(B)
de la famille
cog~n~ra-
&
et qui est universel pour le fonc-
Specu(B )
~, le morphisme
torisation est unique. ~ A'(B)
B
est universelle.
h : B/R ÷ X
~
est isomorphe g
le morphisme
est un objet de
morphisme
~
de
fiddle.
vers un objet de
dr~e par
de
g valeurs dans
dont l'objet quotient
NR : B + B/R
de morphismes
X
B
une sous-cat~gorie
dont le but est dans
Specu(B)
un objet de
sur
~
F : ~ais ~A + B ; c'est-~-dire
adjoint g gauche pleinement
que
6
et
et telle que les morphismes
est une sous-cat~gorie multir~flexive
l'objet des sections globales
me de
et El5].
ferm~e pour les ultraproduits
mine un faisceau
Preuve
Pour la d~finition et des exemples de categories
on peut se reporter ~ ~]
sont dorm's U
est cog~n~-
Nous en ~tudions quelques uns. Pour les categories
arithm~-
88
tiques (4.3 ~ 4.7), on utilise plutSt le th6or~me 3.0. avec lequel on est ramen6 montrer que le foncteur inclusion objet de
~
U :& ÷ ~
est cog6n6rateur, c'est-~-dire que tout
est sous-objet d'un produit d'objets de
&. Or c'est une propri6t6 souvent
bien connue dont la preuve repose essentiellement sur le lemme de Zorn. Les representations (4.8 g 4.12) sont nouvelles. Les faisceaux repr6sentants poss~dent l'originalit6 d'avoir pour bases des espaces topologiques non "spectraux" au sens de Hochster [9] car non To-s6par6s et 6ventuellement non quasi-compacts. 4.0. Repr6sentation d'un anneau cormnutatif par un faisceau d'anneaux locaux [2], [3]. Compte tenu de 7.0 [6], il suffit de montrer que le foncteur
U : Gocc ÷ S n e
est
cog6n6rateur finiment r6gulier. Un ultraproduit d'anneaux locaux 6tant un anneau local, le foncteur classe
~
U
relgve les ultraproduits. Utilisons la proposition 1.3 avec la
des morphismes de la forme
A ÷ A[a-l]. Soit
(A ÷ A~aTl])i~[l,n~ ~
mille U-injective. Pour chaque P c Specu(A), le morphisme -I A ÷ A[ai(p) ] avec i(P) c If,hi ; alors
vers un morphisme
A ÷ Apse
une fa-
factorise
~ tra-
ai(p) ~ P. L'id6al de
A
engendr6 par l'ensemble des 616ments id6al premier de ea famille
A, est ~gal g
(A ÷ A[a71])icD,n]a
a. pour i ~ [l,n] n'6tant eontenu dans aucun l A. La suite a I , ...,an engendre done le A-module A. est donc monomorphique r6guli~re (8.O~5]).
ainsi la repr6sentation classique de
A
par son faisceau structural
On obtient
A.
4.1. Repr6sentation d'un anneau commutatif par un faisceau d'anneaux ind6composables [3], El 3]. Soit
U : /Anclnd ÷/Anc
le foncteur inclusion (7.5 [6]). Pour un anneau
famille des anneaux quotients de l'anneau des idempotents de vers
U. Chaque morphisme
(A ÷ A/PA)
o3
P
A, est une famille universelle de morphismes de
A ÷ A/PA
A. Montrons que le foncteur
U
A ÷ A/Ae
o7
est coggn~rateur finiment rggulier. Un ultra-
ultraproduits. Utilisons la proposition 1.3 avec la elasse A ÷ A/Ae. Soit
est un idempo-
e
produits d'anneaux indgcomposables ~tant indgcomposable, le foncteur
forme
A
est colimite filtrante de morphismes diagonalement
universels de presentation finie relative de la forme tent de
A c ~nc, la
dgcrit l'ensemble des id~aux premiers
0
U
relgve les
des morphismes de la
P c Specu(A), il existe
(A ÷ A/Ae.). [',n] une famille finie U-injective. Pour chaque llXC I i(p) c [ ,3 tel que le morphisme A ÷ A/PA se factorise
travers
i.e tel que
A ÷ A/Aei(p)
tient ~ t o u s l e s
ei(p) c P. Par suite l'idempotent
id6aux premiers d'idempotents de
(A + A/Aei)ic[l,n ]
~ e appari=I i A ; il est donc nul. La famille
est done monomorphique r6guli~re (8.2 [5]).
4.2. Repr6sentation d'un treillis distributif par un faiseeau de treillis locaux Compte tenu de 7.9 [6], il suffit de montrer que le foncteur
U : ~rDLoc ÷ ~rD
~],~] est
cog~n~rateur finiment r~gulier. Ce foncteur rel~ve les ultraproduits puisqu'un ultraproduit de treillis locaux est un treillis local. Utilisons la proposition 1.3 avec la classe
~
principal de
des morphismes quotients de la forme E
engendr6 par
jective. Pour chaque
a. Soit
E ÷ E/(a)
(E ÷ E/(ai))iEEl,n~
~ c Specu(E) , le morphisme
E + E/~
o3
(a)
est le filtre
une famille finie U-infactorise g travers un
59
morphisme
E ÷ E/(ai(~))
a I V ... V an Donc
avec
i(~) e If,n], et par suite
appartient alors g t o u s l e s
ai(~) e ~. L'61~ment
filtres premiers de
E ; il est ~gal ~
I.
E/(a I) ~ ... ~ (an ) = E/(a I V ... V an ) = E. La famille (E ÷ E/(ai))ie[1,n] est
monomorphique rgguli~re d'apr~s 7.O.1, 8.5.1, 8.5.2 [5]. 4.3. Representation d'un anneau com~utatif r~gulier par un faisceau de corps commuta-
[3],
tifs
[10].
Le foncteur inclusion
U : ~c ÷ &ncReg (7.3 [6~) satisfait les hypotheses du th~or~me
3.0. En effet la cat~gorie gorie cog6n~ratrice de r6gulier
A ¢ ~ncReg
(A ÷ A/P)PeSpec(A )
SncReg
est arithm6tique (8.5 [5]), ~c
est une sous-cat6-
/AncReg puisque l'intersection des id~aux maximaux d'un anneau est r~duite ~ z6ro et donc la famille des anneaux quotients
est monomorphique,
~c
est ferm6e pour les ultraproduits et tout
sous-anneau r~gulier d'un corps cormnutatif est un corps. 4.4. Repr6sentation d'un anneau fortement r~gulier par un faisceau de corps [0_]. Le foncteur inclusion 3.0, la cat~gorie
U : ~ ÷ ~nForReg (7.4 [6]) satisfait les hypotheses du th~or~me
/AnForReg
6tant arithm~tique (8.5 [5]) et tout anneau A c /AnForReg
6tant un sous-anneau d'un produit de corps. 4.5. Representation d'un groupe ab~lien r~ticul~ par un faisceau de groupes abgliens totalement ordonn~s ~I]. Le foncteur inclusion
U : ~bTotOrd ÷ ~bRet (7.10 [6]) satisfait les hypotheses du
th~or~me 3.0, la cat~gorie
AbRet
~tant arithm~tique (8.5 ~]).
4.6. Representation d'un anneau commutatif fortement r~ticul~ par un faisceau d'anneaux totalement ordonn~s [II]. Le foncteur inclusion
U : IAncTotOrd ÷ ~ncForR~t
du th~or~me 3.0, la cat~gorie
(7.11
[4)
satisfait les hypotheses
AncForRet ~tant arithm~tique (8.5 [5]).
4.7. Representation d'un anneau commutatif rggulier fortement r~ticul~ par un faisceau de corps commutatifs ordonn~s Le foncteur inclusion
[11].
~cOrd ÷ SncRegForRet
(7.12 [6]) satisfait les hypotheses du
th~or~me 3.0. 4.8. Representation d'un anneau commutatif r6gulier formellement r~el par l'anneau des sections globales d'un faisceau de corps commutatifs ordonngs. On considgre la cat~gorie
~ncRegFormRl des anneaux commutatifs unitaires r~guliers
formellement r~els i.e. qui v~rifient l'axiome :
~Xl, ...,Xn,
1+x~ + ... + x n2
ble, et des homomorphismes d'anneaux et le foncteur oubli de structure AncRegFormRl. Le foncteur A £ SncRegFormRl (O) - I ~ P (x ~ P
ou
U
admet un multiadjoint ~ gauche. Le spectre de
relativement g
(1) P + P C P
vers
U
est l'ensemble des parties
(2) PP C P
(3) P O (-P) = A
P
de
o7
telles que
(A ÷ A/p N (=P))
P, est une famille universelle de morphismes de
U. La topologie spectrale est engendr~e par les parties
{P : -(a~ + ... + a~) e P}
A
(4) Vx C A, ~y c A(xy c - P=>
y c P)) (cf. 7.27 [6]). La famille des anneaux quotients
munis des ordres quotients de A
inversi-
U : ~cOrd ÷
a I ..... a n e A. Elle n'est pas
D(al,...,a n) =
To-s~par6e car toutes
60
les parties
P
qui d~finissent un ordre sur
(-P) = O, sont des points denses de
r~me 2.0 sont satisfaites. Montrons que id~aux maximaux
M
de
A
A i.e. qui v~rifient en plus (5) P
Specu(A). Les hypotheses (1), (2), (3) du th~oU
est coggn~rateur liniment r~gulier. Les
sont r~els, donc les corps quotients
A/M
rgels et par suite ordonnables. La famille des anneaux quotients de maximaux est donc une famille de morphismes de morphique, donc le foncteur duits. Chaque morphisme
U
A
vers
par ses id~aux
U. Or c'est une famille mono-
est cog~n~rateur. Le foncteur
A ~ A/p ~ (_p)
sont formellement A
U
rel~ve les ultrapro-
est colimite filtrante de morphismes quotients
A ÷ A/I
o~ I est un ideal de type fini de A. Si (A ÷ A/I ,...,A ÷ A/in) est une 1 famille finie U-injective, elle est monomorphique donc monomorphique r~guligre puisque
la cat~gorie foneteur
U
AncRegFormReel
est arithm~tique (prop. 7.O.1.
[~).
Ii s'ensuit que le
est cog~n~rateur finiment r~gulier.
4.9. Representation d'un groupe abglien sans torsion par le groupe des sections globales d'un faisceau de groupes ab~liens totalement ordonn~s. Le foncteur
U : &bTotOrd ÷ SbSTor (7.30 et 7.31 E ~ )
est cog~n~rateur finiment r~gu-
lier car il est surjectif sur les objets puisque tout groupe ab61ien sans torsion est totalement ordonnable. 4.10. Representation d'un espace vectoriel r~el par l'espace vectoriel des sections globales d'un faisceau d'espaces vectoriels euclidiens. Montrons que le foncteur
U : ~ucl ÷ ~ec(~)
est coggn~rateur finiment r~gulier (7.32
[6]), en utilisant la proposition 1.2 avec la classe E ÷ E/X E/Xi)ic I
o~
X
e
Xi(l)+...+Xi(n)
et un suppl~mentaire
P la forme
X'
quadratique positive
q
q e Specu(E). II existe
sur
9. Soit
(E ÷
est de dimension finie done poss~de une base
avec E
E. Soit
i(1),...,i(n) c I. L'es-
de codimension finie. Tout ~Igment
x = Xlel+...+Xpep+X'
factorise
des morphismes de la forme
est un sous-espace vectoriel de dimension finie de
une famille U-injeetive de morphismes de
pace vectoriel
D
par
i(n+l) c I
Xl,...,~
c ~
et
x
de
E
el,...,
~tant de
x' c X', on d~finit la forme
q(x) x~+...+x 2. Alors ISO(q) = X', donc P tel que le morphisme quotient E + E/Xi(n+l )
E + E/ISO(q) , c'est-~-dire tel que
Xi(n+l)C
ISO(q)
donc tel que
(Xi(1)+...+Xi(n) N Xi(n+l) = {0}. II s'ensuit que la famille des morphismes quotients (E ~ E / X (• 1) , . . .,E ÷ E/Xi(n+l) ) prop. 6.0 [5], la famille
est monomorphique r~guli~re (8.4 [5~). D'apr~s la
(E ÷ E/Xi)ic I
est monomorphique r~guli~re.
4.11. Representation d'un espace vectoriel rgel par l'espace vectoriel des sections globales d'un faisceau d'espaces vectoriels norm~s. Le foncteur
U : ~orm(~) ÷ ~ect(~)
(7.34 [6])
est eog~n~rateur finiment r~gulier
puisqu'il factorise le foncteur cog~n~rateur liniment r~gulier
Eucl ÷ ~ect(~) (prop.l~)
4.12. Representation d'un ensemble par l'ensemble des sections globales d'un faisceau d'ordinaux finis. Montrons que le foncteur
U : @rdfin ~ ~ns (7.36 ~])
des morphismes de la forme
lier en utilisant la proposition 1.2 avec la classe E ÷ E/R
o~
R
est une relation d'~quivalence sur
est coggn~rateur finiment r~gu-
E
engendr~e par un ensemble fini.
61
Soit
(E * E/Ri)ie I
Soit
R
une famille U-injective de morphismes de
la relation d'gquivalence sur
E engendr~e par
finiment engendrge. La relation d'~quivalence seulement si (x = y o u
%
sur
(lea classes d'~quivalences de
9. Soit i(I) ..... i(n)el.
Ri(1) U ... U Ri(n). Elle eat E x
d~finie par et de
y
x ~ y
suivant
si et R
sont
des singletons)) poss~de un ensemble fini de classes d'~quivalence. L'ensemble quotient E/~
peut ~tre muni d'une structure d'ordre total et eat donc en bijection avec un
ordinal fini. II existe alors
i(n+l) e I
factorise l'application quotient
tel que l'application quotient
E ~ E/b, c'est-~-dire tel que
deux relations d'~quivalence
R
lea relations d'~quivalences
Ri(1),...,Ri(n)
suit que la famille
et
~
Ri(n+l ) C
E + E/Ri(n+1) ~. Or lea
sont premieres entre-elles (8.3 [5]). Donc sont premieres avec
(E ÷ E/Ri(1),...,E ÷ E/Ri(n+l))
Ri(n+l). II s'en-
eat monomorphique rgguli~re
(8.3. [5]). D'apr~s Is proposition 6.0 de ~] is famille
(E ÷ E/Ri)iE I e s t
monomor-
phique r~guli~re. 4.13. Quelques contre-exemples. On montre facilement que lea foncteurs ~c ~ Snc, &ncDifLoc ÷ &ncDif, ~oc ÷ An
~om ~ ~nc,
E6~ ne sont pas cog~n~rateurs propres, donc
ne sont pas cog~n~rateurs finiment r~guliers et par suite ne donnent pas de th6or~mes de representations. REFERENCES [O] R.F. ARENS et J. KAPLANSKY. Topological representatio n of algebra_~s, Trans. Amer. Math. Soc. 63, pp. 457-481, |948. [I] A. BREZULEANU et R. DIACONESCU. Sur la duale de la cat~gorie des treillis, Rev. Roumaine. Math. Pures et Appl. 14, pp. 331-323, 1969. I~1 J.C. COLE. The bicategory of topo~ and Spectra, preprint. M. COSTE. Localisation, spectra and sheaf representation, Lecture Notes in Math. 753, Springer-Verlag. Berlin-New-York, 1979. ~] Y. DIERS. Familles universelles de morphismes, Ann. Soc. Sci. Bruxelles, 93, III, pp. 175-195, ;979. [5] Y. DIERS. Sur lea familles monomorphiques rgguli~res de morphismes, Cahiers de Top Geom Diff, XXI-4, pp. 44"I-425, 1980. [6] Y. DIERS. Une construction universelle des spectres, topologies spectrales et faisceaux structuraux, Archiv der Math, ~ paraltre. [7] P. GABRIEL et F. ULMER. Lokal pr~sentierbare Kategorien, Lecture Notes in Math. 221, Springer-Verlag, Berlin-New-York, 1971. [8] A. GROTHENDIECK, M. ARTIN, J.L. VERDIER. Th~orie des topos et cohomologie ~tale des schemas, Lecture Notes in Math 269, Springer-Verlag, Berlin Heideberg New-York, 1972. [9] M. HOCHSTER. Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142, pp. 43-60, 1969. IOl P.T. JOHNSTONE. Rings, Fields, and spectra. J°ur" AIm" 49, PP" 238-260' 1977" ; K. KEIMEL. The representation of lattice-ordered groups and rings by sections in sheaves. Lecture Notes in Math. 248, Springer-Verlag, Berlin-New-York, 1971. 2] S. MACLANE. Categories for the working Mathematician, Springer-Verlag, New-YorkHeidelberg-Berlin, 1971. I I R.S. PIERCE. Modules over commutative regular rings. Mem. Amer. Math. Soc. 70, 1967. 14 H. SCHUBERT. categories, Springer-verlag, Berlin-Heidelberg-New-York, 1972. 5 A. WOLF. Sheaf representations of Arithmetical Algebras. Mem. Amer. Math. Soc. 148, pp. 87-93, 1974.
Kan
extensions
and
systems
Armin
Given
a diagram
problems: M:
P
When
K
P is M'
~ A? C a n
a similar
way
that
it
in
the paper
we
functors
systems
[F,K]~
fits
k A of
a right Kan
the
Frei
functors
extension
the
of
sole
in the d i s c u s s i o n
apply
general
the
theory
of
some
a given
The
first
for t r e a t i n g
naturally
consider
the
following
functor
M' ~ R a n K M be c l a s s i f i e d
inducing
imprimitivity? reason
we
RanKM
M satisfying
as the r e p r e s e n t a t i o n s
fied by Mackey's answered
M'
> T
of i m p r i m i t i v i t y .
one
of t h e
second.
to a s p e c i a l
are c l a s s i -
question
it a g a i n
in
has been
briefly A t the
situation
is
e n d of
in m o d u l e
theory. All
concepts
gory
and all
mula
as
in
u s e d are
V-concepts,
right Kan
where
extensions
V is a b i c o m p l e t e
are p o i n t w i s e ,
given
closed
cate-
b y the K a n
for-
(2).
In the d i a g r a m
(i)
p
K
P denotes of K,
> T
a small
given
by
} SK
E
category,
ISKI
D is the o b v i o u s E is the
D
=
K any
functor
ITI a n d b y S K ( X , Y )
extension
embedding,
m [p,v]OP
of t h e
identifying
a n d S K the = Nat
identiy
an o b j e c t
shape
category
(T(Y,K.),T(X,K.)).
on objects
to a f u n c t o r
X in S K w i t h
the
and
functor
T(X,K-). Let A be a c o m p l e t e RanKF
for all F in
of t h e
formal
called
indexed
FK = RanKF *)
category
Hom-functor limit
is g i v e n
Supported
[P,A])
by
(it a c t u a l l y a n d F: (see
P
[A],
suffices
) A a functor. or
) a n d the Y o n e d a
[B,K] lemma,
by
the F o n d s
National
that
Suisse
where
A contain
B y the d e f i n i t i o n that
the r i g h t
notion Kan
is
extension
83 (2)
FK(-)
FK admits
= HOmp(T(-,K),F).
a canonical F(~)
We r e c a l l be shape
extension
F K = FD w h e r e
= H O m p ( E ;;,F).
that a f u n c t o r w i t h d o m a i n invariant.
In turn,
F admits
T which
factors
a canonical
over D is said to
extension
F = FE
where A
which
F(*)
= Homp(,,F)
is c l e a r l y
continuous.
The o p e r a t i o n s and Y:
( )K,
(^) is just the P
> [P,V] °p.
RanyF(,)
(-) and functor
( ) extend Rany
faithful
= Homp(*,F)
one has that Y*(^)
embedding
for any c o n t i n u o u s
f u n c t o r M"' : [P,V] Op
~ M"' R a n y Y Z R a n y ( M "
(^)Y* ~
Id(Cont[[P,v]OP,A],where C o n t [ [ P , v ] ° P , A ] [[P,v]°P,A]
Y) as Y is codense;
consisting
= ~(*).
~ Id[P,A].
has M "
of
ways
Indeed
On the o t h e r hand,
category
in o b v i o u s
, w h e r e Y is the Y o n e d a
= HOmp([P,v]°P(*,Y),F)
As Y is f u l l y
to f u n c t o r s
of c o n t i n u o u s
~ A one
thus denotes
the
full sub-
functors.
A
We also o b s e r v e
= Homp((-,K)
that
((ED)*~) (-) = FED(-)
,F) = FK(-).
Summarizing
we have
Theorem
Let A be c o m p l e t e .
gramm
i.
= HOmp(ED(-),F)
Then,
with
the n o t a t i o n s
above,
the dia
64 A
( )
[P,A]
[P,A] commutes
up to n a t u r a l
Corollary M:
2.
> A if and o n l y
tinuous~
furthermore
functor
M = M"' Y is,
M K ~ M'
and M"' ~ ~.
Remark.
From
If a f u n c t o r M'(*)
that
isomorphisms.
M':
> A is of the
if it is of the
for a g l v e n
is,
the p r o o f
of the
M"i: [P,V] °p
exist
in
Hence
the
According candidates
a left
[P,V] °p. functors
When
theorem
Y))
~
adjoint, A :
1 the
M satisfying where
the
M'
unique
M'" R a n y Y of H O m p
and h e n c e
~ M K. As
some
M'"
con-
continuous,
one
the
satisfying
~ RanyM"' Y t h e n
we have
preserves
preserve
all
extensions
all
the
limits
that
is r e p r e s e n t a b l e .
limits.
M'" of M'
[P,V] °p is a r a t h e r
to c l a s s i f y
an i s o m o r p h i s m
[P,v]°P(A(A,~'Y), *)
~
M'"(*) ~ N a t ( M ''iY,*)
continuous
possible,
M'"
[P,V] (*,A(A,M'" Y))
V, then
~ M K for
we have:
definition
in C o n t [ [ P , v ] ° P , A ]
to T h e o r e m
it is p r e f e r a b l e ,
= M'" ED w i t h
> A satisfies
~ A(A,HOmp(*,M"'
M'" has
M'
form M'
form M' ~ M'" ED w i t h
up to i s o m o r p h i s m ,
~ Homp(,,M'" Y) . By the
A(A,M'"(*)
Cont[[P,v]°P,A]
y*
A functor
P
> Cont[[P,v]°P,A]
classify
large
candidates
the
category M by
func-
65
tors h a v i n g d o m a i n S
K
We call a f u n c t o r M": w i t h M'" c o n t i n u o u s [SK,A]
consisting
takes v a l u e s ponding
• We n e x t i n v e s t i g a t e
SK
) A a s y s t e m of i m p r i m i t i v i t y
and d e n o t e of s y s t e m s
in I m p s [ S K , A ] ;
corresponding
by I m p s [ S K , A ]
takes values
functor
The
we use the same symbol Imps[SK,A].
The
functor (
( ) clearly
restricted
and we use E*
~ Imps[SK,A].
of
for the c o r r e s -
f u n c t o r E*,
in I m p s [ S K , A ] ,
Cont[[P,v]°P,A]
if M" = M ' E
the full s u b c a t e g o r y
of i m p r i m i t i v i t y .
functor with codomain
to C o n t [ [ P , v ] ° P , A ]
that p o s s i b i l i t y .
With
for the
this n o t a -
t i o n we have:
Theorem
3.
In the s i t u a t i o n
of d i a g r a m
(1) a s s u m e
be small.
T h e n the f o l l o w i n g
statements
are e q u i v a l e n t :
(i)
E*:
Cont[[P,v]°P,v]
Imps[SK,V]
is an e q u i v a l e n c e .
(ii)
E*:
Cont[[P,v]°P,A]
Imps[SK,A]
is an e q u i v a l e n c e
complete
( ):
[P,V]
Imps[SK,V]
is an e q u i v a l e n c e .
(iv)
( ):
[P,A]
Imps[SK,A]
is an e q u i v a l e n c e
(v)
E is codens e .
(±i) i m p l i e s
We have that
S K to
for all
A.
(iii)
Proof•
T, and h e n c e
(i) and
(E*o ( ? ) ) ( ~ )
(iv) i m p l i e s
= Homp(E~,?)
=
(iii)
for all c o m p l e t e
A.
trivially•
( 3 ) ( ~ ), that is, E*(^)
=
(-).
A
Since
( ) is an e q u i v a l e n c e ,
this e n t a i l s
(iii)
and
to
(v) :
(ii) is e q u i v a l e n t
that
(i) is e q u i v a l e n t
(iv). N e x t we show that
(iii)
to
implies
66
For 9:
any F,G:
P
[P,V](F,G)
> V we have
an i s o m o r p h i s m
) [P,V] (RanEE(F),G)
given
[P,V](F
G)
Imps[S K
V] (F,G)
Imps[S K
V] ( H o m p ( E ~ , F ) , H o m p ( E # , G ) ) ,
RanE([P
v]°P(G,E~))
via
(F),
(-), by the d e f i n i t i o n
by the K a n
[P,v]°P(G,RanEE(F))
by
of
(-),
formula,
as r e p r e s e n t a b l e s
preserve
RanEE
,
[p,v] ( R a n E E ( F ) , G ) , natural
in F and G, which,
isomorphism morphism,
~:
RanEE
hence
to show
the
of R a n E E
with
counit
I:
• Id.
that
we h a v e
jects
of I m p s [ S K , A ] , h e n c e
M'
ciated From
It is c l e a r
> A is of the is i m p r i m i t i v e
with
and,
from
again
and do c h o o s e
1 and
B(1)
is the u n i t
an iso-
that
is,
RanEE
= Id
for B = RanE(BE)
property as the
of RanE(BE) . This
second
arrow that
holds
is an i s o m o r -
E* hits
all ob-
it is an e q u i v a l e n c e .
f o r m M'
and
3 we
> Nat(AE,BE)
the d e f i n i t i o n s
that
= M"D with M"
M'.
Theorems
we m a y
Nat(AE,B(1))
if A is c o n t i n u o u s ,
so is E*.
that
that
of the u n i v e r s a l
phism
T
is ~ E ,
> A
E* ) N a t ( A E , B E )
isomorphism
If M':
of R a n E E
by a n a t u r a l
(ii) . If E is codense,
is an i s o m o r p h i s m ,
[P,V] °p
Nat(A,B)
afortiori
is i n d u c e d
Id E .... ~ E.
for any A:
is the
counit
(v) i m p l i e s
For B in C o n t [ [ P , v ] ° P , A ] and
The
lemma
E is codense.
It r e m a i n s counit
by the Y o n e d a
then
have
M"
is a s y s t e m
in I m p s [ S K , A ]
we
of i m p r i m i t i v i t y
say asso-
67
Corollary M:
P
4.
A functor
) A if a n d o n l y
Corollary dense}
5.
The
in t h i s
M K ~ M'
The
M"
• A is of the
(-)
is an e q u i v a l e n c e
for a g i v e n there
imprimitive
is an M,
used comes
Taking
for K:
P
finite
groups
HCG
V = k-Mod,
the
to the
Mackey.
For
Theorem
3 has
from the
considered
functor as
( )
M' w i t h
some
if E is c o -
associated
the
see
kH
of s y s t e m s [F],
following
system
with
again
taken
as a o n e - o b j e c t
which
takes
t in T to t h e
P-modules
as
functors
(P-Mod) ° p a n d the
functor
o f S ° p to the P - m o d u l e (Z) = H o m p ( T , - ) .
in P - M o d ,
then
codense,
T. T h e
(-)
hence
S °p
between
sense
isoof
s ° P = ( E n d p T ) °p
ringhomomorphism
the u n i q u e
~ s°P-Mod
instance
generated By Theorem
T is d e n s e
ca-
the
[PV] ° p b e c o m e s
(-) : P - M o d
is an e q u i v a l e n c e .
P ---mT
one-object
Interpreting
m (P-Mod) ° p t a k e s
(see for
L e t K:
ring
the
b y t.
the c a t e g o r y
(that is a f i n i t e l y
the P - m o d u l e
theory.
and D becomes
functor theorem
is t h e n
in the
of K is the
left multiplication
A Morita
if T is a p r o g e n e r a t o r
in m o d u l e
category
category
E:
taking
representations
Imps[SK,A]
as a f u n c t o r
f r o m P to Ab,
of
[M].
application
shape
and
to c o i n d u c t i n g Our
of g r o u p s :
group-algebras
categories,
of i m p r i m i t i v i t y
[K] a n d
The
V = Ab.
-> k G of the
~ k-Mod.
tegories.
Here
kH
of r e p r e s e n t a t i o n s
as o n e - o b j e c t
interpreted
is t h e n
if a n d o n l y
u p to i s o m o r p h i s m ,
theory
corresponds
functors
category
details
K
be a r i n g h o m o m o r p h i s m ,
tor)
unique
~ T the embedding
considered
morphic
by
f o r m M' ~ M K for
a n d M ~ M".
terminology
of H,
T
if it is i m p r i m i t i v e .
functor
case
of i m p r i m i t i v i t y
M':
[P])
is g i v e n says
projective 3 the
in P - M o d .
We
object
that genera-
functor thus
E
have
68
Theorem
6.
it is dense
Remark. ( ):
Let P be a ring and T a P-algebra.
If T is a p r o g e n e r a t o r
in P-Mod.
The d i s c u s s i o n
[P,A]
leading
~ Imps[SK,A]
to T h e o r e m
3 points
is an e q u i v a l e n c e
out that when
it g e n e r a l i z e s
a Morita
equivalence.
Bibliography
[A]
.C. Auderset, ries,
[B,K]
Adjonctions
Cahiers
F. B o r c e u x
[F]
A. Frei,
Shape
A. Frei
and induced
sion?, H. Kleisli,
[M]
G.W.
when
Mackey,
B. Pareigis, Teubner,
Soc.
2-categoXV,I(1974).
for e n r i c h e d
12
representations,
A question
L.N.
in Math.
Coshape-invariant
XXII-I
vol.
of limit
Math.
(1975).
to appear
in c a t e g o r i c a l
is a s h a p e - i n v a r i a n t
Springer
theorem,
in
functor
719
functors
shape
a Kan exten-
(1979). and M a c k e y ' s
induced
Cahiers
de Topol.
et G~om. Diff.,
representations
of groups
and q u a n t u m
(1981).
Induced
mechanics, [P]
A notion
Diff.,
des
Mathematicae.
representation Vol.
au niveau
et G~om.
Austral.
and H. Kleisli,
theory:
[K]
Kelly,
Bull.
Quaestiones [F,K]
de Topol.
and G.M.
categories,
et m o n a d e s
Benjamin-Boringhieri
Kategorien (1969).
und Funktoren,
(1968). Math.
Leitf~den,
SMOOTH STRUCTURES
by Alfred Fr~licher
A smooth
structure
on a set S consists
a set F c ]RS of f u n c t i o n s
of a set C c S
such that C and F d e t e r m i n e
tion t h a t FoC c C°°(~R, IR). The sets w i t h smooth ~if
we take as m o r p h i s m s
curves or,
equivalently,
with respect
The m a i n results the a t t e n t i o n
presented
to the e x c e l l e n t
show h o w c l a s s i c a l
calculus
finite dimensional[) singularities,
elementary
and are v a l i d
e.g.
cf
[3] or
properties
for each c a t e g o r y
For c e r t a i n m o n o i d s
[12].
explicitly
condition
Siciak
by d e s c r i b i n g
structure
result
(not~ecessarily
finds also o b j e c t s
with
in this direction. are those w h i c h we call the
is g e n e r a t e d
T h e y are easily o b t a i n e d
in a similar way, r e p l a c i n g B of B . For other examples,
the s m o o t h
structure
[7]. The p r o o f
yields
notion of C -maps.
between
the l i n e a r i t y the usual
by L a w v e r e ~ S c h a n u e l
for and
will be c o n s t r u c t e d
and for m a n y
w i t h the smooth the
theorems
of Boman
is f a c i l i t a t e d convex
of c a l c u l u s
functions
~-morphisms.
is a c a r a c t e r i z a t i o n
form
The
of smooth m a p s [2], of B o c h n a k
and
by u s i n g a m i n i m a l
spaces w h i c h does for
of the d i f f e r e n t i a l
results
and
of the function-spaces.
together
locally
A necessary
of this c o n d i t i o n
for any F r 4 c h e t - s p a c e
spaces w h i c h g e n e r a l i z e s
of c l - m a p s
closed.
first p r o v e d closedness
and that the C°°-maps are e x a c t l y
not r e q u i r e
nevertheless
and was
cartesian
the smooth curves
[I] and of H a i n
caracterization
is c a r t e s i a n
in [3]. The v e r i f i c a t i o n
in order to o b t a i n t h i s
Fr4chet
instance
: first we d r a w a n d then we will
manifolds
and cocompleteness.
which
In §3 it will be shown t h a t
between
there
the c a t e g o r y
was g~ven
The functor y i e l d i n g
C -Fr4chet-manifolds
basic
of~
within~one
(JR, IR) by any s u b m o n o i d
the m o n o i d C (JR, i~] is d i f f i c u l t
a smooth
to the
[4].
sufficient
Zame
set-up,
completeness
by any fixed set B and C
good w i t h respect
properties
but we shall not at all go here
ones,
yield a c a t e g o r y
here go in two d i r e c t i o n s
of C - d i f f e r e n t i a b l e
A m o n g the c a t e g o r i c a l
and
to the functions.
categorical
fits in this
of curves
each other by the condi-
structures
those m a p s w h i c h b e h a v e
~9
at a p o i n t but
and in p a r t i c u l a r
the usual
70
In r e c e n t y e a r s categories
several a u t h o r s u s e d w i t h a d v a n t a g e c a r t e s i a n c l o s e d
in o r d e r to d e v e l o p c a l c u l u s
for n o n - n o r m e d v e c t o r
c o n v e r g e n c e s t r u c t u r e s were u s e d in [6], c o m p a c t l y g e n e r a t e d
spaces
:
spaces in [ ~ ] ,
a r c - d e t e r m i n e d s p a c e s in [ii]. W h i c h c a r t e s i a n c l o s e d c a t e g o r y is the n a t u r a l one for c a l c u l u s ? If one w a n t s to study C~-maps,
the a n s w e r
vector spaces with a compatible
Some ideas and r e s u l t s in
s m o o t h structure.
seems clear
:
this d i r e c t i o n are g i v e n in the last section. The r e s u l t s understand~
p r e s e n t e d here are v e r y easy to f o r m u l a t e and to
h o w e v e r s e v e r a l p r o o f s r e q u i r e h a r d a n a l y s i s and thus can o n l y be
i n d i c a t e d in this e x p o s i t o r y article.
All p r o o f s w e r e c a r r i e d out in d e t a i l
in a s e m i n a r on s m o o t h f u n c t i o n s a t the U n i v e r s i t y of Geneva~ another
p r o o f of the t h e o r e m of Lawvere,
and the a u t h o r was p r e s e n t e d .
in p a r t i c u l a r
S c h a n u e l and Zame due to H. J o r i s
I w i s h to e x p r e s s m y g r a t i t u d e
for a v e r y
a c t i v e p a r t i c i p a t i o n and s u b s t a n t i a l c o n t r i b u t i o n s at this s e m i n a r in p a r t i c u l a r to G o n z a l o Arzabe,
Henri J o r i s and O s c a r P i n o - O r t i z .
§l T H E C A T E G O R Y ~ O F
S M O O T H S P A C E S A N D ITS E L E M E N T A R Y P R O P E R T I E S IR
A s m o o t h s t r u c t u r e on a set S is a c o u p l e F c I R S such that the
(C,F)
where C c S
"duality" C = D,F and F = D*C holds,
with
D.F = {c
: ]R ~ S~ foc 6 C°°(]R, ~R)
for all f £ F}~
D*C = {f
: S ~]R~
for all c 6 C}.
foc 6 C~( IR, JR)
A s m o o t h space is a t r i p l e
and
(S,C,F) w h e r e S is a set and
(C,F) is a
s m o o t h s t r u c t u r e on it. The morphisms
s m o o t h spaces form a c a t e g o r y ~ from
~.(C) c C' or,
(S,C,F)
so
(S',C',F')
for which,
are t h o s e m a p s d
by d e f i n i t i o n ,
: S ~ S' w h i c h s a t i s f y
e q u i v a l e n t l y ~*(F') c F.
The set of s m o o t h s t r u c t u r e s on a fixed set S is o r d e r e d manner
: (C,F)
m o r p h i s m from structure by C O
is c a l l e d finer than (S,C,F) to
(S,C',F').
(C',F')
and is o b t a i n e d as f o l l o w s (C,F)
IR
t h e r e is a f i n e s t
it is c a l l e d the s t r u c t u r e g e n e r a t e d
: F = D*C 0 ; C = D,F. S i m i l a r l y one has S : it is the c o a r s e s t
g e n e r a t e d by any set F 0 c ~ R
s t r u c t u r e w i t h F 0 c F and is o b t a i n e d as C = D , F 0 ; F = D*C. the s m o o t h s t r u c t u r e s of a fixed set S form a c o m p l e t e forgetful
in the u s u a l
if the i d e n t i t y m a p of S is a
For any set C O c S
(C,F) on S such that C O c C~
the s t r u c t u r e
the
f u n c t o r from ~ t o
It f o l l o w s t h a t
lattice,
that the
sets has a left and a r i g h t adjoint,
and that
71
~is
c o m p l e t e and cocomplete,
limits or colimits b e i n g o b t a i n e d
(as in the
c a t e g o r y of t o p o l o g i c a l spaces) by taking them in the c a t e g o r y of sets and then p u t t i n g the initial resp.
final structure on them• We note in p a r t i c u l a r
that the p r o d u c t of smooth spaces
(Si,Ci,Fi), i 6 I, is the object
(S,C,F}
w i t h S = × S. and C c o n s i s t i n g of those curves c : ~R ~ S w h o s e component i6I 1 c i : IR ~ S. b e l o n g 1 An o b j e c t
for all i 6 I, to C i.
(S,C,F) of ~ i s
called separated if for all a # b 6 S
there
exists f 6 F with f(a) ~ f(b).
~ denotes the r e s p e c t i v e full subcategory sep The inclusion functor has an obvious left adjoint and it follows
of ~ . that
~
is also c o m p l e t e and cocomplete. The forgetful functor from sop to sets still commutes w i t h limits, but not with all colimits. The one
sep point set w i t h its u n i q u e smooth structure is o b v i o u s l y
a final object of ~ ,
and it also yields a r e p r e s e n t a t i o n of the forgetful functor from ~ t o Another
important object is the triple
(~R, C~(]R, ~ )
, C°°(IR, ~R} ).
It will be d e n o t e d simply by IR and is generator and c o g e n e r a t o r of For any o b j e c t X = cx =
(SX, C x, F x) of ~ w e JcP(m,
x)
;
have
sets•
sep
:
~x : ~ C ~ ( x
,m)
.
The results of this section do not depend on the nature of the monoid C~(IR, IR) . As it was shown in [3], t h e y hold for the c a t e g o r y ~ g e n e r a t e d a n a l o g o u s l y by any m o n o i d M of maps of any set B to itself of BB). C a r t e s i a n c l o s e d n e s s of ~ h o w e v e r and sufficient condition was given.
(i.e. M a s u b m o n o i d
d e p e n d s on M; in [3] a n e c e s s a r y
In the following section we discuss this
c o n d i t i o n and its v e r i f i c a t i o n for the case B = R, M = C°°(IR, Z{) .
~2 CARTESI~ C~OSEON~SS OF dTPAND
~ep
F r o m the m e n t i o n e d p r o p e r t i e s of the o n e - p o i n t object it follows that if there is a functor
H :~ep
×~yielding
cartesian closedness,
it can be chosen such that the u n d e r l y i n g set of H(Y,Z) ~(Y,Z).
In p a r t i c u l a r one must get on ~ ( ~ ,
structure
(F,~)
such that, w i t h ~(x,y)
F = { ~ : ]R ~ C~°(]R, ~ )
; ~
JR)
then
is the function space
= C°°(]R, JR)
a smooth
: = 7(x) (y),
: m~IR
and } = D*F. Since t r i v i a l l y F c D,(D*F)
~]R a morphism }
the couple
(F,#) will be a smooth
structure iff D,(D*F) c F. Let us d i s c u s s the m e a n i n g of this condition. A c c o r d i n g to the d i s c r i p t i o n of p r o d u c t s i n ~ ,
a map G : 2
~ ] R is a m o r p h i s m
72
~
~
iff for all o,T 6 C°°( ~ , JR)
G is smooth along all smooth curves
one has Go(U,T) 6 C°°(~R, JR) , i.e. iff 2 . A c c o r d i n g to a remarkable
(O,T) of ~
t h e o r e m of Boman [2] this is equivalent to G 6 C ~ ( ~ 2, JR) yields a b i j e c t i o n F ~ C°° (
2, ~ )
and hence 7 ~
. From this and the d e f i n i t i o n of ~ as
D*F we get = {~ : C=°(]R, ~)---~IR; x ~ ~(G(x,-))
is in C ~ ( m ,
We call the elements of ~ smooth functionals explicitly;
for all G 6 C ° ° ( ~ m ) }
. One does not k n o w all of them
h o w e v e r for the linear ones one has
P r o p o s i t i o n 1.
The linear smooth functionals [9 have compact support
for each (p there exists a compact K fl I K
JR)
= f2 ~ K
( i.e.
of IR w i t h the p r o p e r t y
~ t0(f1) = [p(f2 )) and satisfy i9(lim f ) = lim ~ ( f ) n n n-~o n-~o
if fl,f2,..,
is a sequence in C°°(~R, JR)
such that for all k -> 0 the deriva-
tives
f(k)converge locally u n i f o r m l y for n ~ oo . This m e a n s that the linear n smooth functionals are exactly the d i s t r i b u t i o n s of compact support. This result is due to van Que and Reyes [13]
c o n s t r u c t i n g for c o n v e n i e n t subsequences f
, f nI
such that G(i/k,y)
P r o p o s i t i o n 2. I II
= fnk(Y) and G(0,y)
2 If G : IR ~
For all x 6JR,
G(x,-)
; it can be proved by
.... a function G 6 C~ (IR2 JR) n2
= lim fn(y)n-mo
satisfies 6 C~( ~ , ~ )
,
For all linear smooth functionals ~ • x ~ ~ G(x,-)
is in C°°(IR, IR)
then G £ C°°( m 2, JR) . This important result was proved by Lawvere,
Schanuel and Zame [12].
It can be p r o v e d by showing first the c o n t i n u i t y of G and its first o r d e r partial d e r i v a t i v e s ~1 G and D2G. This is quite d e l i c a t e for DIG and we found it useful to show first that DIG is p a r t i a l l y continuous in the second v a r i a b l e and ~IDIG is locally bounded. Once one has o b t a i n e d the continuous different i a b i l i t y of G, the p r o o f is completed by showing that DiG and ~2 G satisfy the same conditions I and II ; for D2G this is easy, and for DIG one m a k e s use of p r o p o s i t i o n i. F r o m p r o p o s i t i o n 2 is follows i m m e d i a t e l y that structure on C°°(~, ~ )
(F,~) is a smooth
. W e remark that for this it w o u l d be enough to prove
73
p r o p o s i t i o n 2 under the a s u m p t i o n that I holds and II holds for all smooth functionals q). However,
if one w o u l d a l l o w n o n - l i n e a r ones in the p r o o f that
G or ~l G or ~)2G are c o n t i n u o u s it w o u l d be
hard to get f u r t h e r , b e c a u s e one
does not haw~ the analogue of p r o p o s i t i o n 1 for n o n - l i n e a r smooth functionals
Theorem. for
The c a t e g o r y ~ o f
smooth spaces is cartesian closed. The same holds
sep The f u n c t i o n - s p a c e structure can be d e s c r i b e d e x p l i c i t l y
objects Y,Z of ~ o n e
d e f i n e s on the f u n c t i o n - s p a c e ~ ( Y , Z )
In fact, for
a structure
(C,F) by C = {d : ]R ~
~(Y,Z);
~
: ]RZ Y ~ Z a morphism}
F = D*C. Using that
(I',~) is a smooth structure on C°°(IR, JR) it is easy to show that
(C,F) is a c t u a l l y a smooth structure. D e n o t i n g the smooth space formed by the set ~ ( Y , Z )
with this structure by H(Y,Z)
it is s t r a i g h t f o r w a r d to show that
one has the universal p r o p e r t y : X -~ H(Y,Z) and
a morphism ~=~
: X Z Y ~ Z a morphism
this yields f u n c t o r i a l i t y of H and cartesian closedness of ~ . Since Z separated implies H(Y,Z) sep
separated, cartesian
closedness of
is o b t a i n e d by r e s t r i c t i o n of the functor H.
B e c a u s e in p r o p o s i t i o n 2 o n l y linear smooth functionals are used it follows easily that the structure of H(Y,Z) ~(Y,Z) support,
~[
of the form ~ ~ ~(f°~oc)
c 6 Cy =
9(IR,Y)
is g e n e r a t e d by the functions
w h e r e ~ is a d i s t r i b u t i o n of compact
and f 6 m Z = ~ ( Z
, m)
.
§3 THE S M O O T H STRUCTURE OF FRECHET SPACES AND MANIFOLDS. F o r ~ n the couple j OO~C(~, n )
, cOO( ~ n , ~ ) )
is a smooth structure. This
is not at all trivial, but it is equivalent to Bomans t h e o r e m [2], w h i c h says that a function o n ~ n
a l r e a d y quoted
is smooth if it is smooth along all
smooth curves. Using p a r t i t i o n s of u n i t y one gets a more general result for any finite d i m e n s i o n a l p a r a c o m p a c t C -manifold V, (c ~ ( ~ , V )
, C
:
(V, ~ ) )
is a smooth structure on V. So every such m a n i f o l d can be c o n s i d e r e d as a smooth space, and the C°°-maps b e t w e e n them are exactly the
~O-morphisms.
In order to get the same results for a greater class of v e c t o r spaces and m a n i f o l d s we need first of all a t h e o r e m w h i c h g e n e r a l i z e s B o m a n ' s result
;
74
this
theorem
and Hain
The useful
will
at t h e
same
time
aeneralize
results
of Bochnak-Siciak
[i]
[7]. following
(E,F w i l l
set-up
in t h i s
for c a l c u l u s
section
always
between denote
locally
convex
separated
spaces
locally
is
convex
spaces).
Definition. class
A map
f : E ~ F between
locally
convex
spaces
is c a l l e d
of
C 1 if for a l l x , h 6 E. df(x,h)
exists
: = w-lim i/~. I~0 a c o n t i n u o u s m a p df
and yields
By w-lim
we mean
Hence
the
df(x,h) 1 lim ~.
for all
1 [ F', We
Proposition
the
i.
limit
((lof) (x+lh)
-
: E × E ~ F.
in F w i t h
is c a r a c t e r i z e d
F' b e i n g
require
in p a r t i c u l a r
(unique)
(f(x+lh)-f(x))
to t h e w e a k
topology.
by
(lof)(x))
the topological
so l i t t l e
because
linearity
of d f ( x , - ) .
= l(df(x,h))
dual
o f F.
it is e n o u g h
to g e t the u s u a l
In f a c t o n e h a s
If f : E ~ F is of c l a s s lim l/l-(f(x+Xh] i~0
respect
- f(x))
C
1
properties,
:
, then
= d f ( x 0 , h 01
X~X 0 h~h 0 We
remark
(and n o t o n l y simultaneous strict
that
here
with
respect
limit
exists
differentiability.
Corollary.
to p r o v e
limit
is w i t h
to t h e w e a k shows
that
As e a s y
If f is o f c l a s s
In o r d e r useful
the
C
1
respect
topology!),
we a r e c l o s e
consequences
, then
proposition
to t h e t o p o l o g y
a n d the to w h a t
we have
f is c o n t i n u o u s 1 the
following
fact
that
of F this
is s o m e t i m e s
called
:
a n d df(x,-) "mean value
is linear, theorem"
is
:
Proposition
2.
interval
I c]R
(special
case
L e t A c E be c o n v e x into E related : d : c').
d(1) The proof For
by
application
f : EI×...xE n
(loc)'(~)
c,d
= l(d(~))
: I ~ E maps for all
for a n y ~ < ~ of I o n e h a s
6 A for ~ < ~ < ~ ~
is a s i m p l e a map
Then
and closed;
c(~)
- c(~)
6
~ 6 I and
1 6 E'
:
(~-~)-A .
of the H a h n - B a n a c h F the n o t i o n
of an o p e n
theorem.
"partially
of c l a s s
C I'' is
75
defined
in t h e u s u a l way;
i.e.
di~.=- Elx...XEnXE.l ~ F h a v e It f o l l o w s 1 C .
as u s u a l
that
the partial
to e x i s t
and
differentials
have
to be c o n t i n u o u s
f is o f c l a s s C 1 if a n d o n l y
in all v a r i a b l e s
if it is p a r t i a l l y
of
class
n now maps of class C : n+l f : E ~ F is o f c l a s s C if it is of c l a s s
Inductively Definition. c l a s s C n.
we d e f i n e
f is of c l a s s
For d n there operator
T behaves
C
1
a n d df is o f
C°~ if it is o f c l a s s C n for all n 6 N.
is a c h a i n
rule.
much better; Tf
It is c o m p l i c a t e d
it is d e f i n e d
and
for t h i s t h e
as
: E×E ~ FXF (x,h) ~ ( f ( x ) , df(x,h))
The
chain
rule then
says
a l s o gof, a n d T n ( g o f ) Since
: if f : E ~ F a n d a
: F ~ G are of class C
n
, then
= TncroTnf.
for f of c l a s s
C
1
t h e m a p df is l i n e a r
(and c o n t i n u o u s )
in t h e
s e c o n d v a r i a b l e , o n l y t h e f i r s t p a r t i a l d i f f e r e n t i a l is of i n t e r e s t a n d y i e l d s 2 2 2 a m a p D f : E x E X E ~ F. D f e x i s t s a n d is c o n t i n u o u s iff f is of c l a s s C , a n d t h e n D 2 f (x,_,_)
is b i l i n e a r
a n d D n+l as t h e
first partial
class
C n if a n d o n l y
Dnf(x,-,...,-)
if D l f ..... D n f
to s h o w t h a t
existence
D n-1 f. If we
suppose
symmetric.
differential
is n - m u l t i l i n e a r
If we w a n t must verify
and
and
a map
continuity
'
In ) ~
on d e f i n e s
D 1 f as df
of Dnf.
shows that
f is o f
One
a n d are c o n t i n u o u s ,
f of c l a s s C n-1 of the g
admits
first partial
basis
L e t E be m e t r i z a b l e
convergent
sequences
U
a sequence
: IN ~ ] N ,
is c o n t i n u o u s .
suppose
that
following
and a n ~ = lima n n-~o
i n E. T h e n t h e r e of r e a l s
of
form
the e x i s t e n c e It is t h e n Under
for the z e r o - n e i g h b o r h o o d s
w e can s h o w t h i s c o n t i n u i t y if we 2 g :~ ~ E of class C , using the
Lemma.
differential
a + l ~ h l + . . . + A n-n h
(x,h) ~ D n f ( x , h , . . . , h )
a denumerable
is e v e n of c l a s s C n, we
: IRn ~ E o f t h e
n t h e c o m p o s i t e m a p fog is o f c l a s s C , we g e t e a s i l y n D f ( x , - , . . . , - ) a n d its m u l t i l i n e a r i t y a n d s y m m e t r y . that the map
and then
symmetric.
for all m a D_s
that
(11'''"
and
exist
Recursively
I
n
exist
(cf.
to s h o w that E
( i . e . i f E is m e t r i z a b l e )
[7])
b 0 = limb n n-~o
a strictly
with a limit
enouqh
the asumption
fog is of c l a s s C n for all
lemma
'
of
:
be l i m i t s
increasing
of
function
10 = l i m ~ ,and a f u n c t i o n n n-x=
78
g
: ]R
2
~ E of c l a s s C°O s u c h t h a t
g(ln,~) This fact
it is f a l s e
3.
E metrizable :n
supposes
that
show that
n~2
. Then
could not
There
is
better
get t h e a b o v e
result
" , which
we
is o n l y a s l i g h t n
f has t h e p r o p e r t y
the norms.
However,
and this
that
but
between
if
of H a i n
[7].
to B a n a c h
classical
spaces)
.
notion
of
"
our notion
: E ~ L
n
C n a n d F r 4 c h e t - C n.
(E;F)
continuity
C n and
is in p a r t i c u l a r n+l
Proposition
Let
g
:
spaces
of
]Rn
(F' t h e t o p o l o g i c a l to a s s u m e
that
still weaker,
f(n)
true
~ Fr~chet-C n ~ C
the
~ F be s u c h t h a t d u a l of F) the
and
with respect
is n o r m - c o n t i n u o u s ,
if f is of c l a s s
coincides
spaces many
them yield
(n)
of f
then
C n+~.
to f
Hence
with the
different
same notion
"of
notions
class
log is of c l a s s C
suppose
Mackey-topology
t h a t F is l o c a l l y
classical
n+l
"of c l a s s
C~ ' '
as
f o r all
t h a t F is c o m p l e t e
(in f a c t
o f F is s e q u e n t i a l l y
complete,(cf
[9]
). T h e n
g is
C n.
Using and Siciak
our who
set-up
the proof
is a l m o s t
gave this proposition
in t h e
:
n
our C -notion
. For not-normable
[i0]).
or,
(in
spaces;
E of c l a s s C
"Fr@chet-C n
imply the
(cf.
complete,
difference
he u s e s t h e here
does not
the one we use
of c l a s s
call
~ F
all
it is e n o u g h
because
result
: IRn+l
: E×...xE
almost
4.
nA2
C~
(who r e s t r i c t s
for all g
shall
convex
C n if a n d o n l y
= Dnf
for B a n a c h
n o t i o n of F r ~ c h e t - C
exist;
locally
: ]Rn ~ E of c l a s s
C n, H a i n
if f is o f c l a s s
C shows
we a s s u m e
that
and this
is F r ~ c h e t - c n ;
proposition
, then the map
(n)
f(n) is c o n t i n u o u s ,
U {0}.
than the respective
f is of c l a s s
: E ~ F is of c l a s s C
1 6 F'
following
f is of c l a s s
f o g is of c l a s s C n+l
" m a p of c l a s s C n
Cn "
in t h e
F is o f c l a s s C n for all g
to
This
for all n £ ~
f : E ~ F be a m a p b e t w e e n
n 6~,
This proposition
If f
why
(n)
for n : i).
Let
;
In o r d e r
He
+ p. 5
also explains
Proposition
fog
= ao(n)
the
s a m e as t h a t g i v e n
c a s e n = i , cf.
[i].
by Bochnak
77
Combining
Boman's
following
theorem,
Theorem
1 .
theorem with propositions announced
in [5]
Let f : E ~ F be a m a p b e t w e e n
that E is m e t r i z a b l e are e q u i v a l e n t
and F
3 and 4 one gets easily the
:
(locally)
locally convex
complete.
T h e n the
spaces
following
and suppose conditions
:
i)
f is of class Cco
2)
f. (C~(IR,E))
3)
f*(C°°{F, IR)) C Cco(E, ]IR)
4)
f*(F')
= Cco( JR, F)
co
Corollary. smooth
For any F r ~ c h e t
structure
and the
m C
(E, IR) space E, the couple
on E. Hence F r @ c h e t
~-morphisms
between
Theorem
2.
of u n i t y
of~from
co
, C
can be c o n s i d e r e d
t h e m are e x a c t l y
If we w a n t to get o b j e c t s sure that p a r t i t i o n s
spaces
(~ (JR,E) the m a p s
(E, IR))
is a
as smooth
spaces
of class C°°.
Fr4chet manifolds
we m u s t m a k e
exist.
Let V be a p a r a c o m p a c t
space E w h i c h has the p r o p e r t y
Fr4chet manifold modelled
t h a t to each n e i g h b o r h o o d
over a F r ~ c h e t
V of zero there
co
exists
a C -function
Then
(JR,V),
between
C
(V, IR)) is a smooth
such spaces are exactly
Remark.
According
Fr4chet-manifolds C
f : E ~IR with
(V,W).
This
the u n i v e r s a l
V , W the n a t u r a l
property
structure
closedness
smooth
can be d e s c r i b e d that
= i and f(x]
= 0 for x ~ V.
on V. The
morphisms
the m a p s of class Cco.
to the. c a r t e s i a n
structure
f(0)
of ~ w e
structure
on the function
explicitly
for any such m a n i f o l d
get for any such space
in a simple way and has
X a Rap
f : X ~ C
(VrW]
is
co
of class C ~ C
(V,W)
iff ~
: X ~ V ~ W is of class C
again a m a n i f o l d
t h i n g s to look at in this vector
spaces
will give
set-up.
More natural
equipped with a compatible
some ideas
. One can then ask
? Of c o u r s e F r 4 c h e t m a n i f o l d s
in this direction.
smooth
: when
is
are not the n a t u r a l
are m a n i f o l d s structure~
modelled
The last
over
section
78
§4 C A L C U L U S
A smooth v e c t o r structure,
i.e.
an a r b i t r a r y
F O R S M O O T H V E C T O R SPACES
space
is a v e c t o r
such that the v e c t o r
object
X :
(S,C,F)
of ~ t h e
w i t h H(X, JR), is a smooth v e c t o r w a y an e q u i v a l e n c e f ~
relation
g ~=~ (f°e)" (0) =
space w i t h a c o m p a t i b l e
space o p e r a t i o n s function
space.
are
set F, being
If we define,
smooth
~-morphisms.
For
identified
for p 6 S, in the usual
"~ " on F by P (goc)" (0) for all c 6 C w i t h c(0)
= p
P then the q u o t i e n t space, tangent not,
is, due to c a r t e s i a n
c a l l e d the c o t a n g e n t
space of X at p as a q u o t i e n t
in general,
smooth v e c t o r evaluation remarks
a vector
space
seem v e r y useful
to o b t a i n
such a result,
E'
= p};
as a subspace
of F w i t h r e s p e c t
in this d i r e c t i o n
is the f o l l o w i n g
indefinitely
Let E be a smooth v e c t o r
{c 6 C; c(0)
here; way.
the
it is
into the
to the
but these It does not
on them.
we want to study spaces
also a smooth v e c t o r
we can introduce
spaces come in a natural
to put a t o p o l o g y
smooth v e c t o r
In o r d e r
of the space
but can be imbedded
formed by the d e r i v a t i o n s
smooth v e c t o r
The question between
space,
at p. We do not go further
show that
c l o s e d n e s s of ~ ,
space of X at p. S i m i l a r l y
differentiable
some r e s t r i c t i o n s
space,
: are the
(CE,F E)
~-morphisms
in the usual
on the spaces
its smooth
sense
seem useful.
structure.
We put
: = E* N F E
where E* notes the a l g e b r a i c real-valued
linear
Definition. points, cf
smooth
dual of E. So E'
functions
The smooth vector
generates
the smooth
is the v e c t o r
space of the
on E.
space E is c a l l e d c o n v e n i e n t
structure
and y i e l d s
if E' s e p a r a t e s
a comolete
bornology
on E;
[8]. This completeness
locally
convex t o p o l o g y
complete; Fr~chet
cf
condition
is e q u i v a l e n t
on E y i e l d i n g
[9]. A c c o r d i n g
space E the natural
E' as t o p o l o g i c a l
to the results smooth
to the c o n d i t i o n
of
structure
dual
t h a t any
is locally
§3 we see that for any (C
(JR,E)
, C
(E~ IR)) is
convenient. If c : IR ~ E is a space E
(i.e. c 6 CE)
hypothesis that
that there
~-morphism
one d e d u c e s exists
from LR to a c o n v e n i e n t
from the s e p a r a t i o n
a u n i q u e map,
denoted
smooth vector
and the c o m p l e t e n e s s
by c',
?
from]]{ to E such
79
loc"= F r o m the o t h e r asumption
(loc)"
(that E' generates the smooth structure of E) it
follows then i m m e d i a t e l y that e" we obtain
~O-morphisms
for all 1 6 E'
: ~ ~ E is also a
c (n) : ]R ~ E for n 6 ~
~morphism.
Inductively
and we see that c is indefini-
tely d i f f e r e n t i a b l e in the usual sense with respect to any locally
convex
t o p o l o g y on E y i e l d i n g E' as t o p o l o g i c a l dual. M o r e o v e r one v e r i f i e s that the (linear) map H(IR,E) ~ H(IR,E)
sending c into c" is a ~ - m o r p h i s m .
U s i n g this we get similar results for the general case T h e o r e m i.
Let d : E 1 ~ E 2 be a ~ Y - m o r p h i s m
:
b e t w e e n c o n v e n i e n t smooth vector
spaces. T h e n the map d~ defined by d~(a,h) ~-morphism.
=
: E1 ~ EI ~ E2
(~OCa,h)" (0) where Ca,h(l) = a + l h
For any a 6 E 1 the map d~(a,-)
is also a
is linear. The
H(E1,E2) ~ H(E 1 rl El, E2) sending ~ into d~ is also a If E~ separates points of E2, then o b v i o u s l y
(linear) map
~-morphism.
(H(E1,E2))'
separates
points of H(E1,E2). And if E~ g e n e r a t e s the smooth structure of E2~ then the remark at the end ~ ~(io~oc)
generate
' certainly
addition
that
H(EI,E2)
one sees
that
for
Theorem
2.
the
of the form
for 1 6 E~, c 6 C and ~0 a d i s t r i b u t i o n of compact support 2 E 1 ' smooth structure of H(E I, E2); since these functions are linear
the
(H(E1,E2))
of §2) the functions H(E1,E 2) ~
(cf
The
~-morphisms
generates satisfies
E 2 convenient
category is,
formed by
the the also
by
restriction
structure
of
H(E1,E2).
is
convenient.
completeness H(E1,
the of
E 2)
condition
convenient the
By
functor
smooth H,
showing
provided Hence
vector cartesian
in E 2 does,
we
have
spaces
:
with
closed.
Other p r o p e r t i e s of that category as well as the c a t e g o r y formed b y the same objects but with only the linear ~ - m o r D h i s m s
are being studied;
in
p a r t i c u l a r d u a l i t y and r e f l e x i v i t y questions. By i n t r o d u c i n g the spaces Ln(E1,E 2) of n - m u l t i l i n e a r
~morphisms
E l ~...~E i ~ E 2 one can of course introduce for a
~P-morDhism ~ ; E i ~ E 2 (n) between convenient smooth vector spaces the maps ~ : E] ~ Ln(EI,E2) w h i c h are also
~-morphisms,
derivatives (n)
and one has the usual relations between the higher
and the higher d i f f e r e n t i a l s dn~.
80
Added in proof. The convenient smooth vector spaces can be identified with the spaces considered by A. Kriegl ("Die richtigen R~ume f~r Analysis im unendlich-dimensionalen", preprint, Vienna 1981, to appear in Monatshefte fur Mathematik), namely the separated locally convex spaces which are
bornological and locally complete.
81
R E F E R E N C E S
[1]
J. B o c h n a k spaces"
2
3
A. F r 6 1 i c h e r
Ac.
9 i0
Notes ii
12
A. Kriegl
XXI/4,
entre
espaces
lisses
engendr@es 1980,
D-
par des
367-375.
et v a r i @ t 6 s
de Fr4che~
p. 125-127. in V e c t o r
of smooth
77,
Spaces w i t h o u t
Norm",
1966.
1979,
p.
functions
d e f i n e d on a B a n a c h
63-67.
and functional
c o n v e x Spaces", Calculus
Springer
analysis",
Mathematics
F.W.
S.H.
Schanuel
Teubner
1981.
in locally
convex
spaces",
Lecture
1974.
glatter
1980.
Mannigfaltigkeiten
and W.R.
Zame
und v e k o r b ~ n d e l " ~
: ~'On C°° F u n c t i o n
Spaces",
1981.
N. V a n Oue and G. Reyes de Whitney",
Recherches
appl.
der F e r n u n i v e r s i t ~ t
1977.
Wien
U. S e i p
diff.
: "Bornologies
: "Eine T h e o r i e
Lawvere,
249-268.
abgeschlossene
Math.
et G4om.
Soe.
Dissertation,
tension
14
Am. Math.
417,
1967, p.
kartesisch
ferm@es
: "Calculus
:"Differential
in Math.
20,
car4siennement
30, S p r i n g e r
26, N o r t h - H o l l a n d
Keller
Preprint 13
in Math.
: "Locally
vector
and of its c o m p o s i t i o n s
Scand.
aus d e m Fachber.
: "A c h a r a c t e r i z a t i o n
H. J a r c h o w
Math.
erzeugte
1981,
and W. B u c h e r
H. H o g b e - N l e n d
H.H.
de Top.
Paris 293,
Proc.
in t o m o l o g i c a l
.
: "Applications
Sci.
Hain
Studies
7-48
Cahiers
A. F r 6 1 i c h e r
R.M.
p.
: "Categories
A. F r 6 1 i c h e r
space", 8
of a function
Seminarberichte
5, 1979,
functions
p. 77-112
of one variable",
Kategorien",
L e c t u r e Notes 7
39, 1971,
: "Dutch M o n o i d e
C.R. 6
"Analytic
: "Differentiability
functions
monofdes", 5
:
A. F r 6 1 i c h e r
Hagen 4
Studia Math.
J. Boman with
and J. Siciak
DMS 80-12,
21,
1981,
des d i s t r i b u t i o n s
8, G4om.
Universit4
: "A c o n v e n i e n t
Algebra
: "Th6orie
Expos@
Settina
diff.
synth,
de M o n t r @ a l
et th@or6~nes d'ex~
fasc.
2, R a p p o r t
de
1980.
for S m o o t h Manifolds".
J. of p u r e and
p. 279-305. S e c t i o n de M a t h @ m a t i q u e s U n i v e r s i t ~ de Gen@ve 2-4, rue du Li6vre CH~I211
GENEVE
24
E n r i c h e d algebras,
spectra and h o m o t o p y limits
John W. Gray O. Introduction.
The purpose of this paper is the same as that of
[5];
to show how certain p r o p e r t i e s of h o m o t o p y limits are consequences of w h a t either are or should be standard facts about categories e n r i c h e d in a closed category. in
The p r o p e r t y to be e x p l a i n e d here is as follows:
[16], T h o m a s o n shows that the d e g r e e w i s e h o m o t o p y limit of a d i a g r a m
of p o i n t e d simplicial spectra is a pointed simplicial spectrum. this the h o m o t o p y limit in the category of such spectra. reasonable to suppose that, category,
in fact,
it is the h o m o t o p y limit in this
but two things have to be proved,
of p o i n t e d simplicial
i) .
The category
Spec K,
spectra is a complete simplicial category,
only such categories have h o m o t o p y jections
He calls
It is e m i n a n t l y
pr n : Spec K, ÷ K,,
e n r i c h e d left adjoints,
limits,
ii) .
for each degree
since
The component pro-
n,
and hence preserve h o m o t o p y
have s i m p l i c i a l ! y limits.
The r e q u i r e d tools are m o s t l y at hand for o r d i n a r y categories in the form of known p r o p e r t i e s of the category for an e n d o f u n c t o r
S
of a category
A.
Dyn S
of algebras
In Section 1 these tools are
s h a r p e n e d and e x t e n d e d to the case of e n r i c h e d categories. Spec A
In Section 2
is d e s c r i b e d for an arbitrary complete V - c a t e g o r y
category)
and a pair of V - a d j o i n t
functors
Z--4Q.
(V
a closed
Finally,
in Section
these results are s p e c i a l i z e d to pointed simplicial spectra.
Note that
the spectra treated here are those for w h i c h phism.
i.
V-categories.
cocomplete,
T h r o u g h o u t this section
symmetric, m o n o i d a l
X n + ~Xn+ 1
V
denotes a complete,
closed category.
category of V - e n r i c h e d categories
is an isomor-
and functors,
V-cat
denotes the
regarded both as a
symmetric, m o n o i d a l closed category itself and as a 2 - c a t e g o r y in w h i c h the 2-cells are V - n a t u r a l transformations; t : F ~> G : A ÷ B the diagrams
between V - f u n c t o r s
A(A,B)
such that for all
commute.
A
and
B,
FA'B > B(FA,FB)
GA,B I B(GA,GB)
i.e., natural t r a n s f o r m a t i o n s
](l'tB) (ti,l)> B(FA,GB)
For basic information,
see
[5],
[8] and references therein.
3,
83
i.i.
Proposition.
Proof:
It is w e l l
to s h o w t h a t If
V-cat
it h a s
A • V-cat
phisms
in
If
• V
that
V-cat
cotensors
then
V.
2 ~ A(f,f')
known
is a c o m p l e t e
2 ~ A
with
2-category.
has
the
limits.
arrow
category
is the V - c a t e g o r y
f : A + B
and
Thus
whose
f'
: A' ÷ B'
d1 - - >
A(B,B')
it is s u f f i c i e n t 2
(cf.
objects
[21]).
are m o r -
are t w o such,
then
is the p u l l b a c k 2 ~ A(f,f')
(f,l)
A(A,A') It is e a s i l y
checked
transformations
t
that
8~ = t
that
8f = f.
1.2.
Proposition.
that
there
: F ~>
where
Q
(i, f,)-> A ( A , B ' )
G
is a n a t u r a l
: A ÷ B
: do ÷ d I
bijection
and V-functors
between
~
in the V - n a t u r a l
V-natural
: A ÷ 2 ~ B
transformation
such such
Let K B
>
B'
Ii be a d i a g r a m F'--4U' tion
~
of V - f u n c t o r s
Then
there
: H U :> U ' K
A
--->
A'
--
H
--
such
that
is a n a t u r a l and
there
are V - a d j u n c t i o n s
bijection
8 # : F ' H =>
KF
between
such
that
F--~U,
V-natural for all
transformaA
and
B
the d i a g r a m s H A(A, UB)
D
:
> A' (HA,HUB)
(I'SB) - - - >
A' (HA,U'KB)
f J
f B_(FA,B)
> B'(KFA,KB) _
- 0-# - i) >
B' (F'HA,KF)
( A' commute Proof: n
(cf. Given
: A =>
the that
[19]) .
UF,
8 : H U ÷ U'K, s : F U => B,
adjunction this
natural
establishes
~'
then
0 # = E'KF
: A' => U ' F ' ,
transformations. a bijection
See
o F'SF
and [4],
as i n d i c a t e d .
s' I, The
o F'HN, : F'U' 6.6
where
÷ _B'
are
for the p r o o f
diagram
D
commutes
84
because bottom
of the are
commute
commutativity
the s i d e s ,
top
of F i g u r e
and bottom
b y the d e f i n i t i o n
1 in w h i c h
of
D.
of V - n a t u r a l i t y .
The The
the
sides,
regions other
top
and
labeled
regions
*
commute
trivially. 1.3.
Definition:
category
S + A
: S A ÷ B.
i)
Let
S
: A ÷ A
is the V - c a t e g o r y
If
~'
: SA'
+ B'
S + A(~,~')
be
whose
then
Pl
+ S
has
The
comma
are m o r p h i s m s E V
is the p u l l b a c k
2 ~ A ( { ,~ ') --
d o
S
A(A,A') A
objects
S + A({,~')
poI Dually
a V-endofunctor.
as o b j e c t s
A(SA,SA')
morphisms
{
: A ÷ SB
and
A + S(~,~')
w
is the p u l l b a c k
of the
diagram A + S(~,~')
.oL
Pl
objects of the
Dyn
S
denotes
are m o r p h i s m s
~
1
d1
2 ~ A(~,~') ii)
> A(B,B')
> A(SB,SB')
the V - c a t e g o r y
: SA + A
and
of S - a l g e b r a s
D y n S(~,W')
--
Dually, objects
coDyn
S
denotes
are m o r p h i s m s
equalizer
of the
~
dlPl
Its
is the e q u a l i z e r
~ A(A,A')
do
the V - c a t e g o r y
: A + SA
and
of S - c o a l g e b r a s coDyn
S(~,~')
two m o r p h i s m s Pl % S(~,~')
A(A,A') d0P 0
1.4.
A.
two morphisms S + A(~,~')
(cf.
e V
in
[6] a n d Remarks:
[i0]) . There
are d i a g r a m s S +A
A_--K--> A_
A+S
A_ ~ >
A_
6 V
in
A. is the
Its
85
-
W v
~
A
A
A
~
v
r~
v
~
r~
A
A
T~I
A
v
! r~ ~J
~D
,r-I ~J
r-~
v
A
f~
.
F 2
S
coDyn
& --K-> & which
are
universal
transformation 1.5.
for
SF =>
Proposition.
isomorphisms Proof.
The
V-functor s
: X ÷ A
F 1 =>SF2)
.
and
a V-
Similarly
there
diagrams
Dyn
S
A
a single
F
Given
s
V-functor
F
F ~>
over
A
transformation A
determines
~ T s
equations
: A
are
and
: A :>
that
¢ T ÷
: X + A (cf.
~
TS
S
there =
• nU I. that
--
~ = F
--i
are
coDyn
determines
such
to
a V-natural
6.4)
then
Dyn
= T9
S ¢ A _
equivalent
and
[6]
S --J T
x A ~
such
S
A
SF)
a V-adjunction
¢ T
: S + A ÷
adjunction
A_
(resp.,
¢ A = A
natural ~
: ST =>
The
FI,F 2
(resp.
T
Vover
A
a unique Similarly,
8~
= sU 2
since,
' S~.
for
instance @S ~ = (sU 2 =cU 2 =8 so
s
n
Clearly
id.
=
• S~)~ • S(Te
• aSU 1
~
and
A
is
~
=
eU2~
• S~
• ~U I)
= ~U 2
SnU 1 =
e •
restrict
to
• ST@
(~S
• S~U 1
• Sn)U 1 =
give
the
@
second
isomor-
phism. 1.6.
It
is
created
by
U
limits.
a complete
well
Recall
indexed
known
: Dyn that
limit
that
S ÷ A if
{G,F}
Dyn
(cf.
Then
the
moment,
there
is
suppose
G
: I ÷ V A
a diagram
for like
S
then
ordinary
limits
has
[i]) .
e
Dyn
S
is
We
and
show F
here
: I ÷ A
that are
which A
has
a
are indexed
V-functorsthen
satisfies
A(A,{G,F}) For
V-category
V-category.
Proof.
the
If
Proposition.
complete
~
[I,V] ( G , A ( A , F ) )
clarity the
one
that in
S 1.2,
: A ÷
is
a V-functor.
87
AoP
S °p
'l
{-,F}
A(-,F)
and hence
t
a bijection
=> B ( - , S F )
and
: A(-,F) = > B ( - , S F )
tA
: A(A,F)
Then
t
cG
° S
÷ B(SA,SF)
corresponds
between
functors
: S({G,F})
÷
with
S
be t h e
~
Dyn
claim S.
that Then
[I,V]
o {-,F}
transformation
Let whose
BoP
components
tA,i = S A , F i
transformation whose
: A(A,Fi)
c : {-,SF}
components
to an e n d o f u n c t o r to a f u n c t o r
: SF = > F .
=>
÷B(SA,SFi) S o {-,F}
are m o r p h i s m s
S F
: A ÷ A.
: I ÷ A
A V-functor
and
a V-natural
Let
I = {G,~} 6 D y n there
B(-,SF)
in
i = {G,~}
We
]
transformations
=> S
components
codomain
corresponds
transformation
V-natural
have
{G,SF}
>
{-,SF}
to a n a t u r a l
Now we return : I ÷ Dyn
id
between
o S
BoP
{-,SF}
[I,V]
A(-,F)
>
o cG S.
: S{G,F} ÷{G,F} Let
~
: SA + A
be a n o b j e c t
of
is an e q u a l i z e r d i a g r a m
Dyn S(~,I)
--> A ( A , { G , F } )
> A(SA,S{G,F) }
t (l'CG) A ( S A , {G,SF})
J (1,{l,~})
[
A(SA, {G,F}) which
is i s o m o r p h i c
to an e q u a l i z e r d i a g r a m (i, t A)
E-->
[I,V] (G,A(A,F)
(i,
>
[I,V] ( G , A ( S A , S F ) )
(y.i i ~ " ~
(i, (i,~)) [I,V] (G,A(SA,F))
the
crucial
[I,V]
one
step being has
given
by the
commutative
diagram
an equalizerdiagram A(~,~)
--> A(A,F)
tA --->
A(SA,SF)
A(SA,F)
in 1.2.
But
in
88
and
[I,V] (G,-)
preserves Dyn
Hence
I = {G,~}.
1.7.
Proposition.
ates
equalizes
so
S(tg,l)
= E -- [I,V] (G,A(%,~))
S
preserves
If
: A ÷ A
coproducts
S
then
gener-
a free V - m o n a d .
Proof.
See
monad
(triple)
is o b v i o u s to h a v e
[i],
[3],
[9] , [12]
problem.
This
that
them
one
gets
value
on o b j e c t s
whose
V-structure
m
is the
is
S(A)
for d i s c u s s i o n s
simplest We
example.
Let
possible
sketch S
:
~m
A(Sr~A,SmB)
of the case
the d e t a i l s
: A ÷ A
I ] Sn(A) , n=0 by the c o m p o s i t i o n
is g i v e n
m >
in m
S.
Define
~
whose
components
are
that
formations
~A
° inm,n
and
S =
is an S - a l g e b r a , valent and
induces 1.8. U
:_ I I
I I smsnA m=On=O
= inm+n"
(S,~,~)
then
the
to the r e c u r s i v e
kn+l
= Ii
°Sln
S + A
has
S
>
Let
D = in 0
: Id =>
to be the t r a n s f o r m a t i o n
~,~
algebra
and
: A ÷ A
If
equations
s
I =
preserves given
s
coproducts
[I P
seen
Hence,
: AS ÷ Dyn
by
:
1 1 : SA ÷ A
Im+n = k m o S m l n ) .
adjoint
are V - n a t u r a l
are e a s i l y
I0 = id,
a V-isomorphism
a V-left
I I sPA p=O
is a V_ - m o n a d .
(actually,
If
Then
conditions:
(by c o m p o s i t i o n )
Corollary.
: Dyn
S
the m a p s DA
such
summand.
: S S =>
S. then
trans-
sPA ÷ A
to be e q u i arbitrary, s
: S =>
89 oo
F(A) Proof:
A~ ÷ A
Composing
has
with
s
=
(fA
a I-left gives
:
co
[ I SnA n=l
adjoint
the m a p
given
fA
:
I I SnA) n=0
by
Fs(A)
: S S A ÷ SA
co
fA
>
= ~A
co
[I n=0
SSnA
>
II n=0
SnA co
satisfies
fA o in I n = inn+l"
Rewritting
for
1.9.
Definition.
coDyn
S
1.10.
Let
If
S
The
inclusion
in
= K(~)
the m a p s SA = lim A
qn+l
° Sn~
given
by
=
] ] SnA, n=l
then
has
coDyn
subcategory
sequential
colimits,
of
coDyn
S.
a V-left
adjoint
K
S
is the
- - ~
- - 9
n
n
sequential let
qn
of
A --~> SA. then
whose
value
on
isomorphism.
: l i m SnA ---~> l i m s n + I A ~ S l i m
in the and
the V - f u l l
~:
subcategory
functor
~: A ÷ SA
Let
denote
preserves
is a V - r e f l e c t i v e
an o b j e c t
where
coDyniS
by i s o m o r p h i s m s
Proposition.
Proof:
SSnA
n > i.
determined
coDyniS
co
If n=0
'
fA ° inn = inn
: S S A ÷ SA.
in w h i c h
SnA >
n
colimit
are
: s n A ÷ SA
given
by
be the m a p
sn~ : S n A ÷ s n + I A .
to the
colimit,
so
n = qn"
The maps
sn+IA
that
=
qn+l
induce
the two
sn+IA
=
qn+l
lim s P A
< -
isomorphisms
h n = hqn
I o qn+l
Sqn
lim s P + I A
B
L
Sh > SB
SSA commutes
are
>
= Sqn"
: S n A ÷ B.
~
S SnA
>
Hence,
in
~ o h o qn+l
= Sh
I
o I o qn+l
iff
~ o hn+ 1 = Sh ° Sq n = S h n.
90
Step
1.
To s h o w
also
the
square
that
K
is left
adjoint
h0 A - -
4
is an i s o m o r p h i s m For
suppose
In p a r t i c u l a r , follows
that
Conversely, sively
by
then
square
square
hl = 4 -1
(~)
one
i.e.,
that
h0
h n + l = 4 -1
h = n 2.
making o Sh
n
.
square
K
o Sh n o s n ¢ =
makes
is V - l e f t
step.
adjoint
We m u s t
also
is an i s o m o r p h i s m , s(sn~,4)
6 V)
square
E --> A(SA,B) II lim _ A(SnA,B) A ( s n + I A , B )
is the e q u a l i z e r ( w h e r e
-> A(SSA,SB) II
S > lira
SB
En ÷ A(SnA,B)_ the
h I o ¢ = h0, (0)
..... > B
can be t a k e n A(sn+IA,sB)
For
(0)
14
sn+IA 4
has
square
consider
sn~[
(n)
also
square
hn+ 1 = 4 -1 o Sh n.
= hn-
square
SnA
coDyn
(0)
iff
Then
Then
= 4 -1
: SA ÷ B
Showing
of the r e c u r s i v e
If
commutes
Since
= 4 -I o S(hn_l)
Step
(~)
commutes.
o Sh0.
hn+ 1 ° sn~
Hence
SB
Sh 0 o ~ = 4 o h0; given
consider
I Sh 0 SA - -
If
inclusion,
B
(o)
commutes.
to the
(1'4-1)
...>. A(SSA,B) tl
> _A(SnA,B) id (I,i)
is omitted) >
A(SA,B) iJ
A($n+IA,sB) (l'4-1)>- lJ_mA(sn+IA,B)
n+l
map.
map
is the
Ei
third
is
and
En+ I. map,
by the
is also
the
diagram (i,~ -I) > A ( S n + I A , B )
~ - - > (i,~ -I)
A(sn+IA,B)
/
S.
is the
coDyn
S
where
SF,
consider
A construction indexed
then
~
limit
very
of
G
and
is an i s o m o r p h i s m
{G,~}.
Let
of V - a d j o i n t topological
1.6
and
transformation
= S{G,F}
S.
indexed
isomorphisms,
complete.
{G,SF}
so is
V-spectra.
A
be a c o m p l e t e
endofunctors situation.)
for
A = < A 0 , A I, • ..,>.
n ~ 0. Let
Prn(-) + = P r n _ 1 A) ;
S
to a V - n a t u r a l ÷
the
of
are
V-function
to that
and hence
that
system
If
are
corresponds {G,~}
2.
inverse
Proposition.
and
pair
K
in the
Let
A.
V-category
: A_~ ÷ _ Am n ~ 1
and
and
(The n o t a t i o n
_ A ~ = --~ A n=0 A n = Prn(A)
Write
(-)+ for
of
be
with and,
PRO(-) + = ~
+ =
Z--~ ~
be
a
to r e f l e c t
projection
V-functors
sometimes,
the u n i q u e
i .e. ,
let
is c h o s e n
V-functor
such
(the i n i t i a l
object
92
÷
÷
4r~
I
A
A
A A
,-4
÷
4-
r./l
t"xl +
"F
A
,~ +
I A
A
~
~
÷
~-
I
2
5
m
r.~
2 c-1
,~ +
+
÷
2 ©
v I A
A
A v
I
U?
r~
2
~
+
4-
4-
,~
v I
÷
A
~
A
93
Similarly
(-)
: A~
_
÷
satisfies
A ~
Clearly
(-)+---~ (-)-
Prn(-)-
= Prn+ 1
so
=
is a V - a d j u n c t i o n .
Now
E
and
~
co
endofunctors
of
A ~,
g i v e n by
Z~ = ~ n=0 co
V_-adjoint.
Hence
Z+ =
~
and
-
(-)+ o Z and
also i n d u c e
oo
~
oo
Q
= --~- ~, n=0
which
are
_
= Q
o (-)
are V - a d j o i n t .
Here Z+
2.1.
Definitions.
subcategory
of
i)
Prespec
Prespec
A
= =
A = Dyn Z +.
corresponding
ii)
to
Spee A
--
isomorphism 2.2. of
Dyn Z+ =
Remark. Aco
together with
u n d e r the
1
c o D y n Q-
An o b j e c t
is the V - f u l l
coDyn. Q-
of
of 1.5.
Prespec A
a map
~
is an o b j e c t
: Z+A ÷ A
whose
A = < A 0 , A 1 .... >
components
are m a p s
~ n + l : EAn ÷ An+l" It b e l o n g s to S p e c _A if the t r a n s p o s e # ~ n : An ÷ ~ A n + l are i s o m o r p h i s m s for n ~ 0.
maps
2.3.
Spec A
Theorem.
Prespec
A
If
~
preserves
are c o m p l e t e
sequential
V-categories
colimits
and there
then
are p a i r s
and
of V - a d j o i n t
functors K Spec A
A~ < --
U complete. such that
The
A
>
__
Prn
left a d j o i n t
(Ln(A)) p = A
if
Ln
is the
p = n
Z-
and
otherwise. ii) adjoint
By 1.6,
it p r e s e r v e s
Prespec
coproducts
A
is c o m p l e t e .
apply,
g i v i n g a left a d j o i n t F such that n UF(A) = I I zPA and the s t r u c t u r e m a p n p=0 n-p c o m p o n e n t s the m a p s n n n+l Z( p=0 I I ZPAn_ p) = p=0 I I zP+IA n-p = I I q=l which
omit
so
~
has a r i g h t
of 1.7 and 1.8
UF(A)=
I I (z+)P(A) . Thus p=o ~UF(A) ÷ UF(A) has as
n+l
÷ I I EqAn+l-q ~qAn+l-q q=0
the f i r s t summand. ii)
limits
Since
so the c o n s t r u c t i o n s
Since
Spec A =
~
has
coDyni~
a V-left
adjoint,
is c o m p l e t e ,
it p r e s e r v e s
by i.ii.
indexed
By h y p o t h e s i s ,
94
preserves
sequential
K
: Prespec
A prespectrum
~:
colimits,
A =Dyn
~+A ÷ A
so 1.10
Z+ ~
applies,
CODyn
corresponds
~- ÷
to
giving
coDyni~-
a left
~ Spec
a coalgebra
~#:A ÷ ~-A
#
components
: An ÷ ~An+l"
A-object
the
The
colimit
reflection
lim(~-)n(A)
into
.
eoDyni~
Thus
has
UK(A) n = l i m
>
The structure
map
UK(A) n = l i m ~ 3 A n + 2.4.
j ~ lim
the
transpose
components
a]+lAn+j+ 1 ~a
i)
The
the
the
isomorphism
isomorphisms
l i r a ~ 3 A n + l + j = a UK(A)n+ 1 >
composition
in h o m o t o p y
FL 0 (A) n ~ 2nA
of
are
>
Remarks.
of i n t e r e s t
is
whose
.
~3An+ j .
j
ZUK(A) ÷ UK(N)
# a-lim(a-)n(A)
as
- - >
n
lim(a-)n(A)
with
-
~n
underlying
adjoint
theory
Q~ = K F L 0
(cf.
[i1]) , is
: A ÷ Spec
given
as
A,
which
is
follows:
so Q~(A) n = lira ~ J ~ n + J A 3
In p a r t i c u l a r ,
Q(A)
= Q~(A) 0 = lim ~ J z J A
is the
stabilizing
functor.
>
ii) indexed H
Let
limits
: I ÷ Spec
Pn = P r n U I
since
A,
all
let
: Spec
three
A ÷ A.
functors
Then
Pn
have
V-left G
H n = pn H.
Then
for any
spectra.
Let
K =
preserves
adjoints.
: I ÷ V,
If
one has
{G,F} n = {G,Fn}.
3.
Pointed
tesian
closed
category with
a map K,
such
that
U
1.6,
x
the
= 1
sets,
and
i.e.,
objects
1 : K ÷ K
denote
sets;
has
a left
follow
coproducts)
Let
(the t e r m i n a l
is c o m p l e t e
not
[A°P,sets] let
denote K, X
the
denote of
K
car-
the together
: 1 ÷ X.
I(X)
K,
does
preserve
of s i m p l i c i a l
simplicial
as a K - c a t e g o r y .
: K, ÷ K
this
category
of p o i n t e d
3.1.
By
simplicial
object
as a K - c a t e g o r y . adjoint
F
given
construction
From
general
.
the
constant
of
K) .
The
underlying
by
f r o m the
the
of
Then
F(X)
= X ~
1.8
since
description
of
K-functor
K, = D y n
i.
functor i,
(although
i(-) D y n S,
does
following: i)
If
Y,,Z,
e K,,
then
K,(Y,,Z,)
e K
not
one has
is the p u l l b a c k
95
- - >
K.(Y,,Z,)
1 = K(l,l)
z[ [(l,z) K_(Y,Z)
- - > (y,l)
ii)
X E K,
then
X ~ Y, 6 K,
1 = K_(X,I) iii)
K,
Z : K(I,Z)
is a l s o
is g i v e n
(l,y~>
tensored
K_(X,Y)
over
X × 1
>
by
being
K, X ~ Y,
the p u s h o u t
1
[ X × Y iv)
One has
the
following
a) K . ( X ~ Y.,Z.) b)
correspond Further,
I,Y,)
as a c l o s e d
K,
X, A
to
If
Y
> X, A
K,
is a s y m m e t r i c
monoidal
by
1 + K,(X,,Y.)
and ®-product
press
the
UK,(X,,Y,) normal
÷ K(X,Y)
and
closed
functor.
products
(i.e.,
functor.
Hence d)
-al
e)
U
(X ~
closed
equipped UX,
The
(Y ~
preserves preserves
with
and
ii) .
1
adjoint i)
=
by
Y,
with
internal
1 + X,A
hereafter.)
The
Y..
hom (We s u p -
underlying
canonical m a p s
× UY, ÷ U ( X ,
left
i) A
category
given
f r o m the n o t a t i o n comes
a)
x y>
given
: K, ÷ K
1 ÷ K,(X,,Y,)
( f r o m iv)
(i x y)
Then
base points
Let
the p u s h o u t
X
U
e K,,
X, ÷ 1 ÷ 1 ~ Y,
denote
(X x i) "
functor
~ Z.)
@ Y,,Z,) 0
category.
Y.
~ K.(Y.,X
= K(X,Y) .
by adjointness
let
formulas
~ K(X,K.(Y.,Z.))
K,(X,,Y,) n = K,(A[n]
c) K , ( X ~ 3.2.
> X @Y,
× Y,) F =
making
(-) ~
(X × Y) ~
i)
1
to
U
into U
a
preserves
so it is a l s o
a closed
tensors;
i.e.,
(X × Y) m
1 = X @(Y ~
cotensors;
i.e.,
U ( X ~ Y,)
= K(X,Y) .
i)
96
3.3.
Definitions.
i)
Let
S1
be the c o e q u a l i z e r
(in
K)
of the two
maps do A[0] _ _ . ~ dI S,1
and let
If
A
3.4.
tially K . - s m a l l
1 S,
~A = S, ~X E A
: A ÷ A
If
A
objects,
in
then
are defined and are K,-
is called s e q u e n t i a l l ~ K , - s m a l l if
has a s t r o n g l y g e n e r a t i n g
then
S2A
preserves
A,
colimits.
K,.
Hence.
if
{A m}
then
1 1 A(Aa,S , ~ lim B i) -~ K,(S..A(Aa, - - >
lim B i) - - >
= lim K,(SI,A(Aa.Bi))
-~ lim A(A
- - >
-~ A ( A
family of sequen-
sequential
is clearly s e q u e n t i a l l y K , - s m a l l in
is the f a m i l y
K,-category.
sequential colimits.
Proposition.
Proof:
and
An object
perserves
SI
>
is a complete and cocomplete
ZA = S ~ - : A ÷ A adjoint. iii)
q
qd 0 : 1 = A[0] ÷ S 1
have the base point
ii)
A(X,-)
A[I]
,S 1 ~ B i)
>
1 , lim S, ~ B i) - - >
so
S
~ lim B. = lim S, ~ B. _ _ >
3.5.
l
Example:
_ _ >
i "
{A[n] m
i}
tially K, small objects in
is a strongly g e n e r a t i n g family of sequenK,.
However.
~K
(-) = K,(S~.-)
preserves
__W
sequential colimits anyway. 3.6.
Definition.
If
A
is a K. -category.
with the same objects and By category,
[5], 2.2.3, if
then
Furthermore, limits in
U,A
by
A
given then,
U,A(A.B)
is a complete and/or cocomplete K,-
U,A.
there is a close r e l a t i o n s h i p b e t w e e n indexed if
H F
Namely,
the K - f u n c t o r s
spond to K , - f u n c t o r s
is the K - c a t e g o r y
= U(A(A.B)) .
r e g a r d i n g the left adjoint
F : K ÷ U,K,.
U.A
is a complete and/or cocomplete K-category.
[5], 2.4.3,
and in
A
then
H
: I ÷ U,A to
: I ÷ H,A
H # : F,I ÷ A
and
and
U
and
G : I ÷ K
are
as a K - f u n c t o r FG
: I ÷ U,K,
(FG) # : F . I ÷ K,
corre-
and
{G,H}u, ~ = { (FG)#,H#}A. If is defined in
~
is any K-category,
then an indexing functor
[5], 4.5.1, which reduces to
free K - c a t e g o r y on an o r d i n a r y category,
N(I/-)
in case
Z I :I ÷ ~ --is the
such that indexed limits over
97
ZI
are homotopy
limits in
U,A i)
limits.
Hence
for a complete K , - c a t e g o r y
holim H = {ZI,H}u,A(= --
ii)
fiN(I/i) )
--
holim H = { ( Z I ~
-
'
the ¢otensore
sors in 3.7.
U,A
(cf.,
Theorem.
generating
If
are actually
A
functors
Example:
by 2.3,
If
S : k-sp ÷ K
of coten-
Pn
k-sp
denotes
is given as follows:
k-sp(X,Y)
= S(Y x) , Z @ X = IZI × X of pointed k-spaces
if
which has a strongly
objects,
has homotopy
then
Spec A
is
limits that are pre-
: U, Spec A ÷ U,A, n ~ 0. and 3.4.
of k-spaces
1 - I : K ÷ k-sp
structure category
K,-small
2.4, ii),
The category
K-category.
ization and
A
is a complete K , - c a t e g o r y
served by the projection
3.8.
i) ~ H#(i))
i
the same by the construction
U, Spec A
cocomplete
H(i))
[5], 2.2.3).
family of sequentially
Immediate,
~
I)#,H#}A(= I (N(!/i)//
a complete K,-category.
Proof:
homotopy
U.A
--
where
A,
are given by either of the formulas
the singular X, Y • k-sp and
is a complete
denotes
functor, and
Z ~ X = X IZl
is a complete
and
geometric
real-
then this
Z • K,
then
Similarly,
the
and cocomplete K, category.
98
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[2]
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G.M.
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Am. Math.
L o c a l i z a t i o n and sheaf reflectors,
210(1975),
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J. P. May,
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D. S. Scott,
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in b i c a t e g o r i e s , 111-160.
J.
Cahiers de Top. et
GENERAL
CONSTRUCTION
IN T O P O L O G I C A L
OF M O N O I D A L
, UNIFORM
Georg
CLOSED
STRUCTURES
AND NEARNESS
SPACES
Greve
Abstract/Introduction: In the f o l l o w i n g p a p e r we c o n s i d e r t o p o l o g i c a l s t r u c t u r e s on f u n c t i o n s p a c e s and c a r t e s i a n p r o d u c t s b e i n g c o n n e c t e d by an e x p o n e n t i a l law of the f o r m C(XeY,Z) ~ C ( X , C ( Y , Z ) ) . T o p o l o g i c a l c a t e g o r i e s p r o v i d e d w i t h s u c h a " m o n o i d a l closed" s t r u c t u r e are s u i t a b l e b a s e c a t e g o r i e s for t o p o l o g i c a l a l g e b r a , a l g e b r a i c t o p o l o g y , a u t o m a t a - or d u a l i t y t h e o r y , in p a r t i c u l a r if ~ is s y m m e t r i c or the u s u a l d i r e c t p r o d u c t . We s t a r t f r o m a p u r e l y c a t e g o r i c a l p o i n t of v i e w p r o v i n g an e x t e n s i o n t h e o r e m w h i c h later turns out to be v e r y c o n v e n i e n t for the c o n s t r u c t i o n of m o n o i d a l c l o s e d s t r u c t u r e s in c o n c r e t e c a t e g o r i e s , n a m e l y in t o p o l o g i c a l spaces, u n i f o r m spaces, m e r o t o p i c s p a c e s and n e a r n e s s spaces. E n l a r g i n g a t h e o r e m of B o o t h and T i l l o t s o n [2] it is s h o w n t h a t t h e r e are a r b i t r a r y m a n y (non s y m m e t r i c ) m o n o i d a l c l o s e d s t r u c t u r e s in t h e s e c a t e g o r i e s , h e n c e t h e r e is a g r e a t d i f f e r e n c e t o the s y m m e t r i c case, w h e r e c l o s e d s t r u c t u r e s s e e m to be u n i q u e (cp. C i n c u r a [3], I s b e l l [8]). A f u r t h e r a p p l i c a tion of the e x t e n s i o n t h e o r e m is a c r i t e r i o n for m o n o i d a l - resp. c a r t e s i a n c l o s e d n e s s of M a c N e i l l e c o m p l e t i o n s . Of c o u r s e a s y m m e t r i c m o n o i d a l c l o s e d s t r u c t u r e is u n i q u e l y d e t e r m i n e d by its v a l u e s on a f i n a l l y and i n i t i a l l y d e n s e s u b c a t e g o r y , but also the c o n v e r s e statem e n t is true, i.e. m o n o i d a l c l o s e d s t r u c t u r e s can be o b t a i n e d by e x t e n d i n g a s u i t a b l e s t r u c t u r e f r o m a s u b c a t e g o r y to its M a c N e i l l e completion.
O. P R E L I M I N A R I E S Throughout
this
paper
we h a v e T - i n i t i a l i.e.
for
every
(fi
: A +
= Tgi'
said
to c a r r y
that and
there for
gi
an i s o m o r p h i s m way maps
Ai,
are m o r p h i s m s
in C.
indexed)
that
every
lifted
structure
sources
16OBC
(cp.
so
[13],[7]),
map
y
: TB ÷
TA with
1 : B +
A.
A is
respect
with
a one
to all x.. We a s s u m e i e l e m e n t u n d e r l y i n g set
to be a m n e s t i c
Topological
are T - f i n a l
functor,
is a s o u r c e
to a m o r p h i s m
with
T is s u p p o s e d
iEObC).
there
a topological
TAi)i£ I there
such
object
reasons
implies
are f a i t h f u l ,
(class
can be
initial
is a u n i q u e
technical
of
Set d e n o t e s
(x i : X ÷
Tf i = xi,
: B ÷ the
: C +
liftings
source
Ai)i6i,
TfiY
T
functors
liftings
I is a r e p r e s e n t i n g
of
defined
sinks,
object
( Ti£ObSet
all
for
in this constant
for T, h e n c e
we
101
can
identify
category
elements
a£TA with morphisms
[
2)
are
direct
obvious.
^
decompositions
D is a W - d i a g r a m
there
Djl
. Djl~
=
the
n
n
morphism ^
D ÷
that
n
^
f:
n
D=
D +
XleDjl
@ S. D. 1=I J1 31
^
@ Eo D. along 1=I J1 Jl
"
@ S. O. ÷ ~ E. O. . M o r e o v e r 1=I J1 J1 1=I J1 J1
the p r o j e c t i v e
^
~ S. O. ~ Ker 1=I J1 31
f
and
^
n
xe
^
obvious
Z D k such kY and kernel
Z --> Y
we have
The first isomorphism theorem is proved to be a consequence of these axicms ([ 5] Theorem i,ii). Finally we ass~ne (HW5)
Any set of normal subobjects of any object possesses a union which is again a normal subobject.
Let us notice that in the presence of the systems of axicms ~WI)-(HW5 ) the normal subobjects of any object form a complete lattice under
n
and
u
For short, the systems of axic~s (ARI)-(AR6) and (HWI)-(HW5) will be denoted by (AR) and (HW),respectively. THEOREM i. (}~) implies (AR), and if any two s.ubobjects of objects in have an intersection, then the cenvers~e also holds.
C
Proof: In view of what has been said above, it is clear that the system (HW) implies (ARI)-(AR5). In order to prove (AR6), let kernel
ker t: Z -->X , and
N
t: X --->Y be a cokernel with
be a normal subobject of
X . By [5] Lenma i.i0
192
t(NU Z) is a normal subobject of u n i o n w e have object of
t(N).
plying that
t(Z)= OS_t(N) , by the definition of t(NUZ)
is a normal sub-
Conversely, by the definition of the image
t(N)=t(NU Z)
is a normal subobject of
Conversely, notice that case of (AR6)
Y . Since
t(N OZ)_~B
through which each
f(A)NB' exists, it contains all the
f~
1
f(A ),
1
containing each A i , f' A ~ f(~> > B , and
f(A)
by
C
be
f(Ai) . Suppose that
factorizes. Then, provided that and since the
f(A ) are normal
1
subobjects of
iE I ,
1
(~R6), they are normal in
f(A)NB',
too, whence
C_C ,
and conversely,
(f')-I(c)>A. 1
for all
iEI
conclude that izes through
, hence f(A')
(f')-I(c)>A' and
C
,
thus
C=f'(f')-l(c)>f'(A')=f(A ') . So we
are equivalent subobjects, whence
fe
also factor-
~ , and we are done.
Remark i. By [ 2] p. 397 we know that the second isomorphism theorem holds under (AR), thus it holds under (HWI)-(HWS) as well. Remark 2. Krer~oa and Terlikowska [ 6] and then Terlikowska-Os~owska [ 12] , [ 13] introduced a self-dual system of axioms which is satisfied in the categories of associative or alternative rings but not in that of not necessarily associative ones. Hence this system of axicms cannot be equivalent to (HW) or (AR), nevertheless there is a strong connection between them. Stlopose we are given a category C
satisfying the system of axic~s (AR). Consider the subcategory
of all objects of cc~positions.
C
C'
consisting
and those morphisms which are kernels or cekernels or their
(For establishing the basic results in the general radical theory
of rings, only these morphisms are really needed. ) Now it is straightforward to
193
check that
C'
satisfies the system of axic~s
AI-A6,
A6 ¢~ and A7 ~'~but not A7
of Terlikowska-Os~owsk~ [12] . Conversely, AI-A7, A6 ~, A7 ~'~of [12] inply most of (AR) but (AR4) and (AR5) only in a weaker form. In the rest of the paper we shall always work in a category
~
C
satisfying
(~). 3. Radicals In his paper [ 3] Carreau presented an elegant treatment of radicals in cate-
gories. What he did in genuine categorical terms is, expressed in the classical language, that radicals can be defined beth by means of a function and of a semisimple class. The same idea is basic in Hoehnke's earlier development [ 4] of radicals in categories of universal algebras. In the category of associative rings Michler [8]
introduced a notion of radical at the same time as Hoehnke did for
universal algebras; these two notions are equivalent for rings. Now we present Carreau's definition of a radical functor in the slightly modified but equivalent version given by Holcombe and Walker [ 5] . By the cokernel subcategory jects are the objects of
C
E(C)
of
C
we mean the subcategory whose ob-
and whose only morphisms are the cokernels of
C .
(In Carreau's terminology t h i s is a special coextensive subcategory. ) A covariant p: E(C)--->C
functor (i)
p
(ii)
is called a radical functor, if
for all
CE C ,
(iii) p(C/~(c))=o
p(C)
is a normal subobject in
for all
C6C
normal subobject
p(C)
in
C
~(p(C))C
is a subfunctor of the inclusion functor
p
assigning to each
CE C
a
satisfying (iii) and for any cokernel
~
frcrn C .
Theorem 2.2 of Carreau [3] states exactly that every radical functor defines a radical and conversely. The most ini0ortant radical functors are cc~plete (which means that if p(B)=B
for some normal subobject
(p(p(A))=pB
p(B')#B'
_p(B)/(p(A)AB)=BI(p(A)nB)=B''
,
195
whence
p(B'')=B''
, a contradiction.
The converse iaplication is obvious.
PROPOSITION 5. If a radical functor radical satisfies
p
is idenlootent , then the corresponding
(M2).
The assertion is obvious. Moreover, under the validity of (~iI) the converse iaplication is also true. THEOPd~4 6. Every cc~plete and idemlootent radical functor defines an
4-
radical and conversely. Proof: In view of Propositions 4 and 5 all we have to prove is that the radical functor
p
defined by an 4 - r a d i c a l
idempotent. Now there is an exists a
B/p(p(A))
= B/p(p(A))
AEC
such that
. By condition
B/O(p(A)) 4 p(A)/p(p(A))
is ideml0otent.
such that B/p(p(A))4
p(A)/p(p(A))
(iii), however, we have condition
St~opose that
p(A)/p(p(A))#O
(M_l) yields
and
p
is not
. By (M2) there p(B/p(p(A))) =
p(p(A)/p(p(A)))=O
p(B/o(p(A)))#B/p(p(A))
and so for ~ a contra-
diction. As usual, to a radical IR
=
S
= [AEC
P
{A6C
: p(A)=
p
we assign two classes A}
and P
: p(A) = 0},
called the radical class and the semisir~le class of
p , respectively.
Knowing the
equivalence of the previous definitions of radicals, the connection between radical and semisir~ole classes as described in Andrunakievi~ and Rjabuhin [ 2] , V, §2, Theorem 3, yields exactly Theorem 3.10 of Holcombe and Walker [5] and its converse. (The latter is the same as [5] Theorem 3.11; in fact the sufficient condition give~l in the note after this theorem, is always satisfied in view of (AR6). ) Thus an
4-radical
p
on any object
A
can be determined both from below and from
above: u (B 4 A
: B 6 IRp) = p(A) = e (C 4 A or C = 0 : A / C E S p )
Till now we have characterized an
.
4 - radical by means of the radical assign-
merit (radical functor) and the radical class. It can also be characterized in terms of the semisimple class and by the pair of radical and semisimple classes, respectively. Such characterizations
for not necessarily associative rings or
~-groups
exist in plenty (see e.g. [ 7] and [ i0] ), and using the tools we already have in our category, their proofs can be carried out word by word in our case, too. Here we pick out just one characterization of each of the latter two types. THEORI!M 7 class of an
(~]itz [i0] Theorem 4 ). A class
4-radical
if and only if
S
S
of objects is the sen/simple
satisfies the following three conditions:
196
(a) i_ff B 4 A 6 $ , then (b)
S
B
has a non-zero factor object in
(c) for all
AEC
, ((A)S)S = (A)S
where
(A)S=N ( B 4 A
THEORI~4 8 (Mlitz [iO] Theorem 2). The classes and semisimple classes of an I~ ~, S
consists of zero objects,
(B)
A E I~
and
(C)
A E S
(D) for any
and
A/B # O S ~ A
A E C
I~
and
or S
B = O : A/B E S). are the radical
4-radical if and only if
(A)
and
$ ,
is closed under subdirect products,
inloly A/B ~ $ ,
i~ply
B~I~
,
there is a normal subobject
B
o_ff A
such that
B 6 I~
A/B ~ S .
References [ i] S. A. AMITSUR, A general theory of radicals, II, Radicals in rings and bicategories, Amer. J. Math. 76 (1954), 100-125. [ 2] V. A. ANDRUNAKIEVI~ and Ju. M. RJABUHIN, Radicals of algebras and structure theory (Russian), Nauka, Mosoow, 1979. [3] F. CARREAU, Sous-cat~gories r~flexives et la th~orie g~n~rale des radicaux, Fund. Math. 71 (1971), 223-242. [4] H.-J. HOEHNKE, Radikale in allg~meinen Algebren, Math. Nachr. 32 (1966), 347383. [5] M. HOLCOMBE and R. WALKER, Radicals in categories, Proc. Edinburgh
Math. Soc.
21 (1978), 111-128. [6] J. ~ A
and B. TERLIKOWSKA, Theory of radicals in self-dual categories,
Bull. Acad. Polon. Sci. S~r. Sci. Math. Astronom. Phys. 22 (1974), 367-373. [7] L. C. A. van ~
and R. WIEGANDT, Radicals, semisimple classes and torsion
theories, ~ t a Math. Acad. Sci. Hungar. 36 (1980), 37-47. [8] G. MICHLER, Radikale und Sockel, Math. Ann. 167 (1966), 1-48. [9] B. MITCH~.T., Theory of categories, Academic Press, 1965. [iO] R. MLITZ, Radicals and semisi~ple classes of
~-groups, Proc. Edinburgh Math.
Soc. 23 (1980), 37-41. [ii] E. G. ~UL'GE~ER, General theory of radicals in categories (Russian), Mat. Sb. 51 (1960), 487-500. [ 12] B. T E R L I K O W S K A ~ S K A ,
Category with self-dual set of axic~s, Bull. ~ a d .
Polon. Sci. S~r. Sci. Math. Astronom. Phys. 25 (1977), 1207-1214. [ 13] B. TERLIKOWSKA-OS~OWSKA, Radical and semisimple classes of objects in categories with a self-dual set of axioms, Bull. Acad. Polon. Sci. S~r. Sci. Math. Astronom. Phys. 26 (1978), 7-13. [ 14] S. VELDSMAN, A general radical theory in categories, Ph. D. Thesis, University of Port Elizabeth, S. A., 1980.
ON THE STRUCTURE OF FACTORIZATION STRUCTURES by A. Melton and G. E. Strecker
For any category on
K.
K
In particular,
we investigate the family of all factorization structures
for each such structure,
lattice of all factorization structures on
K
(E,M), we investigate the complete with left factor a subclass of
E;
this investigation is based on a Galois connection between all such structures and the lattice of all full isomorphism-closed subcategories of
K.
families are precisely all the E-reflective subcategories of
The Galois-closed
K
and all the (E,M)-
dispersed factorization structures of Herrlich, Salicrup and Vazquez.
AMS
(1980) subject classifications:
Secondary:
§0
Primary 18A20, 18A32, 18A40;
06A15, 18A22
Introduction The importance of factorization structures on categories is by now well
appreciated.
Over the years the conditions that have been considered necessary for
an "(E,M)-factorization structure" to carry that name have evolved from those requiring
E
and
M
to be sufficiently nice dual-like classes of epimorphisms
and monomorphisms such that each single morphism has an essentially unique factorization,
(E,M)-
through various stages until the current generally accepted criteria
that (among other things)
E
be a class of morphisms and
sources such that each class-indexed source has an (E,M)-factorization,
M
be a conglomerate of
(even empty or proper class indexed)
m%d, in the category,
(E,M)-diagonalization holds.
To
emphasize that we require diagonalizations as well as factorizations we call such entities "diafactorization structures." The two major references for this paper are
[HSV] and
[Ho], both of which made
significant contributions to the clarification of the nature of
(dia)factorization
structures. In [HSV] Herrlich,
Salicrup and Vazquez introduced a new type of diafactoriza-
tion structure called dispersed and proceeded to show that there is a bijection between all E-reflective subcategories of an (E,M)-category dispersed diafactorization structures on
K.
K
and all
(E,M)-
This was a generalization of the
result that for nice categories such a correspondence exists between the epireflective subcategories of [Sl],
[S2] , [S4]).
K
and all perfect factorizations
(cf. [Hel],[He2],[Na],[Ne],
It also ~aproved and put into the proper context much of the
earlier work on quotient reflective subcategories,
connectedness properties,
corresponding factorizations
[SV2],
(cf. [C], [P], [SVI],
and
[$3]).
In §i, via a modification of the main result of Hoffmann
[Ho] (cf. also Harvey
198
[Ha]), we show that the development classes of a category
K
precisely those classes,
such that
E, for which there exists an
diafactorization structure on problem of
[HSV].
any E
(E,M)
are
(E,M)
is a
This answers the outstanding open
diafactorization structure,
(cf. [HS2] , [T]).
(E,M)-category
E
must be a class of
As a by-product of this theorem we also have, for
K, an internal characterization of all those
for which there exists a
structure
(Th.l.3).
(see [Ne])
The proof of Theorem 1.3 also provides an alternative proof of
the fact that for any epimorphisms
K
M
(Th.l.9).
D
such that
(C,D)
C
contained in
is a dispersed diafactorization
It is interesting to note that such classes are (to within
existence of the colimits) the "standard" classes of E-morphisms introduced in [SI] and investigated further in [$2]. In §2 we describe and investigate a Galois connection that makes precise the nature of the bijection discovered in [HSV]. of an (E,M)-category
K
Namely, the E-reflective subcategorles
and the (E,M)-dispersed diafactorization structures are
precisely the Galois-closed classes and are complete lattices
(in a suitably large
universe)
General Galois
that are anti-isomorphic with each other
(2.6(2)).
results, as well as special properties involved,
are used to investigate in more
detail the structure of the complete lattice
of all diafactorization structures
(C,D)
on
K
with
C
a subclass of
E.
Q
In particular,
partitioned into a family of complete lattices Q
it is shown that
(called levels)
can also be viewed as a union of complete lattices
§i.
Characterization of Diafactorization Structures Definitions and Notation
(i) In all that follows
K
is
(called images) all of which
have a point in common and none of which meets any level non-trivially
i.i
Q
(2.6(i) (i)) and that
will denote a category, and
Mot K, Iso K
(2.6(i) (ii)).
and
Epi
will denote the classes of all morphisms, all isomorphisms and all epimorphisms of
K.
All subcategories will be assumed to be full and isomorphism-
closed. (2) A K__-source with domain
X
empty and possibly proper) domain (3) K
is a pair
i
in
where I
fi
I
is a class
(possibly
is a K-morphism with
X.
is called an (E,M)-cate~or~ and
ture on
(X,(fi)i)
and for each
K
provided that
E
tion with K__-isomorphisms and
(E,M)
is called a diafactorization struc-
is a class of K-morphisms closed under composiM
is a conglomerate of K-sources closed under
composition with isomorphisms such that: (a)
K
has the (E,M)-factorization property;
has a factorization (Z,(mi) I) (b)
K
belongs to
i.e., every K--source
X~Y.
= x--~Z ~-i~Y. l l M, and
has the (E,M)-diagonalization property;
K__-morphisms and
(X,(mi) I)
and
(Z,(hi) I)
where
e
(X, (fi) I)
belongs to
i.e., whenever
e
E
and
are K--sources such that
and
f
are e
199
is in
E,
(X,(mi)i)
is in
and for each
M
there exists a unique m o r p h i s m i
in
d:Z---~X
i
in
I, h.e = m.f, then 1 1 f = de, and for each
such that
I, h. = m.d. l l
(*)
Y--~-~1 Z
I
d''1
l
X -------9-W. m. 1 1 [If only on (4)
(a) is satisfied,
(E,M)
is called a factorization
structure
K° ]
([HSV])
If
(C,D)
on
A
K
of
K K
is an
such that
E;i.e.,c:X---~Y f:X--+A
(E,M)-category,
is called C
is precisely
is in
with
A
C
in
then a diafactorization
(E,M)-dis~ersed
iff
A
c
structure
iff there exists a subcategory
all the A-extendible
is in
E
morphisms
in
and for each K--morphism
there is some K - m o r p h i s m
g:Y---~A
such that
f = gc. (5)
Let
E C
Mor K
(a)
~(E)
then:
will denote the conglomerate
the property e (b)
in
A(E)
E, then
X fi~y = X ~ Z ~Y. • i is an isomorphism.
e
will denote the conglomerate
the property that if square
i
a K__-object e:X--~Z
in
morphism (d)
~
e
(*) c o ~ u t e s ,
for each (c)
of all sources
that if
in
I
is in
1
of all sources
E
then there exists a unique (*) commutes.
(cf.
[S I]
is called an E-injective
E
and each K - m o r p h i s m such that
having
(X,(mi) I)
and if for each
Y
g:Z---~Y
(X,(fi) I)
is a factorization
i
in
d:Z--~X
and
with
having
I
the
such that
[$4])
object iff for each
f:X---~Y, there exists a K--
f = ge.
is the category whose objects
are members of
E
and whose ^
morphisms
(6)
(e)
A0:KE
Let
C
>K
and
E
hOmK_ (e,e)
of
h = gf.
(a)
iso--com~ositive
(b)
left cancellative
(c)
(f,g)
is the functor defined by be subclasses
K_-morphisms for which
belongs to
are pairs
iff
h
w.r.t.
E, then
f
Mor K Then
C
belongs to E
where
~0(f,g) and let
ge = ef. = f.
(cf.
f, g
and
[Ho]). h
be any
is said to be: C
whenever
iff whenever
must belong to
h
{f,g} C
belongs to
C U Iso K; C
and
f
C;
pushout p r o n e iff (i) every K--source pushout
X
(X, (c i) i )
ci > Y.
di ; Z
d (ii) every 2-indexed K--source, out
with each with
d
(X,(k,c)),
ci in
in
C
has a multiple
C; and
with
c
in
C
has a push-
200 X-c
k
~Y
t
I
Z •
(d) a development
c
with
class
(ef.
[Ne] (t))
c
in
C.
iff
(i) C~__ Epi K, (ii) C
is iso-compositive,
(iii) C
(e) an E - s t a n d a r d (i) C
class
(cf.
is a development
(ii) C
1.2 Remark.
and
is pushout prone; IS I],
[S 2] (tt))
iff
class of E-morphisms,
is left cancellative
w.r.t.
and
E.
The following are some w e l l - k n o w n properties
structure
(E,M)
on
K
that we will use in the sequel.
(i)
E
is iso-compositive.
(2)
E
and
M
of any diafactorization
determine
each other;
in fact
M = A(E).
We next obtain an improved version of the main t h e o r e m of Hoffmann that no conditions w h a t s o e v e r
morphisms steps in
E.
are put on the category
Some major steps of the proof,
however,
K
[Ho]
in
or the class of K-
closely
follow analogous
[Ho].
1.3 Characterization For any category
T h e o r e m for Diafactorization K
and any class
E
Structures
of K-morphisms,
the following are
equivalent: (i) There exists a conglomerate
ization (2) E
structure on
is a development
M
of K-sources
for w h i c h
(E,M)
is a diafactor-
K. class.
(3)
(E,A(E))
is a factorization
(4)
(E,~(E))
is a diafactorization
(5) The following hold:
structure on
(a)
E
(b)
A0: ~
K.
structure on
K.
is iso-compositive; ~K
is a topological
functor (%f%).
(t)
In [Ne] Nel d e f i n e d development classes ulation avoids his smallness condition.
(tt)
In [Sl] and [S_] standard classes of epimorphisms are defined more genz erally, without the requirement of the existence of (multiple) pushouts in what corresponds to (6)(c).
(tt+)
A functor
F:A---~X
has a factorization
is called topological
somewhat less generally.
iff each F-source
(X
gi
Our form-
~FAi) I
(X gi y FA.) = (X r F A Ffl ~r FA.) where r is an Xl l -isomorphism and (A--~i~A.) is an F-initial A-source -- or, equivalently, l every F-sink has an (F-final A__-sink, isomorphism)-factorization. (cf. [He3]).
201
Proof: (i)
(4) ---~(i) and ~(5).
Ao-sOurce. tion
(i)
-- - ~ S e t . A l s o , i f k i s a p o a e t w i t h an o b j e c t a i n which t h r e e d i s t i n c t arrows i n i o l a t e and eeoc= ~X~ I ~ r=~_~ i s t h e ~ - p r o d u c t i v e l y i n d e p e n d e n t c o l l e c t i o n o~ o b j e c t s o f Se t k, c o n s t r u c t e d i n I I . 7 , t h e n e v e r y o b j e c t o f S e t k can be embedded i n a c e ~ - s o f t o b j e c t , hence e v e r y commutative s e m i g r o u p has a p r o d u c t i v e r e p r e s e n t a t i o n i n S et k by f u n c t o r s c o n t a i n i n g 8 g i v e n f u n c t o r ~" :k 4 S e t . IV. How l a r g e a r e t h e r e p r e s e n t i n g
objects?
1. I f we i n v e s t i g a t e p r o d u c t i v e r e p r e s e n t a t i o n s i n a c o n c r e t e c a t e g o r y , t h e r e i s a n a t u r a l q u e s t i o n : how l a r g e a r e t h e u n d e r l y i n g s e t s o f t h e r e p r e s e n t i n g o b j e c t s . I n many c o n c r e t e c a t e g o r i e s , t h e f o l l o w i n g e a s y m o d i f i c a t i o n o f t h e b a s i c method p e r m i t s t o d i m i n i s h t h e c e r d i n a l i t y o f t h e u n d e r l y i n g s e t s : one c o n s t r u c t s t h e c o l l e c t i o n c~ = ~ X ~ ~ ~ e o c ~ such t h a t any X~ c o n t a i n s e d i s t i n g u i s h e d p o i n t , say ~ . , . a n d , f o r any f ~ ~ , Xf i s not t h e whole p r o d u c t ~I'~e~cx f ~ J as in II.2, but only its subobJect consisting of ell those points, which differs from
~f = B e~Yoc ~f(~) in at most finitely
many coordinates. The collection ~ and the distinguished points ~/~ , ~ oc , have to be constructed such that any set A c c~°c with card A --~ oc can be recognized from the object X A being a coproduct o f t h e s e new X f ' s . 2. If this modification is combined with the application of Theorem Xll.2, one can obtain, for example, the following assertions: every countable commutative semigroup has s representation by products of a) b) c)
countable topological spaces, containing a ~iven countable space (see[20]), countable posers, ~raphs, tolerance spaces, containing a given countable poset, graph, tolerance space (see [17~), countable unary algebras, containing a given countable unary al~ebra.
S. Let us show an application of this idea on the category (Set~) k, where Setco denotes the category of all countable sets (and k is a poser)° We prove the following
310
_Proposition. The assertion (6) below is equivalent to (1)...(5) in 11.6 and 111.3. (6) Every countable commutative semi~Toup has a productive rePresentation in (Set~) k by functors, containir~ a given functor ~J{: :k ~ S e t ~ . Proof. If • poser k contains k I (or k 2), define X n as in I1.6. Let c~n be its suhfunctor, sending any object of k I (or k 2) to a onepoint set, namely O-n(p) = tO} for every object p of k I (or k 2) and define Xf to be the subfunctor of ~ = ~n n such that, for every object p of k I (or k2, respectively), X~(p) consists of those x e f(n)( p) "at most in finitely coordie ~ ( p ) , which differ fr om ~ O ~ n nates. Then X f x X is still isomorphic to Xf.g end f still can be recognized from Xf ~by the same reasoning as in II.G°). The rest of the Proof is the same ee in II.~. end III.3.
4. Remark. L e t us m e nt i on one t r i c k more, which p e r m i t s t o o b t a i n the following assertions: every countable graph (poset, tolerance space, unary algebra, topological space) can be embedded into a countable graph (poser, tolerance space, unary algebra, topological space) which has 2 o nonisomorphic square roots [17], [18]) and also the assertion if a poser k contains k I or k 2 from 11.6, then every fUnctor ~" :k--,Seto~ is a subfunctor o£ some X:k---> Set~o , which has 2 xo non equivalent square roots. This follows immediately from the above results and the Proposition. The semigroup 5~o~ of all countable subsets of co ~ contains a subset T such that card T = 2 ~° and s + s = s" + e" for every s,s'e T° Proof. Let S be a semlgroup with a countable set of generators, say {shin 6 60 ~ and defining equations s n + s n = s n + an. for ell n,n'~ c~ • By[14~, there exists e disjoint homomorphism h:~ ...... ~ 5 ~ i.e. h(s)~h(s') -- ~ whenever s=@s ". Pu~ T : {~_~eAh(Sn)IA ~ , A4=~,
t h e n T has t h e r e q u i r e d p r o p e r t i e s . References
1. 2°
J . Ad~mek, V. Koubek. On • r e p r e s e n t a t i o n o f eemi~roupa by p r o d u e t s o f a l g e b r a s and r e l a t i o n s , C o l l . Math. 3 8 ( 1 9 7 7 ) , 7 - 2 5 . J° A d ~ e k , V. Koubek, R e p r e s e n t a t i o n o f o r d e r e d commutative s e m i g r o u p s , C o l l . Math. Soc. J e n o s B o l y s i 20, A l g e b r a i c t h e o r y o f s e m i ~ r o u p e , Szeged 1976, 15- 31.
311
3. 4.
5. 6. 7. 8o 9o i0. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.
J. Ad~mek, V. Koubek, V. Trnkov~, Su.~ of Boolean spaces represent every group, Pacific J. Math., 61(1975), i-7. A.L. Corner, On a conjecture of Pierce concerning direct decomposition of Abelian groups, Proc. of Coll. on Abelisn groups, Tihany, 1963, 43-48. Sur l'~quation n = pour types d'ordA.£. C.R. DavisAcsd. (Morel),, des re, Sci. Paris 235(19521 ~, 924-~926. R.H. Fox, On a problem of S. Ulam concerning Cartesian products, Fund. Math., 34(1947), 278-287. W. Hanf, On some fundamental problems conceding isomorphisms of Boolean algebras, Math. Scand. 5~1957), 205-217. J. Ketonen. The structure of countable Boolean algebras, Annsls of Math., I08(1978), 41-89. L. Lov~sz. Direct product in locally finite categories, Acts Sci. Math., 3311972), 319-322. A. Pultr, Isomorphism types of objects in categories determined by numbers of morphieme, Acts Sci. Math., 35(1973), 155-160. A. Tsrski, Cardinal algebras; with an appendix by B. Jonsson and A. Tsrski, Cardinal products of isomorphism types, New York, 194~ V. Trnkov~, X n is homeomorphic to X m iff n..4n, where ~ is a congruence on natural numbers, Fund. Math. 80(19~3), 51-56. V. Trnkov~, Representation of semigroups by products in s category, J. Algebra, 34(1975), 191-204. V. Trnkov~, Isomorphism- of products and representation of commutstive semigroup, Coil. Math. Soc. Janos Bolysi 20, Al~ehraic theory of semigroups, Szeged 1976, 657-683. V. TrnkovA, Productive representations oi" semi~Toups by pairs of structures, Comment. Math. Univ. Carolinae 18(1977), 383-391. V. Trnkov~, Cstegorial sspects ere useful for topology, Lecture N. in Math. 609, Springer-Verlag 1977, 211-225. V. Trnkov~, Cardinal multipllcstion of relational structures, Coll. Math. Soc. Jsnos Bolyai 25, Algebraic methods in Graph theory, Szeged 1978, 763-792. V. Trnkov~, Homeomorphisms of products of spaces (in Russian), Uspechi Math. Nauk 34(1979;, vyp. 6(210), 124-138. V. Trnkov~, Homeomorphisms of powers of metric spaces, Comment. Math. Univ. Carolinse 21(1980), 41-53. V. TrnkovA, Homeomorphisms of products of countable topological spaces, to appear. S. Ulam, Problem, Fund. Math. 20(1933), 285. J. Vin~rek, Representation of countsble commutstive semigroups by products of weakly homogeneous spaces, Comment. Math. Univ. Carolinae 21(1980), 219-229.
CATEGORIES
O F KITS,
COLOURED
GRAPHS,
AND GAS~S
by Antoni
O.
Introduction The main
category
games
between
games,
there them.
an a b s t r a c t
game which
combination
of t w o
automaton Section
and
different one may these
shall
consider
deal here with
between
prove
by Section
graph.
e.g.,
confine ways
categories
ourselves
to
of d e f i n i n g
be way
games,
notion
of
case of the
shown
in
at l e a s t
16
and consequently
of abstract in g a m e
some g e n e r a l
games,
of an output-state It w i l l
abstract
and
a category
the g e n e r a l
in a n a t u r a l
to b e u s e f u l
4 where
of g a m e s ,
as a p a r t i c u l a r
the n o t i o n
to d e f i n e
16 d i f f e r e n t may
natural
of a coloured
of morphisms
categories
if w e
various
notions:
of applying
noncooperative
However,
may be considered
simpler
it is p o s s i b l e
types
illustrated
We
sorts
of g a m e s ,
of c l a s s i c a l
etc.
a method
are many
categories
are a l s o
the n o t i o n
3 that
There
many
a category
of dynamic type,
is to o u t l i n e
theory.
define
games,
of one
morphisms
this p a p e r
in g a m e
one may
of two-person a category
and preliminaries
aim of
theory
accordingly
Wiweger
games.
theory
as
constructions
Some of
it is of p r o d u c t s
and
coproducts of abstract g a n ~ s a r e descri0ed, and b y S e c t i o n 5 w h e r e t h e i n t e r p r e t a t i o n of these constructions i n t h e p a r t i c u l a r c a s e of t h e t w o w e l l - k n o w n t w o - p e r s o n games isgiven.
if
We use
the t e r m i n o l o g y
(At) tC T
is an i n d e x e d
the d i s j o i n t also write identify same
s u m o f the
If
A I + A 2.
TO a v o i d sets w i t h
shall
f :A
, B
A x B
. B denotes
P o w (A) Pow+ set
tacitly
of
o f sets,
At .
Instead
cumbersome their assume
[I]
and
then of
In p a r t i c u l a r , will
StE{I,2}A t
notation
images
[5].
StcTA t
we
in d i s j o i n t
if n e c e s s a r y
that
denote
we
shall
shall
sometimes
sums;
for the
the s e t s
in
are disjoint.
the r e s t r i c t i o n and
sets
considered
reason we
question
and notation family
functor
of
f
will the
denotes and
is a f u n c t i o n
Pow
to
C.
and
C c A,
The canonical
be denoted
by
pr I
then
fI C
projections
and
pr 2
will
denote
A x B
, A
respectively.
set o f all r e a l n u m b e r s . the p o w e r denote
respectively.
set of
A.
the c o v a r i a n t
and
the c o n t r a v a r i a n t
power
313
I.
Output-state
automata,
An output-state
(I)
K =
where
A,X,Y
Y.
are
Every
next-output
A kit
(of s t a t e s , Y
to
output-state 1
A,
inputs, and
automaton
and the
1
and outputs,
respectively),
is a f u n c t i o n
(I)
is a M e a l y
next-state
[4])
function
is a n o u t p u t - s t a t e is the c a n o n i c a l
from
A x X
automaton
function
automaton
projection
any output-state
automaton
(I) w e
= { (x,x') E X x X I V a E A k ( a , x )
It is o b v i o u s
that
A monokit relation
on
~
with
defined
the
as
(I)
onto
such
the
that
first
the
axis,
(2)
M =
where
X
(cf.
graph
is a f u n c t i o n
and
from
the d
from
G =
that
graph.
relation
such
that
on
X. is the i d e n t i t y
is a t r i p l e
D
(of v e r t i c e s into
the
set
(x,x') C p(d)
x
to
and colours
respectively),
P o w ( X × X)
means
that
of all
there
subsets
is an a r r o w
and of
of
x'.
is a 7 - t u p l e
(A,X,Y,~,I,D,p)
UIG =
(A,X,Y,~,I)
The pregame
condition
sets
condition
A pregame
(3)
is a k i t K
}.
(X,D,p),
are
colour
= k(a,x'
is a n e q u i v a l e n c e
[4])
D
X x X;
define
X.
A coloured
such
and games
z o I = p r I. For
p
graphs,
~ = ~ 0 I.
(cf.
next-state
sets from
function
the c o m p o s i t e
i.e.
coloured
is a q u i n t u p l e
(A,X,Y,~,I) ,
is a f u n c t i o n to
kits,
automaton
(3)
is a k i t a n d
is r_egular if
U2G =
for all
(X,D,p)
is a c o l o u r e d
Xl,X~,X2,X ~
(Xl,X ~) C < U I G > & (x2,x ~) E < U I G > & (Xl,X 2) E p(d)
in
X
the
implies
(x{,x~) E p(d) •
UIG
An
a__bstract
is
a monokit.
A pregrame then
~
equivalence
is
a
game
(a ~ame
It (3)
is
is
surjection
classes
for
obvious
short) that
n__on-degenerate and
~-1({a}),
yields a CA.
is
every if
a pregame game
X ~ ~.
a partition
(3)
is If of
such
a regular it
the
is set
that pregame.
the
case,
Y
into
314
Every The
non-degenerate
elements
of
The e l e m e n t s
of
Active lose
players
in t h e
that
y
D
are passive
result
The e l e m e n t s
their
X
means
are
situation
x.
situation
The
players.
condition
(x,x') C p(d)
situation
x
The
over
notion
The
the
F =
where
and
(Ya) a C A
strategies subset
of
A
of
p : D
a.
cases
sense
the p a y o f f Every
and
~acAYa
that various
o f this
k(a,x)
game
and
have
of sets.
that each by
function;
player
d
the
prefers
the
equivalent
to
[6].
notion.
If,
that
A x X
>H(a,x)
noncooperative
and O.Morgenstern
is the
to e a c h
a-th
every
(3) o n e m a y
associate
game.
Y
game
(a g a m e
the e l e m e n t s H(a,x)
as an a b s t r a c t union
of is
game
of the sets
the c o r r e s p o n d i n g
index
x.
the c__anonical p r e s e n t a t i o n
precisely,
the V o r o b ' e v
that
x).
of the e l e m e n t
game has More
X =~acAYa such
(the n u m b e r
is the d i s j o i n t in
coordinate
abstract
f o r m of a V o r o b ' e v
Y y
[3]),
function
situation
(3), w h e r e
on
are
> H ( a , x ' ) ],
(4) c a n be p r e s e n t e d
form
of g a m e s A = D,
a in the
assigns
types
defined
of the p l a y e r
~
as above,
is a n o n - e m p t y
in p a r t i c u l a r ,
is the p a y o f f
game
X
H
(4) is a c l a s s i c a l
H
same meaning
is the s e t o f a l l
.
important
function
Neumann
the Ya
Vorob'ev
Conversely,
game
means
strategies
is e s s e n t i a l l y
it is a s s u m e d
product
[6]
of J . v o n
in the
(a E A ) ,
in t h e
here
. P o w ( X x X)
is a r e a l - v a l u e d
are players,
Ya
a in t h e
is the p r e f e r e n c e
by N.N.Vorob'ev
family
Moreover,
in
the V o r o b ' e v
G = @(F)
o f the
a.
means
x'
V a C A V x , x , C X [ (x,x ') 6 p(a)