Table Of ContentProceedings of the Conference on Categorical Algebra
This Conference was supported by the
United States Air Force Office of Scientific Research
Proceedings of the
Conference on Categorical Algebra
La Jolla 1965
Edited by
S. Eilenberg . D. K. Harrison . S. MacLane . H. Rohrl
Springer-Verlag Berlin . Heidelberg' New York 1966
Professor Dr. S. ElLENBERG
Columbia University, Hamilton Hall. New York 27, N. Y.
Professor Dr. D. K. HARRISON
Department of Mathematics, University of Oregon, Eugene, are.
Professor Dr. S. MACLANE
Department of Mathematics, The University of Chicago, Chicago, JIl.
Profe.sor Dr. H. RilHRL
Department of Mathematics, University of California at San Diego, La Jolla
ISBN-13: 978-3-642-99904-8 e-ISBN-13: 978-3-642-99902-4
001: 10.1007/978-3-642-99902-4
All rights, especially that of translation into foreign languages,reserved. It is also forbidden to repro-
duce this book, either whole or in part, by photomechanical means (photostat, microfilm and/or
microcard) or by other procedure without written permission from Springer-Verlag.
© by Springer-Verlag Berlin' Heidelberg 1966· Library of Congress Catalog Card Number 66-14575'
Softcover reprint of the hardcover 1st edition 1966
Title No. 1339
Preface
This volume contains the articles contributed to the Conference on
Categorical Algebra, held June 7-12,1965, at the San Diego campus of
the University of California under the sponsorship of the United States
Air Force Office of Scientific Research. Of the thirty-seven mathemati-
cians, who were present seventeen presented their papers in the form of
lectures. In addition, this volume contains papers contributed by other
attending participants as well as by those who, after having planned to
attend, were unable to do so.
The editors hope to have achieved a representative, if incomplete, cover-
age of the present activities in Categorical Algebra within the United
States by bringing together this group of mathematicians and by solici-
ting the articles contained in this volume. They also hope that these
Proceedings indicate the trend of research in Categorical Algebra in
this country.
In conclusion, the editors wish to thank the participants and contrib.
utors to these Proceedings for their continuous cooperation and encour-
agement. Our thanks are also due to the Springer-Verlag for publishing
these Proceedings in a surprisingly short time after receiving the manu-
scripts.
S. ElLENBERG
D. K. HARRISON
S. MACLANE
H. ROHRL
Resolved
The members of the Conference on Categorical Algebra wish to ex-
press their deep appreciation to those who have made this conference
possible:
To the University of California at San Diego and especially to Pro-
fessors ECKART, GOLDBERG, STEWART and WARSCHAWSKI for holding
the conference in this idyllic setting,
To the Air Force Office of Scientific Research, and especially to Dr.
R. J. POHRER, for providing the support and encouragement necessary
for the conference,
To Mrs. ILSE WARSCHAWSKI and Mrs. VIVIAN ROHRL, for hospitable
reception and entertainment,
To Mrs. BARI SACCOMAN, for daily assistance with many practical
problems,
Finally and most especially to Professor HELMUT ROHRL, for his
manifest mastery of manifold arrangements.
Unanimously adopted
June 11, 1965
Contents
LAWVERE, F. W.: The Category of Categories as a Foundation for Mathema-
tics . . . . . . . . . . . . . . . . . . . . . . 1
GRAY, J. W.: Fibred and Cofibred Categories ..... 21
LINTON, F. E. J. : Some Aspects of Equational Categories 84
FREYD, P.: Representation in Abelian Categories 95
FREYD, P.: Stable Homotopy . . . . . . . 121
FREYD, P.: Splitting Homotopy Idempotents 173
FABER, R., and P. Freyd: Fill-in Theorems 177
AUSLANDER, 1\1.: Coherent Functors. . . . 189
ISBELL, J. R.: Epimorphisms and Dominions 232
ENGELER, E.: Categories of Mapping Filters 247
HILTON, P.: Correspondences and Exact Squares 254
SONNER, J.: Canonical Categories . . . . . . . 272
WYLER, 0.: Operational Categories . . . . . . 295
GIVE'ON, Y.: Transparent Categories and Categories of Transition Systems 317
WATTS, C. E.: A Homology Theory for Small Categories 331
BARR, M., and J. BECK: Acyclic Models and Triples 336
BARR, M.: Cohomology in Tensored Categories. 344
HELLER, A.: Extraordinary Homology and Chain Complexes . 355
DICKSON, S.: Direct Decomposition of Radicals. . . . . . . 366
CHASE, S. U.: Abelian Extensions and a Cohomology Theory by Harrison 375
WALKER, C. L., and E. A. WALKER: Quotient Categories of Modules. 404
ElLENBERG, S., and G. M. KELLY: Closed Categories ....... 421
List of Participants
AUSLANDER, B. L. FREYD, P. LINTON, F. E. J.
AUSLANDER, M. GIVE' ON, Y. MAC LANE, S.
BASS, H. GRAY, J. W. MORIMOTO, A.
BECK, J. HARRISON, D. K. RINEHART, G. S.
BUCHSBAUM, D. HELLER, A. ROHRL, H.
CHASE, S. U. ISBELL, J. R. SONNER, J.
DICKSON, S. E. KAN, D.M. TIERNEY, M.
DIENER, K. H. KELLY, G.M. VERDIER, J. L.
DUSKIN, J. W. KNIGHTEN, C. M. WALKER, C. L.
DYSON, V.H. KNIGHTEN, R. L. WALKER, E. A.
ElLENBERG, S. LAWVERE, F. \V. WATTS, C. E.
ENGELER, E. LEICHT, J. B. WYLER, O.
FABER, R.
The Category of Categories as a Foundation
for Mathematics *, **
By
F. WILLIAM LAW VERE
In the mathematical development of recent decades one sees clearly
the rise of the conviction that the relevant properties of mathematical
objects are those which can be stated in terms of their abstract structure
rather than in terms of the elements which the objects were thought to
be made of. The question thus naturally arises whether one can give
a foundation for mathematics which expresses wholeheartedly this con-
viction concerning what mathematics is about, and in particular in which
classes and membership in classes do not play any role. Here by "founda-
tion" we mean a single system of first-order axioms in which all usual
mathematical objects can be defined and all their usual properties proved.
A foundation of the sort we have in mind would seemingly be much more
natural and readily-useable than the classical one when developing such
subjects as algebraic topology, functional analysis, model theory of gene-
ral algebraic systems, etc. Clearly any such foundation would have to
reckon with the Eilenberg-MacLane theory of categories and functors.
The author believes, in fact, that the most reasonable way to arrive at
a foundation meeting these requirements is simply to write down axioms
descriptive of properties which the intuitively-conceived category of all
categories has until an intuitively-adequate list is attained; that is es-
sentially how the theory described below was arrived at. Various meta-
theorems should of course then be proved to help justify the feeling of
adequacy. The system to be described is an improved version of the one
sketched in Chapter 1 of the author's doctoral dissertation [Columbia,
1963].
By the elementary theory of aiJstract categories we mean the notions of
formula and theorem defined as follows
O. For any letters x, y, u, A, B the following are formulas
Llo(x)=A, LlI(x}=B, r(x,y;u), x=y.
*
Research partially supported by an NSF-NATO Postdoctoral Fellowship.
** Received September 8, 1965
Conference on Categorical Algebra 1
S. Eilenberg et al. (eds.), Proceedings of the Conference on Categorical Algebra
© Springer-Verlag Berlin · Heidelberg 1966
2 F. W. LAWVERE
These are to be read, respectively, "A is the domain of x", "B is the
codomain of x", "u is the composition x followed by y", and "x equals y".
1. If l/J and lJI are formulas, then
[l/J] and [lJI]
[l/J] or [lJI]
[l /J] => [lJI]
not [l/J]
are also formulas.
2. If l/J is a formula and x is a letter, then
'v'x[l/J] , 3x[l/J]
are also formulas. These are to be read, as usual, "for every x, l/J" and
"there is an x such that l/J", respectively.
3. A string of marks is a formula of the elementary theory of abstract
categories iff its being so follows from 0, 1, 2 above. Of course we im-
mediately begin to make free use of various ways of abbreviating for-
mulas. The notion of free and bound variables in a formula can now be
defined; we mean by a sentence any formula with no free variables, i. e.
in which every occurence of each letter x is within the scope of a quan-
tifier 'v' x or 3x.
The theorems of the elementary theory of abstract categories are all
those sentences which can be derived by logical inference from the
following axioms (it is understood that .10, .11 are unary function
symbols)
Four bookkeeping axioms
.1d.1j (x)) = .1j (x) , i, j = 0, 1.
r(x, y; u) and r(x, y; u') => u = u',
3u[F(x, y; u)] <==> .1dx) = .10(Y) ,
r(x, y; u) => .1o(u) = .10 (x) and .1t{u) = .1dy).
Identity axiom
r(.1o(x),x;x) and r(X,.11(X);X).
Associativity axiom
r(x, y; u) and r(y, z; w) and
r(x, w; f) and r(u, z; g) => f = g.
Besides the usual means of abbreviating formulas, the following (as
well as others) are special to the elementary theory of abstract categories:
A (cid:0) (cid:0)B means .10(/) = A and .1df) = B,
