categorical incunabula

8 Jan 2002

      Herein is an attempt to list all MathSciNet publications through 1958
in that mention categories or functors. There are several
obstructions.

First of course is that we don't really want all mentions of
categories. Besides the sense relevant to this email address there are
the other mathematical senses (e.g. Baire, Lusternik-Schnirelmann),
the ancestral logical sense, and, of course, the ordinary sense. It is
not always completely clear which sense is being used in any given
review, so some of this is a matter of judgment. (As examples of
papers not included: 19,52d by Eilenberg and Tudor Ganea; and 21 #4425
by Ganea and P.J.Hilton. Both use the word "category" but only in the
L-S sense.  But all three authors were categorical in the relevant
sense.)

Next: MathSciNet when asked for all reviews that mention categories or
functor "anywhere" often responds with reviews that look as iff they
ought to be mentioning them but, in fact, do not. Very mysterious. I
have included all these (the topics always look right.)  

Finally, the use of the words in the review may well be on the part of
the categorically inclined reviewer, rather than the author.

By all means tell me what should be changed. Most important, of
course, are the omissions.

I have been surprised by a number of things. First is the 1942 paper
by Sammy and Saunders. Until now I had always assumed that both
functors and categories saw the first light of published day in the
1945 paper. Functors are celebrating their 60th birthday! I have
appended Weil's review of the 1942 paper.

The first person on the list after Eilenberg and Mac Lane is S-T Hu.
His 1947 paper defines "homotopy functor". I've appended Steenrod's
review.

Saunders's 1948 paper (without Sammy) first surprised me 40 years ago.
Buchsbaum in his 1955 paper that introduced abelian categories (under
the name "exact categories") said that he saw no way of defining
infinite products. Which meant that he hadn't seen Saunders's 1948
paper. Is this the first appearance of universal mapping definitions?
I've appended Philip Hall's review.

The 1956 paper by Atiyah is remarkable: it is true categorical
algebra. To quote Eilenberg's review (which I've appended) "The
Krull-Schmidt theorem asserting the existence and essential uniqueness
of direct sum decompositions into indecomposable factors is proved in
[abelian categories] satisfying a suitable chain condition."

In 1958 Andrew Gleason characterized the projective objects in the
category of compact Hausdorff spaces. I've appended Dana Scott's
review. (Which means that Dana was going categorical already as a grad
student.)

Most striking is the stellar nature of these early contributors and
consumers of category theory. At the very end of this posting is a
list of the 92 authors. (I count 4 Fields Medals, 3 National Medals of
Science, 3 Wolf Prizes, 5 Cole Prizes and 10 Steele Prizes.)

**********************************************************************
                                 1942
    4,134d EILENBERG, SAMUEL; MAC LANE, SAUNDERS. Natural isomorphisms
in group theory. Proc. Nat. Acad. Sci. U. S. A. 28, (1942). 537--543.

                                 1945
    7,109d EILENBERG, SAMUEL; MAC LANE, SAUNDERS. General theory of
natural equivalences. Trans. Amer. Math. Soc. 58, (1945). 231--294.

                                 1947
    9,297h HU, SZE-TSEN. An exposition of the relative homotopy
theory. Duke Math. J. 14, (1947). 991--1033.

                                 1948
    10,5e EILENBERG, SAMUEL; MAC LANE, SAUNDERS. Cohomology and Galois
theory. I. Normality of algebras and Teichm|ller's cocycle. Trans.
Amer. Math. Soc. 64, (1948). 1--20.
    10,9c MAC LANE, SAUNDERS. Groups, categories and duality. Proc.
Nat. Acad. Sci. U. S. A. 34, (1948). 263--267.
    10,621d WEIL, ANDRI Variitis abiliennes et courbes algibriques.
(French) Actualitis Sci. Ind., no. 1064 = Publ. Inst. Math. Univ. 
Strasbourg 8 (1946). Hermann & Cie., Paris, 1948 65 pp

                                 1951
    13,314c EILENBERG, SAMUEL; MAC LANE, SAUNDERS. Homology theories
for multiplicative systems. Trans. Amer. Math. Soc. 71, (1951).
294--330.
    13,440a CHEVALLEY, CLAUDE. Deux thiorhmes d'arithmitique. (French)
J. Math. Soc. Japan 3, (1951). 36--44. 

                                 1952
    14,398b EILENBERG, SAMUEL; STEENROD, NORMAN. Foundations of
algebraic topology. Princeton University Press, Princeton, New Jersey,
1952. xv+328 pp.

                                 1953
    14,670b EILENBERG, SAMUEL; MAC LANE, SAUNDERS. Acyclic models.
Amer. J. Math. 75, (1953). 189--199.
    15,53a MORITA, KIITI. Cohomotopy groups for fully normal spaces.
Sci. Rep. Tokyo Bunrika Daigaku. Sect. A. 4, (1953). 251--261.
    16,563b KDHLER, ERICH. Algebra und Differentialrechnung. (German)
Bericht |ber die Mathematiker-Tagung in Berlin, Januar, 1953, pp.
58--163. Deutscher Verlag der Wissenschaften, Berlin, 1953.

                                 1954
    15,816b KEESEE, JOHN W. Sets which separate spheres. Proc. Amer.
Math. Soc. 5, (1954). 193--200.
    16,391a EILENBERG, SAMUEL; MAC LANE, SAUNDERS. On the groups
$H(.Pi,n)$. II. Methods of computation. Ann. of Math. (2) 60, (1954).
49--139.
    16,442c EILENBERG, SAMUEL. Algebras of cohomologically finite
dimension. Comment. Math. Helv. 28, (1954). 310--319.

                                 1955
    16,564g EILENBERG, SAMUEL; MAC LANE, SAUNDERS. On the homology
theory of abelian groups. Canad. J. Math. 7, (1955). 43--53.
    17,579a AUSLANDER, MAURICE. On the dimension of modules and
algebras. III. Global dimension. Nagoya Math. J. 9 (1955), 67--77.
    17,579b BUCHSBAUM, D. A. Exact categories and duality. Trans.
Amer. Math. Soc. 80 (1955), 1--34.
    17,763c GROTHENDIECK, ALEXANDRE. Produits tensoriels topologiques
et espaces nucliaires. (French) Mem. Amer. Math. Soc. 1955 (1955), no.
16,
    18,558B MAC LANE, SAUNDERS. Slide and torsion products for modules.
Univ. e Politec. Torino. Rend. Sem. Mat. 15 (1955--56), 281--309.
    19,974F DUGUNDJI, J. Remark on homotopy inverses. Portugal. Math.
14 (1955), 39--41.

                                 1956
    17,994b MILNOR, JOHN. Construction of universal bundles. I. Ann.
of Math. (2) 63 (1956), 272--284.
    17,1040e CARTAN, HENRI; EILENBERG, SAMUEL. Homological algebra.
Princeton University Press, Princeton, N. J., 1956. xv+390 pp.
    17,1118c WHITEHEAD, J. H. C. Duality in topology. J. London Math.
Soc. 31 (1956), 134--148.
    18,57a ARAKI, SHTRT. On Steenrod's reduced powers in singular
homology theories. Mem. Fac. Sci. Ky{sy{ Univ. Ser. A. 9 (1956),
159--173.
    18,142e KAN, DANIEL M. Abstract homotopy. II Proc. Nat. Acad. Sci.
U.S.A. 42 (1956), 255--258.
    18,375d HARADA, MANABU. Note on the dimension of modules and
algebras. J. Inst. Polytech. Osaka City Univ. Ser. A. 7 (1956),
17--27.
    18,558c EILENBERG, SAMUEL. Homological dimension and syzygies.
Ann. of Math. (2) 64 (1956), 328--336.
    18,662c MCCANDLESS, BYRON H. Test spaces for dimension $n$. Proc.
Amer. Math. Soc. 7 (1956), 1126--1130.
    18,753b POSTNIKOV, M. M. Investigations in homotopy theory of
continuous mappings. III. General theorems of extension and
classification. (Russian) Mat. Sb. N.S. 40(82) (1956), 415--452.
    19,172b ATIYAH, M. On the Krull-Schmidt theorem with application
to sheaves. Bull. Soc. Math. France 84 (1956), 307--317.
    19,440a KAN, DANIEL M. Abstract homotopy. III. Proc. Nat. Acad.
Sci. U.S.A. 42 (1956), 419--421.
    19,522c Siminaire Paul Dubreil et Charles Pisot, 9e annie:
1955/56. Alghbre et thiorie des nombres. (French) Secritariat
mathimatique, 11 rue Pierre Curie, Paris, 1956. ii+213 pp.
    20 #1704 MORITA, KIITI; Tachikawa, Hiroyuki. Character modules,
submodules of a free module, and quasi-Frobenius rings. Math. Z. 65
1956
    20 #4204 DEDECKER, P. Quelques applications de la suite spectrale
aux intigrales multiples du calcul des variations et aux invariants
intigraux. II. (French) Bull. Soc. Roy. Sci. Lihge 25 1956 387--399.

                                 1957
     18,919b SPANIER, E. H.; WHITEHEAD, J. H. C. The theory of
carriers and $S$-theory. Algebraic geometry and topology. A symposium
in honor of S. Lefschetz, pp. 330--360. Princeton University Press,
Princeton, N.J., 1957.
    18,754c COPELAND, ARTHUR H., Jr. On $H$-spaces with two
non-trivial homotopy groups. Proc. Amer. Math. Soc. 8 (1957),
184--191.
    18,815d MILNOR, JOHN. The geometric realization of a
semi-simplicial complex. Ann. of Math. (2) 65 (1957), 357--362.
    19,160a GUGENHEIM, V. K. A. M.; MOORE, J. C. Acyclic models and
fibre spaces. Trans. Amer. Math. Soc. 85 (1957), 265--306.
    19,431d Siminaire "Sophus Lie" de la Faculti des Sciences de
Paris, 1955-56. Hyperalghbres et groupes de Lie formels. (French)
Secritariat mathimatique, 11 rue Pierre Curie, Paris, 1957. 61 pp.
    19,759d FORRESTER, AMASA. Acyclic models and de Rham's theorem.
Trans. Amer. Math. Soc. 85 (1957), 307--326.
    19,759e Kan, Daniel M. On c. s. s. complexes. Amer. J. Math. 79
(1957), 449--476.
    20 #892 MAC LANE, SAUNDERS. Homologie des anneaux et des modules.
(French) 1957 Colloque de topologie algibrique, Louvain, 1956 pp.
55--80 Georges Thone, Lihge; Masson & Cie, Paris
    20 #893 DIXMIER, J. Homologie des anneaux de Lie. (French) Ann.
Sci. Ecole Norm. Sup. (3) 74 1957 25--83.
    20 #894 MORITA, KIITI; Kawada, Yutaka; Tachikawa, Hiroyuki. On
injective modules. Math. Z. 68 1957 217--226.
    20 #896 ROSENKNOP, I. Z. On the H. Cartan algebra of a polynomial
ideal. (Russian) Dokl. Akad. Nauk SSSR (N.S.) 113 1957 1218--1221.
    20 #923 ISBELL, J. R. Some remarks concerning categories and
subspaces. Canad. J. Math. 9 1957 563--577.
    20 #1705 HATTORI, AKIRA. On $.Lambda $-injectivity (problem
6.3.19). (Japanese) S{gaku 8 1956/1957 208--209.
    20 #2392 EHRESMANN, CHARLES. Gattungen von lokalen Strukturen.
(German) Jber. Deutsch. Math. Verein. 60 1957 Abt. 1, 49--77.
    20 #2702 ZEEMAN, E. C. On the filtered differential group. Ann. of
Math. (2) 66 1957 557--585.
    20 #2703 KAN, DANIEL M. On the homotopy relation for c.s.s. maps.
Bol. Soc. Mat. Mexicana 2 1957 75--81.
    20 #2704 KAN, DANIEL M. On c.s.s. categories. Bol. Soc. Mat.
Mexicana 2 1957 82--94.
    20 #3201 BUCHSBAUM, DAVID. A survey of homological algebra. 1957
Report of a conference on linear algebras, June, 1956 pp. 53--59
National Academy of Sciences-National Research Council, Washington,
Publ. 502
    20 #3906 LEGER, GEORGE F., Jr. On cohomology of Lie algebras.
Proc. Amer. Math. Soc. 8 1957 1010--1020.
    20 #3907 SRIDHARAN, R. On some algebras of infinite cohomological
dimension. J. Indian Math. Soc. (N.S.) 21 1957 179--183.
    20 #4540 KUBOTA, TOMIO. Unit groups of cyclic extensions. Nagoya
Math. J. 12 1957 221--229.
    20 #4587 NAKAYAMA, TADASI. On modules of trivial cohomology over a
finite group. II. Finitely generated modules. Nagoya Math. J. 12 1957
171--176.
    20 #5229 EILENBERG, SAMUEL; ROSENBERG, ALEX; ZELINSKY, DANIEL. On
the dimension of modules and algebras. VIII. Dimension of tensor
products. Nagoya Math. J. 12 1957 71--93.
    20 #5485 HELLER, ALEX. Twisted ranks and Euler characteristics.
Illinois J. Math. 1 1957 562--564.
    20 #6449 NAKAYAMA, TADASI. On the complete cohomology theory of
Frobenius algebras. Osaka Math. J. 9 1957 165--187.
    20 #6452 TAKASU, SATORU. On the change of rings in the homological
algebra. J. Math. Soc. Japan 9 1957 315--329.
     22 #1887 WYLIE, S. Intercept-finite cell complexes. 1957
Algebraic geometry and topology. A symposium in honor of S. Lefschetz
pp. 389--399 Princeton University Press, Princeton, N.J.
    20 #6449 NAKAYAMA, TADASI. On the complete cohomology theory of
Frobenius algebras. Osaka Math. J. 9 1957 165--187.
    20 #6452 TAKASU, SATORU. On the change of rings in the homological
algebra. J. Math. Soc. Japan 9 1957 315--329.
    21 #1328 GROTHENDIECK, ALEXANDER. Sur quelques points d'alghbre
homologique. (French) Tthoku Math. J. (2) 9 1957 119--221.
    21 #2675 AMITSUR, S. A. The radical of field extensions. Bull.
Res. Council Israel. Sect. F 7F 1957/1958 1--10.
    21 #4417 PETERSON, FRANKLIN P. Functional cohomology operations.
Trans. Amer. Math. Soc. 86 1957 197--211.
    22 #12127 Siminaire A. Grothendieck; 1re annie: 1957. Alghbre
homologique. (French) Secritariat mathimatique, 11 rue Pierre Curie,
Paris 1958 42 pp. (mimeogiaphed).
    23 #A3163 LOONSTRA, F. Erweiterungen von Grenzgruppen. (German)
Nederl. Akad. Wetensch. Proc. Ser. A 60 = Indag. Math. 19 1957
548--559.
    23 #A3579 NAKAMURA, TOKUSI. Minimal complexes of fibre spaces. J.
Math. Soc. Japan 9 1957 1--19.
    25 #109 GOPALAKRISHNAN, N. S.; RAMABHADRAN, N.; Sridharan, R. A
note on the dimension of modules and algebras. J. Indian Math. Soc.
(N.S.) 21 1957 185--192.

                                 1958
    20 #895 TACHIKAWA, HIROYUKI. Duality theorem of character modules
for rings with minimum condition. Math. Z. 68 1958 479--487.
    20 #1661 BUZBY, B.; WHAPLES, G. Quadratic forms over arbitrary
fields. Proc. Amer. Math. Soc. 9(1958), 335--339; erratum 10 1958 174.
    20 #1712 GRIFFITHS, H. B. On limits of systems of groups. Proc.
Amer. Math. Soc. 9 1958 118--129.
    20 #2393 OHKUMA, TADASHI. Duality in mathematical structure. Proc.
Japan Acad. 34 1958 6--10.
    20 #2705 SHIH, WEISHU. Sur la suite exacte d'homotopie. (French)
C. R. Acad. Sci. Paris 246 1958 2833--2835.
    20 #3183 MORITA, KIITI. Duality for modules and its applications
to the theory of rings with minimum condition. Sci. Rep. Tokyo Kyoiku
Daigaku Sect. A 6 1958 83--142.
    20 #3202 DOLD, ALBRECHT; Puppe, Dieter. Non-additive functors,
their derived functors, and the suspension homomorphism. Proc. Nat.
Acad. Sci. U.S.A. 44 1958 1065--1068.
    20 #3203 AUSLANDER, MAURICE; BUCHSBAUM, DAVID A. Homological
dimension in noetherian rings. II. Trans. Amer. Math. Soc. 88 1958
194--206.
    20 #3537 DOLD, ALBRECHT. Homology of symmetric products and other
functors of complexes. Ann. of Math. (2) 68 1958 54--80.
    20 #3805 MAL.CPRIME CEV, A. I. Defining relations in categories.
(Russian) Dokl. Akad. Nauk SSSR 119 1958 1095--1098.
    20 #4257 GUTIIRREZ-BURZACO, MARIO. Extension of uniform
homotopies. Nederl. Akad. Wetensch. Proc. Ser. A 61 = Indag. Math. 20
1958 61--69.
    20 #4262 BAUER, FRIEDRICH-WILHELM. .ber Fortsetzungen von
Homologiestrukturen. (German) Math. Ann. 135 1958 93--114.
    20 #4264 BRAHANA, THOMAS R. Axioms for local homology theory. Duke
Math. J. 25 1958 381--399.
    20 #4588 HILTON, P. J. Homotopy theory of modules and duality.
1958 Symposium internacional de topologma algebraica International
symposium on algebraic topology pp. 273--281 Universidad Nacional
Autsnoma de Mixico and UNESCO, Mexico City
    20 #4658 CARTAN, HENRI. Espaces fibris analytiques. (French) 1958
Symposium internacional de topologma algebraica International
symposium on algebraic topology pp. 97--121 Universidad Nacional
Autsnoma de Mixico and UNESCO, Mexico City
    20 #4837 CARTAN, HENRI; Eilenberg, Samuel. Foundations of fibre
bundles. 1958 Symposium internacional de toplogma algebraica
International symposium on algebraic topology pp. 16--23 Universidad
Nacional Autsnoma de Mixico and UNESCO, Mexico City
    20 #5192 AURORA, SILVIO. On power multiplicative norms. Amer. J.
Math. 80 1958 879--894.
    20 #5194 NORTHCOTT, D. G. A note on polynomial rings. J. London
Math. Soc. 33 1958 36--39.
    20 #5228 MAC LANE, SAUNDERS. Extensions and obstructions for
rings. Illinois J. Math. 2 1958 316--345.
    20 #5800 MATLIS, EBEN. Injective modules over Noetherian rings.
Pacific J. Math. 8 1958 511--528.
    20 #5979 SPENCER, D. C. A spectral resolution of complex
structure. 1958 Symposium internacional de topologma algebraica
International symposium on algebraic topology pp. 68--76 Universidad
Nacional Autsnoma de Mixico and UNESCO, Mexico City
    20 #6092 MILNOR, John. The Steenrod algebra and its dual. Ann. of
Math. (2) 67 1958 150--171.
    20 #6414 AUSLANDER, MAURICE; BUCHSBAUM, DAVID A. Codimension and
multiplicity. Ann. of Math. (2) 68 1958 625--657.
    20 #6450 NAKAYAMA, TADASI. Note on complete cohomology of a
quasi-Frobenius algebra. Nagoya Math. J. 13 1958 115--121.
    20 #6451 HOCHSCHILD, G. Note on relative homological dimension.
Nagoya Math. J. 13 1958 89--94.
    20 #6453 KAPLANSKY, IRVING. Projective modules. Ann. of Math (2)
68 1958 372--377.
    20 #6460 BAER, REINHOLD. Die Torsionsuntergruppe einer Abelschen
Gruppe. (German) Math. Ann. 135 1958 219--234.
    20 #6461 ERDVS, JENV. On the splitting problem of mixed abelian
groups. Publ. Math. Debrecen 5 1958 364--377.
    20 #6694 ECKMANN, BENO; HILTON, PETER J. Groupes d'homotopie et
dualiti. Groupes absolus. (French) C. R. Acad. Sci. Paris 246 1958
2444--2447.
    20 #6698 PUPPE, DIETER. Homotopiemengen und ihre induzierten
Abbildungen. I. (German) Math. Z. 69 1958 299--344.
    20 #7045 HARADA, MANABU; KANZAKI, Teruo. On Kronecker products of
primitive algebras. J. Inst. Polytech. Osaka City Univ. Ser. A 9 1958
19--28.
    20 #7048 BER.V STE.U.I N, ISRAKL. On the dimension of modules and
algebras. IX. Direct limits. Nagoya Math. J. 13 1958 83--84.
    20 #7049 KAPLANSKY, IRVING. On the dimension of modules and
algebras. X. A right hereditary ring which is not left hereditary.
Nagoya Math. J. 13 1958 85--88.
    20 #7050a HILTON, P. J.; LEDERMANN, W. Homology and ringoids. I.
Proc. Cambridge Philos. Soc. 54 1958 152--167.
    20 #7050b HILTON, P. J.; LEDERMANN, W. Homological ringoids.
Colloq. Math. 6 1958 177--186.
    20 #7051 HELLER, ALEX. Homological algebra in abelian categories.
Ann. of Math. (2) 68 1958 484--525.
    21 #77 NORGUET, FRANGOIS. Sur l'homologie associie ` une famille
de dirivations. (French) C. R. Acad. Sci. Paris 247 1958 1081--1083.
    21 #79 HARADA, MANABU. The weak dimension of algebras and its
applications. J. Inst. Polytech. Osaka City Univ. Ser. A 9 1958
47--58.
    21 #1317 HARADA, MANABU. A note on Hattori's theorems. J. Inst.
Polytech. Osaka City Univ. Ser. A 9 1958 43--45.
    21 #1583 GODEMENT, ROGER. Topologie algibrique et thiorie des
faisceaux. (French) Actualit'es Sci. Ind. No. 1252. Publ. Math. Univ.
Strasbourg. No. 13 Hermann, Paris 1958 viii+283 pp.
    21 #1598 .V SVARC, A. S. The genus of a fiber space. (Russian)
Dokl. Akad. Nauk SSSR (N.S.) 119 1958 219--222.
    21 #2233 WHITEHEAD, J. H. C. Duality between $CW$-lattices. 1958
Symposium internacional de topologma algebraica International symposi
um on algebraic topology pp. 248--258 Universidad Nacional Autsnoma de
Mixico and UNESCO, Mexico City
    21 #2234 SPANIER, E. H. Duality and the suspension category. 1958
Symposium internacional de topologma algebraica International symposi
um on algebraic topology pp. 259--272 Universidad Nacional Autsnoma de
Mixico and UNESCO, Mexico City
    21 #2236 VAN EST, W. T. A generalization of the Cartan-Leray
spectral sequence. I, II. Nederl. Akad. Wetensch. Proc. Ser. A 61 =
Indag. Math. 20 1958 399--413.
    21 #2680a BOCKSTEIN, MEYER. Sur le spectre d'homologie d'un
complexe. (French) C. R. Acad. Sci. Paris 247 1958 259--261.
    21 #2680b BOCKSTEIN, MEYER. Sur la formule des coefficients
universels pour les groupes d'homologie. (French) C. R. Acad. Sci.
Paris 247 1958 396--398.
    21 #2980 HU, SZE-TSEN. Algebraic local invariants of topological
spaces. Compositio Math. 13 1958 173--218 (1958).
    21 #3471 NAKAYAMA, TADASI. On algebras with complete homology.
Abh. Math. Sem. Univ. Hamburg 22 1958 300--307.
    21 #3838 DARBO, GABRIELE. Teoria dell'omologia in una categoria di
mappe plurivalenti ponderate. (Italian) Rend. Sem. Mat. Univ. Padova
28 1958 188--220.
    21 #3850 SPANIER, E. H.; WHITEHEAD, J. H. C. Duality in relative
homotopy theory. Ann. of Math. (2) 67 1958 203--238.
    21 #4176 ROSENBERG, ALEX; ZELINSKY, DANIEL. Finiteness of the
injective hull. Math. Z. 70 1958/1959 372--380.
    21 #4421 SCHUBERT, HORST. Semisimpliziale Komplexe. (German) Jber.
Deutsch. Math. Verein 61 1958 Abt. 1, 126--138.
    21 #4960 LANG, SERGE; TATE, JOHN. Principal homogeneous spaces
over abelian varieties. Amer. J. Math. 80 1958 659--684.
    21 #5189 ECKMANN, BENO. Groupes d'homotopie et dualiti. (French)
Bull. Soc. Math. France 86 1958 271--281.
    21 #5196 POENARU, VALENTIN. Considirations sur les variitis
simplement connexes ` $3$ dimensions. (French) Rev. Math. Pures Appl.
3 1958 139--156.
    21 #5668 NAKAYAMA, TADASI. Note on fundamental exact sequences in
homology and cohomology for non-normal subgroups. Proc. Japan Acad. 34
1958 661--663.
    22 #1898 KAN, DANIEL M. On homotopy theory and c.s.s. groups. Ann.
of Math. (2) 68 1958 38--53.
    22 #1899 KAN, DANIEL M. An axiomatization of the homotopy groups.
Illinois J. Math. 2 1958 548--566.
    22 #1900 KAN, DANIEL M. On monoids and their dual. Bol. Soc. Mat.
Mexicana (2) 3 1958 52--61.
    22 #61 BERSTEIN, I. Geometric dimension of abelian groups.
(Russian) Rev. Math. Pures Appl. 3 1958 93--99.
    22 #6817 BOREL, ARMAND; SERRE, JEAN-PIERRE. Le thiorhme de
Riemann-Roch. (French) Bull. Soc. Math. France 86 1958 97--136.
    22 #6818 GROTHENDIECK, ALEXANDER. La thiorie des classes de Chern.
(French) Bull. Soc. Math. France 86 1958 137--154.
    22 #6835 KUNIYOSHI, HIDEO. Cohomology theory and different. Tthoku
Math. J. (2) 10 1958 313--337.
    22 #6836 KUNIYOSHI, HIDEO. On the cohomology groups of ${.germ
p}$-adic number fields. Proc. Japan Acad. 34 1958 609--611.
    22 #12509 GLEASON, ANDREW M. Projective topological spaces.
Illinois J. Math. 2 1958 482--489.
    23 #A3569 KAN, DANIEL M. Minimal free c.s.s. groups. Illinois J.
Math. 2 1958 537--547.
    24 #A1301 KAN, DANIEL M. Adjoint functors. Trans. Amer. Math. Soc.
87 1958 294--329.
    24 #A1720 KAN, DANIEL M. Functors involving c.s.s. complexes.
Trans. Amer. Math. Soc. 87 1958 330--346.
    24 #B416 ROSEN, ROBERT. The representation of biological systems
for the stand-point of the theory of categories. Bull. Math. Biophys.
20 1958 317--342.
    27 #4851 HEATON, R.; WHAPLES, G. Polynomial cocycles. Duke Math.
J. 25 1958 691--696.
    31 #233 YONEDA, NOBUO. Note on products in ${.rm Ext}$. Proc.
Amer. Math. Soc. 9 1958 873--875.

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4,134d 20.0X 
Eilenberg, Samuel; Mac Lane, Saunders 
Natural isomorphisms in group theory. 
Proc. Nat. Acad. Sci. U. S. A. 28, (1942). 537--543. 

A vague idea of covariance and contravariance is often met with in
group-theory, topology, etc.; that is, one feels that the character-
group is contravariant to the group, that the homology and co-homology
groups of a complex are, respectively, covariant and contravariant to
the complex. This is of special importance in the building up of
limits of direct and inverse systems ("projective" and "inductive"
limits) of groups, spaces, etc. The authors have succeeded in finding
for this a precise definition, which is likely to be helpful in
classifying and systematizing known results and also in looking for
new relations between groups. In this note, they give a brief sketch
of their method, for groups only. The main idea is that of a functor,
which will best be explained by an example: for them, the definition
of the character-group to an Abelian group $G$ is only one half of the
definition of a functor, which they call $Ch (G)$, the other half
being the (obvious) rule by which any homomorphism of $G$ into another
group $H$ determines a homomorphism of the character-group of $H$ into
the character-group of $G$. Generally speaking, a functor, associated
with some groups $G\*sb 1,G\sb 2,\cdots$, consists of the definition
of some associated group, together with a rule indicating that the
latter behaves in a certain prescribed fashion under homomorphic
transformations affecting $G\sb 1,G\sb 2,\cdots$. Examples are given
to illustrate this concept; in particular, the authors use it to
derive some interesting relations concerning Whitney's "tensor-
product" of groups, and clarify the nature of the latter.

                                                  Reviewed by A. Weil 

9,297h 56.0X 
Hu, Sze-tsen 
An exposition of the relative homotopy theory. 
Duke Math. J. 14, (1947). 991--1033. 

Although a large amount of knowledge has accumulated about the
homotopy groups of Hurewicz, this is the first organized account of
the topic. Both the absolute and relative homotopy groups are defined
and their basic group properties established. The "homotopy sequence"
of a pair $(Y,Y\sb 0)$ is proved to be exact, and is shown to be a
covariant functor under mappings. The operations of $\pi\sp 1(Y\sb 0)$
on $\pi\sp n(Y,Y\sb 0)$ are defined and the question of simplicity is
studied. The Hurewicz theorem is proved in the relative case: $\pi\sp
n(Y,Y\sb 0)\approx H\sp n(Y,Y\sb 0)$ if $(Y,Y\sb 0)$ is $r$-aspherical
for $r<n$, and is $n$-simple. The Hurewicz isomorphisms relating the
groups of $Y$ to those of certain function spaces over $Y$ are
extended to the relative case. Indeed the isomorphisms provide an
isomorphism of the homotopy sequences. The paper does not consider the
products defined by J. H. C. Whitehead [Ann. of Math. (2) 42, 409--428
(1941); these Rev. 2, 323] or the torus homotopy groups of R. H. Fox
[Proc. Nat. Acad. Sci. U. S. A. 31, 71--74 (1945); these Rev. 6, 279].

                                            Reviewed by N. E. Steenrod

10,9c 20.0X 
Mac Lane, Saunders 
Groups, categories and duality. 
Proc. Nat. Acad. Sci. U. S. A. 34, (1948). 263--267. 

The direct product and the free product of two groups are defined
abstractly in terms of homomorphisms, the two definitions being
formally deducible one from the other by applying the following
"duality rules": invert the direction of each homomorphism, invert the
order of all products of homomorphisms, interchange homomorphisms onto
with isomorphisms into. The same duality is observed to hold between
free Abelian groups and infinitely divisible Abelian groups. The
author aims to formulate these and other similar duality relations of
group theory axiomatically. This is done by a refinement of the notion
of category, originally introduced by Eilenberg and MacLane [Trans.
Amer. Math. Soc. 58, 231--294 (1945); these Rev. 7, 109]. A category
is a class of entities called "mappings" (e.g., homomorphisms) in
which the products of certain pairs of mappings are defined and
satisfy certain axioms (conditional existence and associativity of
products, existence of "identities"). A bicategory is now defined to
be a category with two (dual) distinguished classes of mappings,
called injections and projections, which satisfy certain simple
additional postulates. A group can be interpreted as a bicategory with
one identity, the mappings of the category being the elements of the
group. A lattice can be interpreted as a bicategory whose mappings are
all injections: the mappings are the pairs $[a,b]$ of lattice elements
such that $a\supset b$, with product $[a,b][b,c]=[a,c]$. The author
states that "most of the phenomena of universal algebra and of
(axiomatic) group duality have appropriate and simple formulations in
terms of lattice-ordered bicategories." Here, lattice-ordered
bicategories are a special class of bicategories which include both
groups and lattices (interpreted as above).

                                                  Reviewed by P. Hall 

19,172b 53.3X 
Atiyah, M. 
On the Krull-Schmidt theorem with application to sheaves. 
Bull. Soc. Math. France 84 (1956), 307--317. 

The Krull-Schmidt theorem asserting the existence and essential
uniqueness of direct sum decompositions into indecomposable factors is
proved in exact categories in the sense of Buchsbaum [Trans. Amer.
Math. Soc. 80 (1955), 1--34; MR 17, 579] satisfying a suitable chain
condition. As an application it is shown that the Krull-Schmidt
theorem holds in the class of vector bundles over a connected complete
algebraic variety or over a connected compact complex manifold. This
is achieved by showing that the exact category whose objects are
suitable sheaves satisfies the chain conditions.

                                             Reviewed by S. Eilenberg 

22 #12509 54.00 
Gleason, Andrew M. 
Projective topological spaces. 
Illinois J. Math. 2 1958 482--489. 

In the usual terminology of homological algebra the author shows that
in the category of all compact Hausdorff spaces and all continuous
maps the projective objects (i.e., spaces) are precisely the
extremally disconnected spaces (i.e., those spaces where the closures
of open sets are open). Further, every object in this category is the
image of a projective object, namely, the compact space $X$ is a
continuous image of the Stone space of the complete Boolean algebra of
all regular closed subsets of $X$ (a set is regular closed, or a
closed domain, if it is equal to the closure of its interior). This
specific projective resolution is shown to be unique in a suitable
sense. The method of proof reminds the reviewer of the Bourbaki use of
ultrafilters in characterizing compact spaces. Also Lemma 3.1 (the
regular closed sets form a complete Boolean algebra) is not new [see,
e.g., J. C. C. McKinsey and A. Tarski, Ann. of Math. (2) 47 (1946),
122--161; MR 7, 359]. The analogous results are stated for the
category of locally compact spaces (warning: do not use all continuous
maps) and then by duality for commutative $C\sp *$ algebras. The dual
result for the category of compact totally disconnected spaces (i.e.,
for Boolean algebras) had been obtained earlier by R. Sikorski, as the
author points out.

                                                Reviewed by Dana Scott

**********************************************************************

Amitsur, S. A. 57
Araki, Shtrt 56
Atiyah, M. 56
Aurora, Silvio 58
Auslander, Maurice 55, 58 (2)
Baer, Reinhold 58
Bauer, Friedrich-Wilhelm 58
Ber\v ste\u\i n, Israkl 58
Berstein, I 58
Bockstein, Meyer 58 (20
Borel, Armand 58
Brahana, Thomas R. 58
Buchsbaum, D. A. 55, 57, 58 (20
Buzby, B. 58
Cartan, Henri 56, 58 (2)
Chevalley, Claude. 51
Copeland, Arthur H. 57
Darbo, Gabriele 58
Dedecker, P 56
Dixmier, J. 57
Dold, Albrecht 58 (2)
Dugundji, J. 55
Eckmann, Beno 58 (2)
Ehresmann, Charles 57
Eilenberg, Samuel 42, 45 48, 51, 52, 53, 54 (2), 55, 56 (2), 57, 58  [t = 13]
Erdvs, Jenv 58
Forrester, Amasa 57
Gleason, Andrew M 58
Godement, Roger 58
Gopalakrishnan, N. S. 57
Griffiths, H. B. 58
Grothendieck, Alexander 55, 57, 58
Gugenheim, V. K. A. M. 57
Gutiirrez-Burzaco, Mario 58
Harada, Manabu 56, 58 (3)
Hattori, Akira 57
Heaton, R. 58
Heller, Alex 57, 58
Hilton, P. J. 58 (4)
Hochschild, G. 58
Hu, Sze-tsen 47, 58
Isbell, J. R. 57
Kan, Daniel M  56, 57 (2), 58 (6) [t = 9]
Kanzaki, Teruo 58
Kaplansky, Irving 58 (2)
Kawada, Yutaka 57
Kdhler, Erich 53
Keesee, John W 54
Kubota, Tomio. 57
Kuniyoshi, Hideo 58 (2)
Lang, Serge 58
Ledermann, W. 58 (2)
Leger, George F., Jr. 57
Loonstra, F. 57
Mac Lane, Saunders 42, 45, 48 (2), 51, 53 ,54, 55 (2), 57, 58  [t = 11]
Mal\cprime cev, A. I. 58
Matlis, Eben 58
McCandless, Byron H. 56
Milnor, John 56, 57, 58
Moore, J. C. 57
Morita, Kiiti 53, 56, 57, 58
Nakamura, Tokusi 57 (4), 58 (4) [t = 8]
Norguet, Frangois 58
Northcott, D. G. 58
O'Neill, Barrett 58
Ohkuma, Tadashi 58
Peterson, Franklin P. 57
Poenaru, Valentin 58
Postnikov, M. M. 56
Puppe, Dieter 58 (2)
Ramabhadran, N. 57
Rosen, Robert 58
Rosenberg, Alex 57, 58
Rosenknop, I. Z. 57
Schubert, Horst 58
Serre, Jean-Pierre 58
Shih, Weishu 58
Spanier, E. H. 57, 58 (3)
Sridharan, R. 57 (2)
Steenrod, Norman. 52
Tachikawa, Hiroyuki 56, 57, 58
Takasu, Satoru 57 (2)
Tate, John 58
Weil, Andri 48
Whaples, G. 58 (2)
Whitehead, J. H. C.  56, 57, 58 (2)
Wylie, S. 57
Yoneda, Nobuo 58
Zeeman, E. C. 57
Zelinsky, Daniel 57, 58
\v Svarc, A. S 58
van Est, W. T. 58

10-Jan-2002 10:02:06 -0400,1470;000000000000-00000000

categorical incunabula

Peter Freyd