## general theory of natural equivalences - Semantic Scholar

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GENERAL THEORY OF NATURAL EQUIVALENCES BY

SAMUEL EILENBERG AND SAUNDERS MacLANE

Contents Page 231

Introduction. I. Categories

and functors.

237

1. Definition of categories.

237

2. Examples of categories. 3. Functors in two arguments.

239 241

4. Examples of functors. 5. Slicing of functors. 6. Foundations. II. Natural equivalence of functors. 7. Transformations of functors. 8. Categories of functors.

242 245 246 248 248 250

9. 10. 11. 12. 13.

Composition of functors. Examples of transformations. Groups as categories. Construction of functors by transformations. Combination of the arguments of functors.

250 251 256 257 258

III. Functors and groups. 14. Subfunctors. 15. Quotient functors. 16. Examples of subfunctors. 17. The isomorphism theorems.

260 260 262 263 265

18. Direct products of functors.

267

19. Characters.

270

IV. Partially

ordered sets and projective

limits.

272

20. Quasi-ordered sets.

272

21. Direct systems as functors. 22. Inverse systems as functors.

273 276

23. The categories ®tr and 3n». 24. The lifting principle.

277 280

25. Functors

281

which commute

with limits.

V. Applications to topology. 26. Complexes. 27. Homology

and cohomology

283 283 groups.

284

28. Duality. 29. Universal

287 coefficient theorems.

288

30. Cech homology groups.

290

31. Miscellaneous remarks. Appendix. Representations of categories.

292 292

Introduction. The subject matter of this paper is best explained by an example, such as that of the relation between a vector space L and its "dual" Presented

to the Society, September

by the editors

231 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

May 15, 1945.

232

SAMUEL EILENBERG AND SAUNDERS MacLANE

[September

or "conjugate" space T(L). Let L be a finite-dimensional real vector space, while its conjugate TiL) is, as is customary, the vector space of all real valued linear functions t on L. Since this conjugate T{L) is in its turn a real vector space with the same dimension as L, it is clear that L and T(L) are isomorphic. But such an isomorphism cannot be exhibited until one chooses a definite set of basis vectors for L, and furthermore the isomorphism which results will differ for different choices of this basis. For the iterated conjugate space TiT{L)), on the other hand, it is well known that one can exhibit an isomorphism between L and T(TiL)) without using any special basis in L. This exhibition of the isomorphism L=T{TiL)) is "natural" in that it is given simultaneously for all finite-dimensional vector spaces L. This simultaneity can be further analyzed. Consider two finite-dimensional vector spaces Li and L2 and a linear transformation Xi of Li into L2;

in symbols

(1)

Xi: Li-*Lt.

This transformation Xx induces a corresponding linear transformation of the second conjugate space F(L2) into the first one, TiLi). Specifically, since each element t2 in the conjugate space T{L2) is itself a mapping, one has two transformations

Li->L2->

R;

their product ¿2Xiis thus a linear transformation of Li into R, hence an element ¿i in the conjugate space TiLi). We call this correspondence of t2 to h the

mapping F(Xi) induced by Xi; thus F(Xi) is defined by setting [r(Xi)]¿2 = ¿2Xi, so that

(2)