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General Theory of Natural Equivalences Author(S): Samuel Eilenberg and Saunders Maclane Source: Transactions of the American Mathematical Society, Vol American Mathematical Society General Theory of Natural Equivalences Author(s): Samuel Eilenberg and Saunders MacLane Source: Transactions of the American Mathematical Society, Vol. 58, No. 2 (Sep., 1945), pp. 231- 294 Published by: American Mathematical Society Stable URL: http://www.jstor.org/stable/1990284 Accessed: 08-10-2015 12:00 UTC Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://www.jstor.org/page/ info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. American Mathematical Society is collaborating with JSTOR to digitize, preserve and extend access to Transactions of the American Mathematical Society. http://www.jstor.org This content downloaded from 128.151.244.46 on Thu, 08 Oct 2015 12:00:15 UTC All use subject to JSTOR Terms and Conditions GENERAL THEORY OF NATURAL EQUIVALENCES BY SAMUEL EILENBERG AND SAUNDERS MAcLANE CONTENTS Page Introduction ......................................................... 231 I. Categories and functors...................................................... 237 1. Definitionof categories................................................... 237 2. Examples of categories.................................................... 239 3. Functors in two arguments................................................ 241 4. Examples of functors..................................................... 242 5. Slicing of functors........................................................ 245 6. Foundations......... 246 II. Natural equivalence of functors .............................................. 248 7. Transformationsof functors............................................... 248 8. Categories of functors.................................................... 250 9. Composition of functors........................ 250 10. Examples of transformations........................ 251 11. Groups as categories................. 256 12. Constructionof functorsby transformations........ ......................... 257 13. Combination of the argumentsof functors................................... 258 III. Functors and groups....................................................... 260 14. Subfunctors............... 260 15. Quotient functors............... 262 16. Examples of subfunctors.................................................. 263 17. The isomorphismtheorems .. , . .................................. 265 18. Direct products of functors.. , . .................................. 267 19. Characters......... 270 IV. Partially ordered sets and projective limits.................... ... ............. 272 20. Quasi-orderedsets . ........................................................ 272 21. Direct systemsas functors.......... ...................................... 273 22. Inverse systemsas functors............... 276 23. The categoriesZir and jnt............................................... 277 24. The liftingprinciple ... ............. 280 25. Functors which commute with limits .......... ............... .............. 281 V. Applications to topology.................................................... 283 26. Complexes.............................................................. 283 27. Homology and cohomologygroups . 284 28. Duality ................................................................. 287 29. Universal coefficienttheorems ............................................. 288 30. tech homology groups. ................................................... 290 31. Miscellaneous remarks ..................... 292 Appendix. Representationsof categories. ..............................,,,.292 Introduction.The subject matter of this paper is best explained by an example, such as that of the relationbetween a vector space L and its "dual" Presented to the Society, September 8, 1942; received by the editors May 15, 1945. 231 This content downloaded from 128.151.244.46 on Thu, 08 Oct 2015 12:00:15 UTC All use subject to JSTOR Terms and Conditions 232 SAMUEL EILENBERG AND SAUNDERS MAcLANE [September or "conjugate" space T(L). Let L be a finite-dimensionalreal vector space, while its conjugate T(L) is, as is custoiiary, the vectorspace of all real valued linear functionst on L. Since this conjugate T(L) is in its turn a real vector space with the same dimensionas L, it is clear that L and T(L) are isomor- phic. But such an isomorphismcannot be exhibited until one chooses a defi- nite set of basis vectorsfor L, and furthermorethe isomorphismwhich results will differfor different choices of this basis. For the iterated conjugate space T(T(L)), on the other hand, it is well known that one can exhibitan isomorphismbetween L and T(T(L)) without using any special basis in L. This exhibitionof the isomorphismL T(T(L)) is "natural" in that it is given simultaneouslyfor all finite-dimensionalvector spaces L. This simultaneitycan be furtheranalyzed. Consider two finite-dimen- sional vector spaces L1 and L2 and a linear transformationX1 of L1 into L2; in symbols (1) X1: L1-+L2. This transformationX1 induces a correspondinglinear transformationof the second conjugate space T(L2) into the firstone, T(L1). Specifically,since each elementt2 in the conjugate space T(L2) is itselfa mapping,one has two trans- formations X L L2 R; theirproduct t2X1 is thus a lineartransformation of L1 into R, hence an element t1 in the conjugate space T(L1). We call this correspondenceof t2 to t1 the mapping T(X1) inducedby Xi; thus T(X1) is definedby setting [T(X1)]t2 =t2X1, so that (2) T(Xi): T(L2) -+ T(L1). In particular,this induced transformationT(X1) is simply the identitywhen X1 is given as the identity transformationof L1 into L1. Furthermorethe transformationinduced by a product of X's is the product of the separately induced transformations,for if X1maps L1 into L2 while X2 maps L2 into L3, the definitionof T(X) shows that T(X2X1) = T(X1)T(X2). The process of formingthe conjugate space thus actually involves two differ- ent operations or functions.The firstassociates with each space L its con- jugate space T(L); the second associates with each linear transformationX between vector spaces its induced linear transformationT(X)(1). (1) The two differentfunctions T(L) and T(X) may be safely denoted by the same letter T because theirarguments L and X are always typographicallydistinct. This content downloaded from 128.151.244.46 on Thu, 08 Oct 2015 12:00:15 UTC All use subject to JSTOR Terms and Conditions 1945] GENERAL THEORY OF NATURAL EQUIVALENCES 233 A discussion of the "simultaneoils" or "natural" character of the iso- morphismL_T(T(L)) clearly involves a simultaneous considerationof all spaces L and all transformationsX connectingthem; this entails a simultane- ous considerationof the conjugate spaces T(L) and the,induced transforma- tions T(X) connectingthem. Both functionsT(L) and T(X) are thus involved; we regard them as the componentparts of what we call a "functor"T. Since the induced mapping T(X1) of (2) reverses the direction of the original Xi of (1), this functorT will be called "contravariant." The simultaneous isomorphisms r(L): L >~ T(T(L)) compare two covariantfunctors; the firstis the identityfunctor I, composed of the two functions 1(L) = L, I(X) = W the second is the iterated conjugate functorT2, with components T2(L) = T(T(L)), T2(X) = T(T(X)). For each L, r(L) is constructedas follows.Each vector xCL and each func- tional tET(L) determinea real number t(x). If in this expressionx is fixed while t varies, we obtain a linear transformationof T(L) into R, hence an elementy in the double conjugate space T2(L). This mapping r(L) of x to y may also be definedformally by setting [[i-(L)]x]t=t(x). The connectionsbetween these isomorphismsr(L) and the transforma- tions X: L1-+L2 may be displayed thus: 7r(Li) L1- r() T2(L1)2 I(X) } T2(X) I z7(L2) 2 L2 --E T (L2) The statementthat the two possible paths fromL1 to T2(L2) in this diagram are in effectidentical is what we shall call the "naturality"or "simultaneity" condition for t; explicitly,it reads (3) r(L2)I(X) - T2(X)r(Li). This equality can be verifiedfrom the above definitionsof t(L) and T(X) by straightforwardsubstitution. A functiont satisfyingthis "naturality" condi- tion will be called a "natural equivalence" of the functorsI and T2. On the other hand, the isomorphismof L to its conjugate space T(L) is a comparisonof the covariant functorI with the contravariantfunctor T. Sup- pose that we are given simultaneousisomorphisms This content downloaded from 128.151.244.46 on Thu, 08 Oct 2015 12:00:15 UTC All use subject to JSTOR Terms and Conditions 234 SAMUEL EILENBERG AND SAUNDERS MAcLANE [September en(L): L:r-?T(L) foreach L. For each linear transformationX: L1-+L2 we then have a diagram L, - aT(LO) > T(L1) 1(X) { ~L2) { T(X) L2 - a(LT(L2) The only "naturality"condition read fromthis diagram is o(L1) = T(X)o-(L2)X. Since o-(L1) is an isomorphism,this condition certainlycannot hold unless X is an isomorphismof L1 into L2. Even in the more restrictedcase in which L2- L1,=L is a single space, there can be no isomorphismu: L-+T(L) which satisfies this naturality condition o-= T(X)o-Xfor every nonsingularlinear transformationX(2). Consequently, with our definitionof T(X), there is no "natural" isomorphismbetween the functorsI
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