Algebra I 18ed by Armando Rojo

By Armando Rojo

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T (TI) I,re. The set of iterations of the functor II is the smallest set of functors F:stn-+st, which contains I,re and is closed relative to the product of functors. t (II) H. 35 CATEGORIES Let T:i2~i2 be a transposition. The functor TI is commutative, while the category i is a category with commutative multiplication, if there exists the natural equivalence Every object AFi defines the functors All, llA: i~i: All (B) =A (ll) B; llA (B)=B (11) A. re). re' (the natural Every set consisting of certain natural equivalences a, T, er and their inverses, generates certain equivalences between the iterations of functor n (in the iterations we admit any permutation of the arguments and the identification of certain arguments with the object A).

1. M. Gel'fand and G. E. Shilov, "Categories of finite-dimensional spaces, " Vestn. Mosk. , No. 4, 27-48 (1963). N. 1. Glebov, "The structure of the class of R-tests for equivalence," in: Problems of Cybernetics, No. 9, Fizmatgiz, Moscow (1963), pp. 103-122. A. Grothendieck, "Sur quelques points d'algebre homologique," Tohoku Math. , 9: 119-221 (1957). Kh. N. Inasaridze, ·On the theory of extensions in categories," Soobshch. Akad. Nauk GruzSSR, 30(5): 537-544 (1963). Kh. N. Inasaridze, "Universal functors," Soobshch.

Re' (the natural Every set consisting of certain natural equivalences a, T, er and their inverses, generates certain equivalences between the iterations of functor n (in the iterations we admit any permutation of the arguments and the identification of certain arguments with the object A). Let the set of equivalences be fixed. This set is called coherent if between any two iterations of functor II it generates not more than one natural equivalence. MacLanefs paper [174] was devoted to finding the coherency conditions for various groups of natural transformations.

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