Saunders Mac Lane, "Duality for groups", Bulletin of the American Mathematical Society 56 (1950): 485--516.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....from P to both X;Y , in 7 We here break our notational convention of denoting arrows by lower case Greek letters, in order to emphasize that what matters here is that the arrow Upsilon is the unique arrow satisfying the stated property. 8 The concept of a UMP was introduced by Mac Lane in [18]. 16 CHRIS HILLMAN the sense that any object Z which also has arrows from Z to both X;Y must have an arrow into P . We pause here to give a few examples of how UMP s may be used to generalize familiar results concerning set theoretic constructions to more general categories. Example 1: Recall ....

Saunders Mac Lane, "Duality for groups", Bulletin of the American Mathematical Society 56 (1950): 485--516.

....observed by G.M. Kelly, it frequently happens that study of a more general correct concept is facilitated by study of its strict counterpart and of the relationship between the two. It has been said that Mac Lane s early definition of factorization system called a bi category structure in [MAC] suffered from an attempt to axiomatize too closely the notion of inclusion function, rather than injective function. Probably because inclusions are closed under composition while decomposition functions are not, it was anticipated that any attempt to capture inclusions would fail ....

S. Mac Lane. Duality for groups. Bull. Amer. Math. Soc., 56:485--516, 1950.

....to develop a theory of separable and purely inseparable morphisms which includes particularly the question of factorizablity into these types of morphisms. 0. 2 Orthogonal factorization systems of morphisms have been investigated and used in Category Theory almost since its very beginnings (see [ML]) and are nowadays presented by two classes E and M of morphisms, both closed under composition with isomorphisms from left and right, respectively, such that (I) every morphism f factors as f = m f Delta e f with e f 2 E and m f 2 M, and (II) every solid arrow commutative diagram Delta Delta ....

S. Mac Lane, Duality for groups, Bull. Amer. Math. Soc. 56 (1950) 485-516.

....and has many applications in computing science: 4.1 Products. An early achievement of category theory was to give a precise definition for the notion of product, which was previously known in many special cases, but only understood vaguely as a general concept. The definition is due to Mac Lane [46]. 4.2 Product Types. Given types T 1 and T 2 , their parallel composition is their product in the category T of types. Thus, a morphism f : T 1 Theta T 2 T takes two inputs in parallel, of types T 1 and T 2 , and returns one output, of type T . It is usual to assume that a category of ....

....the pair (C; C) and sending a morphism c : C C 0 in C to (c; c) C; C) C 0 ; C 0 ) in C ThetaC. Moreover, C has coproducts (also called sums ) iff Delta has a right adjoint. This beautifully simple way to formalise two mathematical concepts of basic importance is due to Mac Lane [46], and extends to general limits and colimits. 5.2 Freebies. Another beautifully simple formalisation gives a general definition of free constructions: they are the left adjoints of forgetful functors. For example, the path category functor Pa : Graph Cat of Example 2.6 is left adjoint to the ....

Saunders Mac Lane. Duality for groups. Proceedings, National Academy of Sciences, U.S.A., 34:263--267, 1948.

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S. Mac Lane, Duality for groups, Bull. Amer. Math. Soc. 56 (1950) 485-516.

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