S.Eilenberg and G.M.Kelly. A generalization of the functorial calculus.-J.of Algebra, 1966.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....which prevents, in general, the composing of dinaturals. Kelly [13] in his abstract treatment of coherence in categories, discusses this very situation in detail for a general calculus of transformations, while special cases of the general problem were already resolved in Eilenberg Kelly [5]. Cut elimination theorems were successfully applied to coherence questions by Lambek [15, 16, 17, 18] and Mints [22] Of course, for the simple types of this paper, normalization or cut elimination poses no problems. But even for these types, we shall obtain more: cut elimination implies that the ....

S. Eilenberg and G. M Kelly. A generalization of the functorial calculus, J. algebra 3 (1966), pp. 366-375.

....which prevents, in general, the composing of dinaturals. Kelly [13] in his abstract treatment of coherence in categories, discusses this very situation in detail for a general calculus of transformations, while special cases of the general problem were already resolved in Eilenberg Kelly [5]. Cut elimination theorems were successfully applied to coherence questions by Lambek [15, 16, 17, 18] and Mints [22] Of course, for the simple types of this paper, normalization or cut elimination poses no problems. But even for these types, we shall obtain more: cut elimination implies that the ....

S. Eilenberg and G. M Kelly. A generalization of the functorial calculus, J. algebra 3 (1966), pp. 366-375.

....for specific purposes, as in those papers, but does not describe the situation in the full generality of a model theoretic approach. On the other hand, this issue of contra covariant functors was partly at the origin of relevant generalizations of the notion of functor in mathematics, for example [EK66]; see also [Mac71] In this line of work, Bainbridge et al. propose to interpret terms as dinatural transformations, yet another elegant categorical notion derived from tensor algebra and algebraic topology. The rub is that, in general, dinatural transformations do not compose, while terms do; ....

S. Eilenberg, G.M. Kelly. "A Generalization of the Functorial Calculus." Journal of Algebra 3, pages 366--375, 1966.

....is an action of V on A for which we have the adjunction (2. 1) For X 2 V and T : A A, the set [A; A] fX; T ) A; A] X ; T ) is the set of natural transformations ( A : X A TA) which is isomorphic to the set of natural transformations ( A : X A(A; TA) now in the generalized sense of [EC]; and to say that this set admits a representation of the form V(X; Y ) is, by the de nition of an end, to say that the end R A2A A(A; TA) exists. In other words: 4.1. Proposition. For an action F = f; e f ; f ) of V on A, the functor f : V [A; A] has a right adjoint g if and only if, ....

....by the adjunction a natural transformation j A : I A(A; TA) 4.4) which factorizes as A j for a unique morphism j : I gT: 4.5) Next, the composite gT gT TA A ## A(TA;TTA) A(A; TA) M ## A(A; TTA) A(A;mA ) ## A(A; TA) 4. 6) is natural in A by the Eilenberg Kelly calculus of [EC], so that it factorizes as An for a unique morphism n : gT gT gT ; 4.7) and it is (gT; n; j) that is the monoid (MonG)T of MonV. There is a similar formula for the right adjoint r of q : V (A; A) where now V is symmetric monoidal closed; once again, it involves a large end, and thus is ....

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S. Eilenberg and G.M. Kelly, A generalization of the functorial calculus, J. Algebra 3 (1966), 366-375.

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S.Eilenberg and G.M.Kelly. A generalization of the functorial calculus.-J.of Algebra, 1966.

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S.Eilenberg and G.M.Kelly. A generalization of the functorial calculus.-J.of Algebra, 1966.

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