Closed Categories Generated by Commutative Monads
نویسندگان
چکیده
The notion of commutative monad was denned by the author in [4]. The content of the present paper may briefly be stated: The category of algebras for a commutative monad can in a canonical way be made into a closed category, the two adjoint functors connecting the category of algebras with the base category are in a canonical way closed functors, and the frontand end-adjunctions are closed transformations. (The terms 'Closed Category' etc. are from the paper [2] by Eilenberg and Kelly). In particular, the monad itself is a 'closed monad'; this fact was also proved in [4]. In section 1 and henceforth, Y is a symmetric monoidal closed category; in this setting, the construction of the fundamental transformation k : (Afr\B)T -*• Ai\\(B)T can take place (rh denoting the inner hom-functor of Y, and T an arbitrary ^-endofunctor on Y). Some equations involving k are proved. These are used in sections 2 and 3 for the main construction. We shall stick to the terminology and notation of [4], which is the same as the terminology of [2] except that the hom-object of A and B is denoted A<\\B instead of (AB) or hom Y(A, B).
منابع مشابه
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Introduction. This note is concerned with "categories with internal horn and | and we shall use the terminology from the paper [2] by EIL~.NBERG and Kv.Imy. The result proved may be stated briefly as follows : a Y/--monad ("strong monad") on a symmetric monoidal closed category ~ carries two canonical structures as closed functor. I f these agree (in which case we call the monad commutative), t...
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