Title: Embedding of Closed Categories Into Monoidal Closed Categories
Abstract:defined a closed category as a category with internal homomorphism functor, left Yoneda natural arrows, unity object and suitable coherence axioms.A monoidal closed category is a closed category with ...defined a closed category as a category with internal homomorphism functor, left Yoneda natural arrows, unity object and suitable coherence axioms.A monoidal closed category is a closed category with an associative tensor product which is adjoint to the int-hom.This paper proves that a closed category can be embedded in a monoidal closed category: the embedding preserves any associative tensor product which may exist.Besides the usual tools of the theory of closed categories the proof uses the results of B. Day on promonoidal structures.Introduction.The concept of closed category has been extensively developed in [6, and involves basically a category V, an internal homomorphism functor [ ]: Vop X V -» V and natural arrows of Yoneda type,If there is also a tensor product with -®B left adjoint to [B -] we have a monoidal closed category [6, 471-548], [7].A biclosed category is a monoidal closed category with an additional int-hom, { }, such that B ® -is left adjoint to {B -}.Within the framework of the monoidal closed categories the "psychological dual" of a closed category is a monoidal category [1], [8], [6] whose structure reduces to the tensor product.The object of this paper is to prove that any closed category can be embedded structure-preservingly in a biclosed category.Moreover this embedding preserves any associative tensor product which may exist: we speak of preservation up to isomorphisms compatible with the structure under consideration.Our basic tool is the concept of promonoidal structure of B. Day [2], [3] and the calculations require a frequent use of the representation theorem for closed categories [4, Theorem 1.8.6, 457].The argument goes as follows: For V a closed category the map A **-» [A -] defines a functor from Vop into the category E of the V-endofunctors of V and V-natural transformations.The category E is monoidal for the composition of endofunctors and by [2], [3] induces a biclosed monoidal structure on the category, [E, S], of set-valuedRead More