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Let $\mathcal C,\otimes$ be a monoidal category, i.e. $\otimes : \mathcal C \times \mathcal C \to \mathcal C$ is a functor, and there's a bit more structure and properties. Suppose that for each $X \in \mathcal C$, the functor $X \otimes - : \mathcal C \to \mathcal C$ has a right adjoint. I will call this adjoint (unique up to canonical isomorphism of functors) $\underline{\rm Hom}(X,-) : \mathcal C \to \mathcal C$. By general abstract nonsense, $\underline{\rm Hom}(X,-)$ is contravariant in $X$, and so defines a functor $\underline{\rm Hom}: \mathcal C^{\rm op} \times \mathcal C \to \mathcal C$. If $1 \in \mathcal C$ is the monoidal unit, then $\underline{\rm Hom}(1,-)$ is (naturally isomorphic to) the identity functor.

Then there are canonically defined "evaluation" and "internal composition" maps, both of which I will denote by $\bullet$. Indeed, we define "evaluation" $\bullet_{X,Y}: X\otimes \underline{\rm Hom}(X,Y) \to Y$ to be the map that corresponds to ${\rm id}: \underline{\rm Hom}(X,Y) \to \underline{\rm Hom}(X,Y)$ under the adjuntion. Then we define "composition" $\bullet_{X,Y,Z}: \underline{\rm Hom}(X,Y) \otimes \underline{\rm Hom}(Y,Z) \to \underline{\rm Hom}(X,Z)$ to be the map that corresponds under the adjunction to $\bullet_{Y,Z} \circ (\bullet_{X,Y} \otimes {\rm id}) : X \otimes \underline{\rm Hom}(X,Y) \otimes \underline{\rm Hom}(Y,Z) \to Z$. (I have supressed all associators.)

Question: Is $\bullet$ an associative multiplication? I.e. do we have necessarily equality of morphisms $\bullet_{W,Y,Z} \circ (\bullet_{W,X,Y} \otimes {\rm id}) \overset ? = \bullet_{W,X,Z} \circ ({\rm id}\otimes \bullet_{X,Y,Z})$ of maps $\underline{\rm Hom}(W,X) \otimes \underline{\rm Hom}(X,Y) \otimes \underline{\rm Hom}(Y,Z) \to \underline{\rm Hom}(X,Z)$? If not, what extra conditions on $\otimes$ are necessary/sufficient?

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2 Answers 2

up vote 13 down vote accepted

It is associative. Consider the evaluation cube drawn here. Four of the faces commute by definition of the composition map, and one by functoriality of the tensor product. The commutativity of these five faces implies that any of the maps $W \otimes \operatorname{Hom}(W, X) \otimes \operatorname{Hom}(X, Y) \otimes \operatorname{Hom}(Y, Z) \to Z$ are equal, so by adjunction, the two composites of compositions are equal.

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In S. Eilenberg and G. M. Kelly Closed categories, in Proc. C. O. C. A.. (La Jolla, 1965),

These is a comprehensive study about Monoidal and Close structure on a category, and the relation and equivalence between these.

EDIT (I explain better):

Let given a monoidal category $\mathscr{C}, \otimes , I, l, r)$ if for each $B, C\in \mathscr{C}$ I have natural isomorphism $\pi: (A \otimes B, C) \cong (A, [B, C])$, we can get (by Yoneda lemma) naturally a functor $[-, ?]: \mathscr{C}^{op}\times \mathscr{C} \to \mathscr{C} $ and a parametric adjunction described by the above isomorphism.

From [EK] subsection 3, p. 477 (with a condition about a functor $V: \mathscr{C}\to Set$ that is not essential, you can chose $V= (I, -)$ with little modification to the [EK] esposition), we have that $\mathscr{C}, \otimes , I, l, r)$ is a monoidal closed category (see [EK] p. 475). From [EK] T.5.2 p.445, $\mathscr{C},$ is a enriched category enriched on itself considered as closed category. From [EK] T. 6.4 p. 468, $\mathscr{C}$ is also enriched on itself considered considered as monoidal category and in particular you get the associativity condition VC3' of [EK] p. 496.

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Do you mean that being monoidal is equivalent to being closed? –  Fernando Muro Sep 16 at 8:26
I mean the subsection 4 "Relations Between the Axioms" p.482, in the (big) article indicated. –  Buschi Sergio Sep 16 at 22:18
not helpful at all. –  Fernando Muro Sep 16 at 23:39
I did a mistake about (after a lot of time..). I'll edit my above answer and explain better. –  Buschi Sergio Sep 17 at 13:04

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