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If $M$ is a monoidal category, an enriched category over $M$ is a category $C$ whose hom-sets are viewed as objects in $M$. And a monoidal category $M$ is said to be closed if the tensor product functor has a right adjoint, known as the internal Hom functor.

My question is, are “monoidal category enriched over itself” and “closed monoidal category” equivalent? To put it another way, do all monoidal categories with internal Hom sets have internal Hom functors, and do all monoidal categories with internal Hom functors have internal Hom sets?

If not, does anyone know of counterexamples in either direction?

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The two concepts are distinct.

In general it is true that every monoidal closed category is enriched over itself, and this fact is exploited for studying enriched presheaves.

The converse does not hold as the following example will show.

Consider the category $\mathbf {Top}$ of topological spaces and continuous maps between them, with the cartesian monoidal structure. This category is not catesian closed (if I am not mistaken that is because products do not commute with colimits in $\mathbf {Top}$, hence products cannot have right adjoints).

Nevertheless you can enrich $\mathbf{Top}$ over itself by regarding every set $\mathbf{Top}(A,B)$ as a topological space with the trivial topology (i.e. the topology in which every subset is open). With this topology the compositions are clearly continuous and it is trivial to verify that this is indeed a $\mathbf{Top}$-enriched category.

So while it is true that $\mathbf{Top}$ can be enriched over itself it is not cartesian closed.

Edit: from a discussion with the OP I have noticed that there is a misunderstanding on the notion of monoidal closed category.

Being monoidal closed is not equivalent to just having an internal $\hom$-functor, you also need for the covariant functors $\hom[X,-]$ to be right adjoint to the products $-\otimes X$.

The example above shows this: the products $-\times X$ in $\mathbf{Top}$ do not have, generally, right adjoints, but the $\hom$-functor can be turned into a $\mathbf{Top}$-valued functor.

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  • $\begingroup$ So why is it that you can’t use the internal Hom sets to define an internal Hom functor in this case? $\endgroup$ – Keshav Srinivasan Mar 10 at 19:35
  • $\begingroup$ @KeshavSrinivasan I am not sure I understand your question. What do you mean by internal Hom-functor and internal Hom-set? $\endgroup$ – Giorgio Mossa Mar 10 at 19:56
  • $\begingroup$ In a category enriched over itself, an internal Hom-set is a Hom-set viewed as an object of the category. In your case, $Top(A,B)$ viewed as a tooological space with the trivial topology. An internal Hom functor is a functor of the form Top(A,_):Top -> Top. See here for details: ncatlab.org/nlab/show/closed+category#definition $\endgroup$ – Keshav Srinivasan Mar 10 at 20:46
  • $\begingroup$ This link also gives a basic idea of what an internal Hom functor looks like: en.wikipedia.org/wiki/Hom_functor#Internal_Hom_functor $\endgroup$ – Keshav Srinivasan Mar 10 at 20:48
  • $\begingroup$ So, if I understand correctly by internal hom-set you mean a function that associate an object of the category to every pair of objects, while by internal hom-functor you mean an extension of the former function to a functor in the category. Am I right? $\endgroup$ – Giorgio Mossa Mar 10 at 20:52

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