Dear all,

I recently got involved with enriched category theory and I want to apply the machinery in a computer science environment. I am interested in non-complete enrichments and here particular in functor categories.

I am aware that if $\mathcal{V}$ is a complete symmetric monoidal closed category and $\mathcal{A}$ and $\mathcal{B}$ are $\mathcal{V}$-categories, then $[\mathcal{A},\mathcal{B}]$ can be enriched over $\mathcal{V}$. In Kelly it is then shown that under this assumption the enriched Yoneda lemma and enriched Yoneda embedding hold. There is also a short explanation of what happens when $\mathcal{V}$ is not necessarily small. I would now be interested what happens when $\mathcal{V}$ is not necessarily complete.

In Borceux (Handbook II Chapter 6) this is made a bit more precise. Here, it is made clear that we do not actually need completeness for the enriched Yoneda lemma. Still, for the enriched Yoneda embedding, we appartently need completeness. But it is not made precise why we actually need it. I assume that this is related to the functor category “problem”. Nevertheless, there is a difference between giving a recipe to get an enrichement when $\mathcal{V}$ is complete, but this does not mean that we cannot find an enrichment when $\mathcal{V}$ is not complete. Is anyone aware of results in this direction?

To reduce the problem, it would be enough to consider functor categories $[\mathcal{A},\mathcal{V}]$, where $\mathcal{V}$ is our enrichment. Here, the enriched Yoneda lemma indicates that $[\mathcal{A},\mathcal{V}](H^{A}, F)$ has a hom-object. In Borceux we can find a definition for an object of $\mathcal{V}$-natural transformations between two functors $F,G:\mathcal{A} \rightarrow \mathcal{V}$ when $\mathcal{V}$ is not complete. This means in the special case we do not have problem to find the right hom-object, but what about a general $\mathcal{V}$-functor $F$. I have not found nice examples that points out a problem. Maybe for some enrichements we can still get an enrichable $[\mathcal{A},\mathcal{V}]$ functor category.

I would appreciate any comment or reference to the literature that might answer some of these questions.

Kind regards,

franeb

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