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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
Welcome, franeb! (Do you have a name?)
Have you looked at the process of “universe enlargement” described in sections 2.6 and 3.11–3.12 of Kelly’s book? That provides a way to get something like a $\mathcal{V}$-functor category when $\mathcal{V}$ is not complete, and it will admit a Yoneda embedding. Some of the hom-objects in this functor category (such as those whose domain is representable, by the Yoneda lemma) will lie in $\mathcal{V}$, but in general not all of them will.
Dear Mike,
thanks for your reply. I have had a look at 2.6 before, but I was and I am still quite certain that this only handles the case when $\mathcal{A}$ is not small. I agree that in 2.6. Kelly never explicitly mentions the need for completeness, but I am convinced that he implicitly assumes it throughout Chapter 2. In 3.11 and 3.12 the focus is more on taking a non-closed $\mathcal{V}$ and turn it into a closed extension. I also do not see how something incomplete would become complete by enlarging the universe. At least this is not generally possible.
I can also be wrong with my claim that this is not applicable for completeness, but at the moment I just do not see how this would work.
What do you mean by “… something like a $\mathcal{V}-functor category$”? If not all the hom-object lie in $\mathcal{V}_{0}$ then it would not be $\mathcal{V}$ generally enrichable.
Kind regards,
Frank
You are right that Kelly assumes $V$ to be complete all through chapter 2 (the first paragraph of chapter 2 says “Henceforth we add the assumption that $V_0$ is complete”) and that 2.6 only talks about the problem of non-small $A$. However, the reason that non-small $A$ is a problem is that $V$ cannot be expected to admit non-small limits. If $V$ lacks small limits, then small $A$ has the same problem — existence of insufficiently many limits in $V$.
You’re right that enlarging the universe in the naive sense of $Set \mapsto Set'$ is unlikely to help with completeness, but the enlargements constructed in 3.11–3.12 using the Yoneda embedding are complete and cocomplete. In 3.11 the enlargement is the presheaf category $V'=[V_0^{op},Set']$, which is complete (even $Set'$-complete) while in 3.12 it is a reflective subcategory of that which is therefore also complete. Thus, you can always get a $V'$-category of functors between any two $V$-categories.
Thank you for pointing it out. I should have been more careful reading it.
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