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I want to be able to point to category of V-enriched categories, so I created an entry, so far just with a brief Idea-paragraph.
For the 2-category of enriched categories, you anticipate a separate entry, or one should add the information here ?
I expanded the entry to discuss $\mathcal{V} Cat$ as a $2$-category. Although I didn't do it, a move to VCat would probably also be good.
[malformed duplicate]
The entry as it stands is hard to follow, especially the section Structure of the category of V-enriched categories for various contexts, which is quite unmotivated and peters out.
As a general question, what does one require on $\mathcal{V}$ for $\mathcal{V}Cat$ to be enriched over itself?
Well, one needs $\mathcal{V}$ to be symmetric monoidal, so that $\mathcal{V}Cat$ has a tensor product (section 1.4 of Kelly’s book), and this I think is the tensor product I would like to enrich $\mathcal{V}Cat$ over.
EDIT: Hmm, and then in section 2.2 it seems that taking $\mathcal{V}$ to be complete is enough to get the internal hom for $\mathcal{V}Cat$, and I presume this enriches $\mathcal{V}Cat$ over itself…
OK, so just before equation (2.29) on page 33 of Kelly’s book it is stated that the symmetric monoidal 2-category $\mathcal{V}Cat$ is closed (with the standing assumption that $\mathcal{V}$ is complete and symmetric monoidal). I’m slightly wary of this being about a symmetric monoidal 2-category, but it’s probably best to think of it just as a symmetric monoidal enriched category.
Yes, it’s a symmetric monoidal 2-category in the strictest reasonable sense of a symmetric monoidal $\mathrm{Cat}$-enriched category. This should be true whenever $\mathcal{V}$ is complete and symmetric monoidal closed.
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