Enriched 2-categories

Question

I have been lead to believe, due to various conversations and presentations, that there is a standard notion of an enriched 2-category (indeed, even an enriched n-category). However, after searching I can find no reference for such a construction (although I did find a paper where the authors mention that they know of no reference). So what would the data of an enriched 2-cateogry be?

Edit: Rereading my question, it feels unsatisfactory. Perhaps I should be more direct and simply ask the following: if I have a 2-category who's Hom sets naturally carry an algebraic strucutre (say that of a topological space) are there any natural questions I can ask that might allow me extend this structure to the whole 2-category?

Answer 1

There is a standard notion of a bicategory enriched in a monoidal bicategory, which is a categorification of the notion of category enriched in a monoidal category. The standard reference would be Garner–Shulman's Enriched categories as a free cocompletion. The strict variant would be what most people would refer to as an enriched 2-category.

However, the notion referenced in the paper of Shapiro that you link is a different notion, which might be called a 2-category locally enriched in a monoidal category. This is a special case of the former notion, when the base of enrichment is taken to be the monoidal category $V\text{-}\mathbf{Cat}$, viewed as a locally discrete monoidal 2-category (this assumes that $V$ is at least braided, so that we can define a tensor product of $V$-categories).