Pointer to literature on double enrichment and functors among enriching categories?

Question

I'm currently working with the following two situations:

1. $\mathbb A$ is a monoidal category, $\mathbb B$ is an $\mathbb A$-enriched monoidal category, and $\mathbb C$ is a $\mathbb B$-enriched category (and ${\mathbb A}\ncong {\mathbb B}$, ${\mathbb B}\ncong {\mathbb C}$, ${\mathbb A}\ncong {\mathbb C}$).

2. $\mathbb V$ and $\mathbb W$ are monoidal categories, $v$ is a $\mathbb V$-enriched category, $w$ is a $\mathbb W$-enriched category, and there is a monoidal functor $F:{\mathbb V}\to{\mathbb W}$.

I haven't been able to find much in Kelly's Basic Concepts of Enriched Category Theory, but surely others have come across these two situations before; there are lots of easy results (like, the image of $v$ under $F$ is a $\mathbb W$-enriched category) which must have been known for ages. Are there terms for the situations above? That would help me search the literature. If not, even just one paper on either of these cases with a bibliography would be a good foothold.

So far everything I've found on enriched category theory seems to focus on the case where there's only one enriching category in which all of the enriched categories are enriched (or the self-enrichment case where $\mathbb D$ is isomorphic to a $\mathbb D$-enriched category).

Thank you!

Answer 1

One key phrase to look for is "change of base" (which of course means many other things, but category theorists have also used it to mean your #2). I don't know a canonical reference at a commonly used level of generality, but all these facts are certainly folklore in categorical circles.

Your #1 might be called "iterated enrichment," and in good situations, it is a special case of #2. If A is a monoidal category and B is a monoidal A-category, then (the underlying ordinary category of) B is monoidal, and we have a (lax) monoidal functor B→A given by the A-valued $\underline{Hom}(I,-)$ where I is the unit object of B. In particular, every B-enriched category becomes an A-enriched category in a canonical way. If B is cocomplete, then this functor has a left adjoint (given by copowers of I) which is (automatically) colax monoidal, and strong monoidal if B is closed. Conversely, given a lax monoidal functor R:B→A where B is closed and hence a B-enriched category, then B becomes A-enriched according to #2.

Edit: I remembered one reference that I like: Geoff Cruttwell's thesis.

Answer 2

The thesis of Dominic Verity treats base change in a very abstract 2-categorical setting - looks like a lot of specialization is necessary until you arrive at the sort of situation sketched in the question.