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

### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

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!

-

## 2 Answers

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.

-

Thanks Mike!  Sadly the phrase "change of base" means (as you mention) so many things that it isn't really workable as a search phrase.  I like the term "iterated enrichment" quite a bit, and will adopt it immediately.  As you guessed, the two scenarios are related, but unfortunately I can't assume that $\mathbb B$ is self-enriched (closed).  I voted your answer up, but I'm going to hold off on accepting an answer in order to see if somebody can come up with a citation of some sort. – Adam Sep 27 2010 at 4:28
I remembered one reference. – Mike Shulman Sep 27 2010 at 20:59
I think that  Eilenberg, S. & Kelly, G.M. Closed categories  in: Proceedings of the Conference on Categorical Algebra. (La Jolla, 1965)  is the earliest reference treating change of base in the sense you mention. – Daniel Schäppi Mar 11 2012 at 3:16

### You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.

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.

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 22, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9465652704238892, "perplexity_flag": "middle"}