Relationship between enriched, internal, and fibered categories

Question

In this question, let $(\mathcal{V}, \otimes, [-,-], e)$ be a nice enough symmetric monoidal closed bicomplete category.

The usual set-based Category theory has been generalized in many directions, let me mention some of them.

1. $\mathcal{V}$-enriched categories.
2. Internal categories in $\mathcal{V}$.
3. $\mathcal{V}$-fibered categories.

The literature provides comparisons between these approaches.

• Streicher's Fibered Categories, Sec. 4 provides a strategy to construct a fibered category out of an internal category.
• Enriched and internal categories: an extensive relationship by Cottrell, Fujii and Power show that when $\mathcal{V}$ is cartesian, $\mathcal{V}$-Cat and the category $\text{Cat}_d(\mathcal{V})$ of internal categories in V with a discrete object of objects are equivalent.
• Verity's PhD thesis Enriched Categories, Internal Categories and Change of Base, Chap.2.2 discusses how to internalize an enriched category.

Even common generalizations have been suggested, like enriched indexed categories, or enriched internal categories.

Question. Could you help me in understanding the most cutting edge results that relate enriched, internal, and fibered categories? I am not interested in common generalizations, just in processes, functors, adjunctions, that relate $\mathcal{V}$-categories, internal $\mathcal{V}$-categories and fibered $\mathcal{V}$ categories, as in Cottrell et al's paper.

Answer 1

Too long for a comment. Thanks for compiling this bibliography, Ivan! It seems you've got the literature at your fingertips to get a sense for the available tools.

One thing to say is that in contrast to the way you've presented things, usually internal category theory and fibered category theory are viewed as being tightly connected: essentially, internal categories are to small categories as fibered categories are to locally small categories. When using either framework, one gets used to passing back and forth with the other as a matter of course. For examples of this way of thinking, see e.g. this book; I'm thinking in particular of Pare and Schumacher's article therein. In particular, the construction of a fibered category from an internal one is very standard -- it should be the same whereever you read about it.

When it comes to comparing internal / fibered categories to enriched categories, there's a bit more of a conceptual jump. I actually haven't delved into this into too much detail, but I think the references that you and others have mentioned are probably good starting points.

One thing I do think is worth mentioning is that the enriched vs. internal comparison comes up in $\infty$-category theory, where Segal categories are a sort of "enriched" approach to $\infty$-categories whereas complete Segal spaces are a sort of "internal" approach. The fact that the equivalences in complete Segal spaces need to be "normalized" via the completeness / univalence condition echoes the basic fact that categories enriched in categories (i.e. 2-categories) are not the same thing as categories internal to categories (i.e. double categories); in order to use the latter to model the former you need to impose some condition. In a more literal sense, the enriched vs. internal distinction is also seen when comparing simplicial categories to Horel's model for $\infty$-categories, which uses categories internal to simplicial sets.

(Side note: I'd actually be interested in seeing, in non-$\infty$-land, an approach to viewing enriched categories internally which goes the route of complete Segal spaces rather than Segal categories, normalizing objects to encode isomorphism information rather than being discrete -- if this is even possible!)

All of that is mostly to provide context, but also to point out that if you're interested in comparison methods, it may be worth looking at the various Quillen adjunctions which are used to compare these different models of $\infty$-categories. Similarly, $(\infty,n)$-categories can be modeled as $n$-fold complete Segal spaces, which runs into the same issue of "normalizing away the double category / 2-category distinction" issue, and again the comparisons to other more "enriched-type" models like $n$-quasicategories will be confronting enriched vs. internal comparison issues.

Finally, I'll add that Cruttwell and Shulman's virtual equipments -- one of these frameworks accommodating both enriched and internal settings -- have been used quite literally to do real work $\infty$-categorically by Riehl and Verity in their $\infty$-cosmos work, where they construct a virtual equipment out of an $\infty$-cosmos. They are quite concerned with addressing model comparison results systematically, so for example they study when a morphism of $\infty$-cosmoi gives rise to a morphism of the associated virtual equipments, and when notions like pointwise Kan extension are preserved. So they're going beyond just constructing Quillen adjunctions, and working to get comparisons of the category-theoretic concepts internal to their $\infty$-cosmoi. The virtual equipment framework of course owes a lot to the proarrow equipments of Wood, which also deserve to be mentioned in your general bibliography. For that matter, Koudenburg has a number of papers developing the formal category theory of virtual equipments and change-of-base questions between them -- e.g. with the goal of things like comparing monoidal Kan extensions to ordinary ones.

The constant idea which I would keep in mind, which I think is stressed in most of these works and of which you're probably well aware, is that in order to adequately translate categorical concepts from one setting to another, it's often not enough to have a 2-functor between the appropriate 2-cateogories, because the notion of discrete fibration used in the internal / fibered setting does not translate directly to the notion of $V$-presheaf in the enriched setting; for that reason you get structures like (virtual) proarrow equipments and so forth which explicitly encode both the structure of functors and the structure of bimodules.

And all of that is just a long way of saying that I feel irresponsible for knowing of so many places where these issues matter and yet not myself having an answer to the question of "What are the main things one needs to do to compare different category theories, and how can one do them?"

EDIT: But on second thought, maybe I've seen this done enough times that I can give you a hitlist of the kinds of things one needs to think about when performing change of base from $\mathcal C$ to $\mathcal D$, where $\mathcal C$ and $\mathcal D$ are places where one can "do category theory":

1. You probably need at least a 2-functor $\mathcal C \to \mathcal D$ to be sure that notions like adjunctions and monads transfer (though note that in $\infty$-categorical contexts, monads are an exception to the rule that all the basic concepts of $\infty$-category theory can be expressed in a metatheory that doesn't go to $\infty$ -- you can't say what a homotopy coherent monad is in an $\infty$-cosmos just using the underlying virtual equipment. But this is more an issue of the "$\infty$" part than the "category theory" part).

2. You need some way of keeping track of distributors/presheaves/bimodules/correspondences/profunctors in order to formulate notions like representability, pointwiseness of Kan extensions, etc. Your comparison mapping needs to act on this data too.

I'd say that if you can adequately translate both types of information, then you can adequately translate "all the category theory" from $\mathcal C$ to $\mathcal D$, at least according to the wisdom that seems to have accrued through the work of all the papers cited in the questions, comments, and this answer.

Answer 2

Earlier this year Lyne Moser gave a talk at an online workshop for double categories which discussed some recent (joint) work that I think is relevant to this question (if I understand it correctly). In particular, she discussed how repeated enrichment and internalisation starting with the case $\mathcal{V}=\mathsf{Set}$ relate to one another, so e.g. how this explains the relation between $2$-categories (as $\mathsf{Cat}$-enriched categories) and double categories (as internal $\mathsf{Cat}$-categories) and so on for higher $n$. The methods pass through the theory of fibrations, so these enter the story too somehow. I definitely recommend checking out the recording or the slides, since they do a much better job of explaining than I could!