### Mixing Internalization with Virtualization: a Terminological Problem

#### Posted by Mike Shulman

I used to complain sometimes that $n$-categories (including $(\infty,n)$-categories) get all the press at the expense of other higher categorical structures, but thankfully I don’t think that’s true any more (to the extent that it ever was). Double categories, “virtual” (multicategory-like) structures, and other higher-categorical notions are increasingly recognized as useful alongside the more traditional $n$-categories. However, this larger zoo of categorical structures brings with it problems of terminology: how can we consistently name all of these things?

Some 10 years ago in this preprint, I suggested the term “$(n\times k)$-category” for an n-category internal to k-categories (strictness or weakness in all ways is negotiable). Thus a double category is a 1x1-category, while we can also talk about 2x1-categories, 1x2-categories, and so on. For instance, there is a 2x1-category of rings, algebras over rings, and bimodules over algebras, and a 1x2-category of monoidal categories and bimodules over them. A triple category can be called a 1x1x1-category, and similarly an $n$-fold category could I suppose be called a $1^n$-category although that starts to look a bit iffy.

A lot of these structures do have *underlying* $n$-categories. A double category has an underlying bicategory, while both 2x1-categories and 1x2-categories usually have underlying tricategories. At higher dimensions one needs to assume lifting/fibrancy properties for this to work. Making this precise when relating monoidal double categories to monoidal bicategories (possibly braided or symmetric) was the point of the preprint I mentioned above; Linde Wester Hansen and I recently posted an improved version that also makes it functorial, so that (for instance) monoidal functors between double categories also induce monoidal functors between bicategories.

Our proof uses yet another kind of higher-categorical structure, which mixes the “internalization” and “enrichment” approaches. Classically an $(n+1)$-category is a category enriched (perhaps weakly) over $n$-categories; if we allow this “$n$” to be something more general than a natural number, like the symbol $n\times k$, we can talk for instance about ((1x1)+1)-categories: categories enriched over double categories. These were introduced by Garner and Gurski under the name “locally cubical bicategories”, which is nicer-sounding than “((1x1)+1)-category” and mostly not misleading (although I think it might suggest instead categories enriched over *triple* categories, since those actually have cubes in their hom-sets), so for most of the paper we adopted that terminology.

You might have guessed by now that I tend to be a little overzealous in trying to use, or at least suggest, consistent terminology. In another paper from 10 years ago, Geoff Cruttwell and I tried to impose a little more order on yet a third direction of categorification, namely passage to generalized multicategories. The main purpose was to unify the various frameworks for defining them, but we also proposed a more general terminology: we refer to “$T$-multicategories”, for a monad $T$, as “virtual $T$-algebras”. This is most useful when $T$-algebras already have another name, allowing us to refer to their virtualization without giving the monad $T$ a name; thus for instance we can talk about virtual double categories, a name that seems to be catching on. Ordinary multicategories could then also be called “virtual monoidal categories”, although the word “multicategory” is shorter and not misleading so there’s no reason to stop using it.

A related idea that’s been coming up frequently recently is the idea of generalized polycategories, in which both domains and codomains can be more interesting than just a single object. Generalized polycategories certainly exist (this paper of Garner points the way), although to my knowledge no one has written down any general definition yet. But what should the polycategorical analogue of “virtual” be? Can we decide on a good red-herring adjective X such that an “X symmetric monoidal category” is a (colored) prop while an “X linearly distributive category” is a polycategory?)

That’s not the most burning question on my mind right now, though; what I’m worried about is what happens when we combine virtualization with internalization. Eugenia, Nick, and Emily used internal categories in multicategories to talk about multivariable mates. They referred to these as *double multicategories*, but that terminology could be misleading: since “doubling” something generally means having two of it, “double multicategory” sounds like something with two directions of multicategory, such as an internal *multicategory* in multicategories. Is there a better word? The taxonomical impulse tempts me to write something like “(1 x multi)-category” (or just “$1\times$multicategory”), but I expect no one would like that. Would they? Plus, even that doesn’t generalize to other kinds of virtual structures.

In fact the specific case that I need a name for right now is internal categories in *polycategories*, which I used recently to give a different way of talking about multivariable mates. In my preprint I lazily called these “double polycategories” in analogy to Cheng-Gurski-Riehl’s double multicategories, but a referee has rightly objected that this has the same problem: it sounds like an internal polycategory in polycategories. Can anyone think of a name for an internal category in polycategories that’s better than “$1\times$polycategory”?

## Emily

This is not a helpful comment but I’m finding it amusing to imagine what the “laws of categorical arithmetic” might be in Mike’s proposed taxonomy. Note $(1 \times 1) + 1 \neq 2$.

More seriously this makes me wonder if we need to introduce a different symbol than “$\times$” to separate the $n$ and the $k$ in an “$n$-by-$k$ category” so that we’d be less-inclined to expect an associativity property that moves a dimension variable between the two arguments: an “$(n \cdot m)$-by-$k$ category” is not the same as a “$n$-by-$(m \cdot k)$ category.”