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October 29, 2019

Mixing Internalization with Virtualization: a Terminological Problem

Posted by Mike Shulman

I used to complain sometimes that nn-categories (including (,n)(\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 nn-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×k)(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 nn-fold category could I suppose be called a 1 n1^n-category although that starts to look a bit iffy.

A lot of these structures do have underlying nn-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)(n+1)-category is a category enriched (perhaps weakly) over nn-categories; if we allow this “nn” to be something more general than a natural number, like the symbol n×kn\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 “TT-multicategories”, for a monad TT, as “virtual TT-algebras”. This is most useful when TT-algebras already have another name, allowing us to refer to their virtualization without giving the monad TT 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 “1×1\timesmulticategory”), 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 “1×1\timespolycategory”?

Posted at October 29, 2019 11:50 AM UTC

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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 (1×1)+12(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 nn and the kk in an “nn-by-kk category” so that we’d be less-inclined to expect an associativity property that moves a dimension variable between the two arguments: an “(nm)(n \cdot m)-by-kk category” is not the same as a “nn-by-(mk)(m \cdot k) category.”

Posted by: Emily Riehl on October 29, 2019 7:35 PM | Permalink | Reply to this

Dan Freed’s category number

While I’m on a role with amusing asides, I was recently re-acquainted with Dan Freed’s notion of “category number”:

The category number of a mathematician is the largest integer n such that he/she can ponder n-categories for a half hour without developing a migraine.

See slide 16 of

Posted by: Emily Riehl on October 29, 2019 7:40 PM | Permalink | Reply to this

Re: Emily

Do I get a generalized category number for being kk collaborations away from a mathematician with a category number?

Posted by: Blake Stacey on October 29, 2019 8:44 PM | Permalink | Reply to this

Re: Emily

I haven’t really worried about people thinking that (nm)×k=n×(mk)(n\cdot m)\times k = n \times (m \cdot k) because ×\times is so rarely used (after elementary school) for multiplication of natural numbers. So I would expect most mathematicians to read the ×\times in n×kn\times k as some more abstract kind of “multiplication”, analogous to its other uses such as vector cross product or cartesian product of sets. But if you have suggestions for better notation (or better terminology), I’d love to hear them.

Are there any “laws of categorical arithmetic”? That is, are there any nontrivial equations that hold between symbols of this sort?

It might make sense to regard some of the functors between different structures as “inequalities”. The fact that we can regard a 1-category as a 2-category with identity 2-cells could be a manifestation of 121\le 2, and so analogously the fact that we can regard a 2-category as a double category with identity vertical arrows could be denoted 2(1×1)2 \le (1\times 1).

Posted by: Mike Shulman on October 29, 2019 10:27 PM | Permalink | Reply to this

Re: Emily

Having spent a lot of time with both matrices and cross products, ×\times is both still in use and definitely multiplication-y for me. But if we want 1×11\times 1 to be at least 2 in some fashion, that makes the appropriate operator more like addition than multiplication. Since ++ is already taken, maybe just using \in? That’s closer to how we’re saying it: an nkn\in k-category, pronounced “n-in-k category”, is an nn-category in kk-categories. I think I could accept the statement 2(11)2 \leq (1 \in 1), although I wouldn’t be able to parse it immediately, whereas 2(1×1)2 \leq (1 \times 1) strikes me as simply false.

Posted by: Layra on October 29, 2019 11:28 PM | Permalink | Reply to this

Re: in

The motivation for n×kn\times k-category is geometric: it has (n+1)(k+1)(n+1)(k+1) cells naturally arranged in a rectangle. Note that it is actually multiplicative, so addition and membership symbols have the wrong intuition. The unnaturalness of the ordering 2(1×1)2 \le (1\times 1) is just due to the off-by-one indexing that an nn-category has n+1n+1 kinds of cell. But anyway, bringing up inequalities was just a lark; I don’t think the weirdness of the result should be considered as an argument against geometrically natural naming.

Posted by: Mike Shulman on October 29, 2019 11:59 PM | Permalink | Reply to this

Re: Mixing Internalization with Virtualization: a Terminological Problem

It sounds like no one is interested in the question I actually asked, but another possible answer that I just thought of is multi double category (and poly double category). Just as a monoidal double category is “monoidal in only one direction” (at least in a certain sense – e.g. the coherence isomorphisms go in only one direction), so a multi double category is multi in only one direction.

Posted by: Mike Shulman on November 3, 2019 6:05 AM | Permalink | Reply to this

Re: Mixing Internalization with Virtualization: a Terminological Problem

How about “cat 1cat^1-polycategory?”

Posted by: Richard Garner on November 14, 2019 9:49 PM | Permalink | Reply to this

Re: Mixing Internalization with Virtualization: a Terminological Problem

Is there a general principle behind that? What does the “1” mean?

In general, just as a “VV-category” is a category enriched over VV, I would expect a VV-polycategory to be a polycategory enriched over VV, and so in particular a catcat-polycategory would be a polycategory enriched over categories, i.e. a “2-polycategory”.

Posted by: Mike Shulman on November 14, 2019 9:52 PM | Permalink | Reply to this

Re: Mixing Internalization with Virtualization: a Terminological Problem

Well that was by analogy with cat1^1-group, which is an internal category in the category of groups, or catn^n-group, which is an internal nn-fold category in the category of groups.

In general you could make an argument that for “theories”, juxtaposition should denote tensor product, eg “group scheme”. It doesn’t bear close scrutiny, but it is at least an argument.

Posted by: Richard Garner on November 26, 2019 2:17 AM | Permalink | Reply to this

Re: Mixing Internalization with Virtualization: a Terminological Problem

Yes, the referee actually also suggested “polycategory category” – or maybe it should be “category polycategory”? But I don’t think I can handle either of those.

Posted by: Mike Shulman on November 26, 2019 3:13 AM | Permalink | Reply to this

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