I gave a talk in the TallCat seminar based on the document I had linked in comment #33. I linked the video on the page.

]]>Re comment #30. I tried to improve the situation, putting material in named sections, adding definition environments and linking to those environments.

Re comment #28. I now also added a formal candidate definition. We can see whether this makes sense to have or not.

It looks like I would need to invest much more time to penetrate the definition you have in mind.

It might look like this, but I don’t think it need be the case! Certainly, it’s not a negligible amount of maths (unsurprising, since it deals with manifold diagrams). But, actually it’s not a lot either, and the math involved is very elementary (in particular, computer implementable). I’ve just written up a small document for you: from zero to manifold-diagrammatic categories. It takes less than four, reasonably-line-spaced pages to write out the definition from scratch (modulo basic category theory…) :-)

(any feedback appreciated)

]]>proof reading: fixed typos + minor improvements

]]>clarification of ’combinatorial-ity’ of approach, draft of formal definition

]]>actually, what I had written up on the manifold-diagrammatic n-category page is already meant to be a fully combinatorial description.

I don’t think this is becoming clear. The page starts out asking the reader to note that

Embeddings of manifold $n$-diagrams are framed stratified maps whose…

and then that

Quotients of manifold diagrams are framed stratified maps whose

and the surrounding discussion and diagrams occupy the better part of the page.

Now I gather that I should have paid more attention to the sentence that starts with:

Combinatorially, embeddings of combinatorial manifold diagrams T,S are described by spans…

which on first (and second reading) seemed like a parenthetical side remark (to me, at least, and I couldn’t and can’t parse anyways), so that I had ignored it. But now I take it you are saying now that this sentence is meant to hint at the actual definition you are advertising.

I admit that I am busy with something else and just looking at the writeup here only in stolen minutes. It looks like I would need to invest much more time to penetrate the definition you have in mind.

]]>So, actually, what I had written up on the manifold-diagrammatic n-category page is already meant to be a fully combinatorial description. (e.g. I write “We give geometric insight into these classes below, but ultimately only care about their combinatorial definitions.”, “Combinatorially, …”, as well as “combinatorial manifold diagrams” in other places). In particular, that applies to the definitions of $\mathrm{MDiag}$ and $\mathbb{M}\mathrm{Diag}$.

It is absolutely possible to make the sketch more precise, and I could write out candidates for the “sheaf condition”. Indeed, it’s really not hard (or tedious) to write out *some* reasonable condition. What’s hard is to *use* it, and in particular, to compare the resulting definition to other models of higher categories. This is why I’m not sure whether there’s value in going into more detail than what’s currently on the page.

edit: I see you asked in particular about dimension 1. short answer: yes that should be possible :-)

]]>See this theorem.

Oh. Thanks for saying, I would not have spotted that connection any time soon.

Would that equivalent combinatorial description then not be the suitable basis on which to state your definition of $n$-categories?

Is there a direct combinatorial description of a simple example, say of $MDiag_1$ with its Grothendieck topology, which would make the “Sketch definition” (here) into a definition?

]]>Indeed, you need not ever think about stratifications. See this theorem. It says manifold diagrams are classified by combinatorial manifold diagrams (at least, when working up to framed stratified homeomorphism, but since framed stratified homeomorphism classes are contractible spaces we really don’t care about working ’up to’ them). Combinatorial manifold diagrams are, in particular, finite combinatorial objects. These can be encoded on a computer!

]]>geometric computads are very practical things in my opinion :-) (see e.g. homotopy.io!)

This may be just the kind of example that I am missing. Currently I am under the impression that in order to get a concrete handle on manifold-diagrammatic $n$-categories I need first a handle on the double category $MDiag_n$ of, quoting from *manifold diagram* “conical, compactly triangulable stratification of standard n-dimensional directed space” and even ignoring all the adjectives this sounds like a formidable task. I wouldn’t even know where to start.

So I guess I am missing something. How do we connect from the definitions concerned with stratified spaces that you have been sketching out to a combinatorics that can be taught to a computer?

]]>No problem, it was enjoyable writing this out again (the double category idea only existed in some old notes, and the nLab is now the only place where this is written up in a readable way).

And yes, definitely a bit of a mouthful. Of course, once the combinatorics of trusses starts to feel natural to you (…after several years of staring at it in my case) the approach will feel less like a mouthful.

I actually haven’t spend proper time thinking about the practicality of the (sketch) definition. Working it out in dim 2 or 3 should still be possible I expect. There are definitely some people who are actively exploring manifold diagrams (or trusses) as shapes for higher categories, and they’ve already made more progress than I had.

There are the obvious trade-offs about the combinatorics of manifold diagrams which may still be worth mentioning:

- you do get “unbiased” shapes with manifold diagrams, but now you have to deal with all of them simultaneously… and there are many of them (in contrast, there’s just one simplex in each dimension!)
- you do get coherences for free, but now you have to actually put in work to understand the coherences you are getting, e.g. what are the higher analogs of braids (in contrast, with, say, inner horn fillers you know pretty well what you’re getting!)

I’ll also add two (subjective and only tangentially-related) opinions on practicality:

Understanding the “elementary” (i.e. perturbation-stable) coherences and, in the context of invertible morphisms, the “elementary” singularities, is interesting: together, elementary coherences (braids, etc.) and singularities (cusps, swallowtails, etc.) generate all homotopical behaviour. On paper this is a tall order, but breaking the problem into these two classification questions makes things seem much more tractable (also, in manifold diagrams, we can phrase these classification questions in exact (and

*combinatorial*) terms for the first time). The real power of the manifold-diagrammatic approach might relate more to these classification questions, and their consequences, than to the current higher-categorical needs of contemporary mathematics.The “non-practicality” I’m alluding to also exclusively applies to “non-free” higher categories; in contrast geometric computads are very practical things in my opinion :-) (see e.g. homotopy.io!). I was just talking to Mike Shulman last week about how one could make use of this for the purpose of working with higher directed (inductive) types. (spoiler: ensuring canonicity properties is hard without understanding the elementary coherences)

Thanks for adding details! This gives an idea of the definition you have in mind.

It’s a mouthful of a definition, even in its skechy form. Can one work with this in practice?

Did you try to spell out low-dimensional cases, such as showing in detail that the 2-category of manifold 1-categories is equivalent to ordinary $Cat$?

]]>fixed typos, added links, slightly improved images

]]>added sketch definition of manifold-diagrammatic higher categories, highlighting the combinatorics available for such endeavour (embeddings, quotients, bundles)

(DRAFT!)

]]>Added it to my to-do list :-) will report back when I get around to actually doing it!

]]>Thanks. But could you just add to the entry here a paragraph on what the definition you imagine is like? I really don’t know.

]]>you are right: the “nine short stories” document only provides a definition of free geometric n-categories, aka geometric computads. I’d say that’s not as bad as it seems, because any n-category should be the n-truncation of an (n+1)-computad (truncation is meant in the sense of left adjoints to inclusions, see here). But it does mean the title of the document is misleading. I actually added a disclaimer in the file.

(As for the nLab page itself, I agree it’s not optimal. The intro does say “broadly refers” though, and two concrete definitions are linked. I’m sure, more definitions will be added soon, as I know of some people working on them.)

]]>It would be helpful if the page *manifold-diagrammatic n-category* could indicate the actual definition, or point to an exact place where this can be found. I have glanced over the “Nine short stories”-document provided, but I haven’t spotted the definition of “geometric n-category” that it’s all about (?) Which page should I turn to?

first attempt at implementing ongoing nforum discussion

]]>Potentially actionable items from this discussion:

- rename “geometric n-category” to “manifold-diagrammatic definition of higher categories” (but emphasise in the article that different terminology exists in the wild. and link to this discussion maybe)
- (?) rephrase “geometric definition of higher categories” to “spatial definition of higher categories”
- (?) amend subsection in “higher category theory” with remark contrasting articles on “algebraic/spatial/manifold-diagrammatic” definitions of higher categories?

Sounds good to me.

]]>(I had actually looked up “stratic” already before your reply! :-) )

I think “manifold-diagrammatic” is my favorite. It’s longer, but its descriptive and pretty accurate, and it’s not making up new words for now. It’s in line with other terminology (“diagrammatic methods”, “diagrammatic calculi”, “string-diagrammatic”, etc.). And it could be shortened (to “diagrammatic” or more artistic choices) if need be.

]]>It sounds like quite generally you may want a new term specific to your notion of stratified manifolds?

How about *stratometric* or *geostratic*?

Well, there is stratifold for instance. Their theory is of course well described already, and so the term has a very specific meaning by now.

I think the terminology needs to reflect the “manifold character” of what’s going on. (“Geometry” does that job, I guess “strati*fold*” does it too; but “stratified” and “space” by themselves don’t… I think!)

Further down in the math word bucket, you find words like “string” or “brane” :-)

I’d be happy if, for now, we pick terminology roughly according to the algebra/topology/geometry divide (as argued in my previous comment). And then assign the task of finding a good name for (more concrete) models to future research.

]]>“geometric/stratified-geometric/stratified-manifold-based definition of higher categories” would be all good choices

Yes. I would prefer keeping the “stratified” in there, since “geometric” means so many other things.

If there were a short term for “stratified space”, that would be good. Like “stratispace” (?)

]]>