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Where the idea-section says “stratified geometric notion” I guess it makes sense to link to stratified space? Or else that entry could be boosted to be worthy of pointing to for this purpose.
On another note, I am little worried that the term “geometric $n$-category” is not so suggestive of what is actually meant and more suggestive of notions (such as “Lie category”) that are definitely not meant. Maybe the alternative “manifold-stratified” suggested in the note (p. 2) is better in this respect, but generally I feel the terminology is off: The $n$-categories here are themselves neither geometric nor stratified, instead they are encoded by stratified topological data, which is different.
Re #3, that goes along with the the fact that where at higher category theory we raise the topological versus algebraic/combinatorial distinction, we don’t have a separate ’algebraic n-category’ entry, or ’topological n-category’.
Re #3, I agree overall. Let me try to explain the idea for the term. It was loosely inspired by how adjectives are sometimes adjoined to subjects.
For instance
Then:
Similarly,
… at least, that was the rough idea.
Re #4, I think I misunderstood the comment. I added a subsection to higher category theory which I thought would address this. But I don’t think that was required, feel free to delete the subsection again.
Continuing the above interpretation;
Having thought about this more, @Urs: what do you think of the terms “manifold-diagrammatic” or just “diagrammatic” n-category? That has historically been used with a similar meaning. In the end, I think until someone actually writes down a satisfying theory of “geometric”/”diagrammatic” higher categories maybe the choice of terminology is not so important yet. For now, “geometric” is just meant to vaguely highlight the connection to manifolds and it has been used in a few places in that way…
PS: Coincidentally, I just learned from a colleague that they indeed use “geometric (n-)categories” to mean a certain categories with Lie groups as objects. :-)
Thanks for all the feedback.
The adjectives themselves are fine, but the issue I mean is that they do not qualify the $n$-categories, but their models.
For example, a topological 1-type presents a groupoid but not a topological groupoid. Instead a topological 1-type serves as a topological model for a plain groupoid. This distinction is important (the shape modality…). It carries over to notions such as “geometric homotopy type” which are $\infty$-groupoids equipped with geometry (aka $\infty$-stacks).
Instead, the terminology you are after should be something like the following, I’d suggest:
topological model for higher categories
algebraic model for higher categories
stratified-manifold model for higher categories.
We have it reflected roughly this way on the nLab already as:
These entry titles and their content could do with fine-tuning — and will have to if we bring in your new sratified-space model — but the general perspective makes sense to speak of different qualities of models for higher categories. The whole point of the exercise is that the resulting higher categories are in fact all equivalent to each other, not remembering the algebraic/topological/geometric tools that were used to present them.
At the moment we have geometric definition of higher categories described in such a way that at quasi-category
The notion of quasi-category is a geometric model for (∞,1)-category.
Would that be better as ’topological definition of higher categories’?
I recall discussions at the $n$-Category Café of a spectrum between the purely algebraic and the purely spatial/topological with several intermediate points. With the stratified-manifold models here, how now to characterise the space of kinds of model?
I’ll start with a tangential comment that tries to verbalize some intuition (i’ll get back to the actual discussion in a moment). In my world view, an object is of ‘geometric nature’ if important parts of its defining data are represented geometrically, with ‘geometrically’ meaning in terms of (preferably smooth) manifolds or stratified manifolds. Lie Groups and many concepts from classical Algebraic Geometry are examples. (See also: Geometry of tensor calculus and A geometric approach to homology theory which roughly follow the same ideology.)
From that perspective, some of the well-established terminology is, of course, not that great. For instance, “geometric realization” realizes combinatorial structures as topological spaces, and that is a priori unrelated to stratified-manifold-thinking (one could argue cell complexes are also stratified manifolds, but there’s no diversity here: all manifolds are disks). Similarly, geometric definition of higher categories is not that great of a choice from that perspective. (As David actually suggests, I’d personally prefer “topological definition of higher categories”).
Back to the main discussion (of naming geometric $n$-categories). The data of geometric $n$-categories, like all $n$-categories, will be primarily combinatorial (sets of higher morphisms + conditions/additional structure to describe composites). So indeed, they are not of ’geometric nature’ as you point out. But their data should have geometric ‘semantic interpretations’. (I guess similarly, you could ‘semantically interpret’ certain filler conditions as topological contractibility conditions for some topological space of composites… so again, as David suggested, “topological/spatial definition of higher categories” does seem consistent!)
Thus, I think “geometric/stratified-geometric/stratified-manifold-based definition of higher categories” would be all good choices. (In a perfect world, I guess I’d also want a single (catchy) adjective to do the same job … as was my intention with “geometric” or “diagrammatic” :-) )
“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” (?)
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 “stratifold” 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.
It sounds like quite generally you may want a new term specific to your notion of stratified manifolds?
How about stratometric or geostratic?
(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.
Sounds good to me.
Potentially actionable items from this discussion:
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?
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.)
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.
Added it to my to-do list :-) will report back when I get around to actually doing it!
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 $Cat$?
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:
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)
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?
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!
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?
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 :-)
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.
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)
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.
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