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• CommentRowNumber1.
• CommentAuthorChristoph Dorn
• CommentTimeMar 14th 2023

Article stub, basic ideas and related concepts

• CommentRowNumber2.
• CommentAuthorDavid_Corfield
• CommentTimeMar 14th 2023

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeMar 14th 2023

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.

• CommentRowNumber4.
• CommentAuthorDavid_Corfield
• CommentTimeMar 14th 2023

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’.

• CommentRowNumber5.
• CommentAuthorChristoph Dorn
• CommentTimeMar 15th 2023
• (edited Mar 15th 2023)

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

• Geometry = study of, say, (stratified) manifolds
• Topology = study of, say, spaces
• Algebra = study of, say, algebraic structures

Then:

• Algebraic Geometry = study of stratified geometric things, but with algebraic methods
• Algebraic Topology = study of topological things, but with algebraic methods
• Geometric Topology = study of topological things, but with geometric methods

Similarly,

• Geometric Higher Category Theory = study of higher category things, but with geometric methods

… at least, that was the rough idea.

• CommentRowNumber6.
• CommentAuthorChristoph Dorn
• CommentTimeMar 15th 2023

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;

• Algebraic Higher Category Theory might be somewhat tautological (if you take Higher Category Theory $\simeq$ Higher Algebra)
• Topological Higher Category Theory would not necessarily be a tautology, if, say, along the homotopy hypothesis, you’d use topological arguments to argue about higher categories
• CommentRowNumber7.
• CommentAuthorChristoph Dorn
• CommentTimeMar 15th 2023

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. :-)

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeMar 16th 2023

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.

• CommentRowNumber9.
• CommentAuthorDavid_Corfield
• CommentTimeMar 16th 2023

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?

• CommentRowNumber10.
• CommentAuthorChristoph Dorn
• CommentTimeMar 16th 2023

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” :-) )

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeMar 16th 2023
• (edited Mar 16th 2023)

“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” (?)

• CommentRowNumber12.
• CommentAuthorChristoph Dorn
• CommentTimeMar 16th 2023

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.

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeMar 16th 2023

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

• CommentRowNumber14.
• CommentAuthorChristoph Dorn
• CommentTimeMar 16th 2023

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.

• CommentRowNumber15.
• CommentAuthorUrs
• CommentTimeMar 16th 2023

Sounds good to me.

• CommentRowNumber16.
• CommentAuthorChristoph Dorn
• CommentTimeMar 16th 2023

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?
• CommentRowNumber17.
• CommentAuthorChristoph Dorn
• CommentTimeMar 16th 2023

first attempt at implementing ongoing nforum discussion