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Stream: learning: questions

Topic: The Zoo of geometrical shapes for higher structures

Ruby Khondaker (she/her) (Jul 10 2025 at 14:47):

I've come to realise recently that translating between different "shapes" for higher categories (whether that's cubical or globular or opetopic or...) is one of the hardest parts - apparently this involves some difficult combinatorics? I'd just assumed it was analogous to "choosing a coordinate system" the way you'd do so in physics, but clearly not :sweat_smile:

So, what shapes are useful for what sorts of situations/operations? The ones I've collected so far are:

• Globular. These are good when you want to talk about things with a single source and a single target. For example, comparing whether two cells are "equivalent" seems to be most naturally done with a globular shape.
• Cubical. These seem to be good for doing constructions like tensor products.
• Simplicial. These seem to be good because there's a huge amount of theory already built into working with simplices, and "horn fillers" are a convenient way to let the shapes themselves witness composition.
• Opetopic - I came across these in Leinster's book as a good way to model "many-to-one" operations?
• Disks (in the sense of Joyal's $\Theta$ category). Apparently these are helpful in defining $\Theta$-spaces and letting globular shapes "witness" composition?
• Atoms - these give rise to diagrammatic sets, which apparently have good tensor products and other good operations like "joins"?

Are my descriptions of "which shapes are good for which purposes" accurate? What other sorts of shapes are useful? I guess I'm going with the analogy of a choice of shape being a choice of "coordinate system", and things that are easy in one coordinate system might be hard in another, or vice-versa

Ruby Khondaker (she/her) (Jul 10 2025 at 14:55):

Presumably there isn't "one shape to rule them all"

Ruby Khondaker (she/her) (Jul 10 2025 at 14:56):

And I suppose there's a tradeoff between how expressive your shape is, and how hard it is to define a presheaf on that shape category

Amar Hadzihasanovic (Jul 10 2025 at 15:40):

Chapter 9 of my book is devoted to classes of shapes (seen as subclasses of atoms/molecules), so probably most I have to say about the topic is in there.

Amar Hadzihasanovic (Jul 10 2025 at 15:49):

In a nutshell, every class of shapes is "good" for operations under which it is stable. Globes, cubes, and simplices are classes generated by points&arrows under a single operation, which is what makes them particularly uniform: globes under suspension, cubes under Gray products, simplices under joins. So these are minimalist choices. They give you one of these operations for free and you give up having an easy description of the others.

Amar Hadzihasanovic (Jul 10 2025 at 15:50):

Atoms are the "maximalist" choice: they are in a certain sense the largest well-behaved class that is closed under all such operations.

Amar Hadzihasanovic (Jul 10 2025 at 15:52):

So I agree that there is a tradeoff between minimalism vs expressiveness.

Amar Hadzihasanovic (Jul 10 2025 at 15:54):

Opetopes are a nice class somewhere in the middle, but attempts to use them for models higher categories have always hit a wall at some point. I think they may be better adapted to other higher-algebraic structures.

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Ruby Khondaker (she/her) (Jul 10 2025 at 21:23):

One further question I had - are there nice dense subcategories of the category of atoms?

By this I mean the following - take, for example, Joyal's disk category $\Theta$. Then, if I've understood this correctly (which I may very well have not!), the globe category $\mathbb{G}$ is a dense subcategory of $\Theta$ - after all, since the objects of $\Theta$ should be "globular pasting diagrams", it seems to me that every object of $\Theta$ can be obtained by pasting together objects of $\mathbb{G}$. Perhaps what I'm referring to is a different notion of density, though, since as far as I can tell a dense functor picks out a specified colimit, whereas this seems to be more about just the existence of one.

So what I thought one could do is - suppose you want to define a presheaf on $\Theta$. It should suffice to just define a presheaf on $\mathbb{G}$, and then "extend by colimits" to a presheaf on $\Theta$. Of course not every presheaf on $\Theta$ can be obtained this way, only those satisfying some kind of "Segal condition" - but that's usually what you want for higher categories anyway!

Ruby Khondaker (she/her) (Jul 10 2025 at 21:32):

I'm wondering whether a similar thing can be done here? The category of atoms is, as you say, very "maximalist", in terms of being very expressive with the number of shapes it has. But I guess intuitively, I would expect this to mean that it's "harder" to define presheaves on the atom category, which I think are called diagrammatic sets? Since there's just a huge amount of information you have to supply, as far as I can tell. You have to know how to probe your space $X$ by every regular directed complex with a maximal cell.

However, maybe you don't need to supply all this information? Suppose there was a nice subcategory of the atom category such that every atom can be built up in some "canonical" way by pasting together objects of this nice subcategory; I'm not quite sure whether this is the same sort of "canonical" as for dense functors, but morally I want something like "an atom is a pasted-together collection of objects of the nice subcategory".

Then, instead of defining a presheaf on the atom category, you could just define it on this nice subcategory, and then "extend by colimits" to the full atom category! And again, this wouldn't give you the generic presheaf on the atom category, only one satisfying an appropriate "segal condition" - but, I assume you'd already want to restrict to the class of such presheaves anyway, if you want to focus on the diagrammatic sets that model $\omega$-categories.

I guess my idea is to "have my cake and eat it too." This way, you can get the expressiveness of the atom category in terms of tensor products, joins, suspensions etc, but without needing to supply a huge amount of data from the get-go - instead just specifying it on a nice subcategory, and then extending.

Ruby Khondaker (she/her) (Jul 10 2025 at 21:37):

What would be particularly nice (but I am not familiar enough with regular directed complexes to tell whether this actually works) is if the "minimalist" categories of shapes like globes, or cubes, or simplices, somehow formed "dense" subcategories in this sense - that you can get a general atom by pasting together a bunch of globes, or a bunch of cubes, or a bunch of simplices. That way, there's a nice "automatic" way to convert a globular, or cubical, or simplicial, set into a diagrammatic one, via "extend by colimits". Which won't yield a generic diagrammatic set, but instead one satisfying an appropriate "segal condition" - but that's what you want anyway.

Apologies if this is already covered in your book and I haven't read the section!

Amar Hadzihasanovic (Jul 11 2025 at 04:48):

1. There is exactly the same number of atoms as there are simplices, cubes, and globes: namely, a countable infinity. You don't specify a presheaf on these categories by giving every single component of the functor individually, of course, so it is not at all the case that somehow diagrammatic sets are "more data". If you need to say something about all atoms, you use an induction principle.
2. In fact, when you are actually presenting finite diagrammatic sets, like finite cell complexes, having more shapes means that you need in principle fewer cells to construct your cell complex. A diagrammatic model of a finitely-generated space is usually smaller than a simplicial or cubical model.
3. The globe category is dense in Theta but it is not dense in omegaCat --- density is not a transitive property.
4. The relation between globes and thetas (objects of Theta) is exactly like the relation between atoms and molecules. Like globes, atoms are atomic--they are by definition indecomposable.
5. In general, a representable presheaf is “indecomposable”. In other words, a dense subcategory of a presheaf category has to contain the representables. So there is no dense subcategory of diagrammatic sets smaller than the atoms.
6. Diagrammatic higher categories is not a Segal-type model, the higher categories are not “satisfying a Segal condition”. It is more like the complicial and cubical models.

Amar Hadzihasanovic (Jul 11 2025 at 04:50):

By the way it is true that diagrammatic sets can also be seen as certain continuous presheaves on a larger category which is still “combinatorial” and includes also the molecules---this is the category of regular directed complexes---and we use this all the time.

Amar Hadzihasanovic (Jul 11 2025 at 04:51):

The relation between these two would satisfy your intuition that “oh, you can probe a diagrammatic set with all regular directed complexes, but actually it only suffices to probe it with atoms”. But you can't go smaller than atoms.

Ruby Khondaker (she/her) (Jul 11 2025 at 05:49):

In that case, it seems like “atoms” are aptly named! They’re precisely the indivisible pieces of your diagrammatic set.

Ruby Khondaker (she/her) (Jul 11 2025 at 05:55):

I do wonder whether you could use a segal-type model for this - could one consider “complete segal space objects” within the category of presheaves on regular directed complexes?

Amar Hadzihasanovic (Jul 11 2025 at 06:25):

One could do that, but Segal-type models have a kind of different, somewhat incompatible "ideology" behind, and if you want conceptual cleanliness then it's best not to create ideological hybrids.
Namely, Segal-type basically sees a higher category as "a strict higher category, internal to spaces"—so you have first a notion of spaces/higher groupoids, which gives you notions of equivalence, contractibility, etc. and then, roughly, you get weak higher categories by systematically interpreting the axioms of strict higher categories in the "up-to-homotopy" sense derived from your theory of spaces.

Amar Hadzihasanovic (Jul 11 2025 at 06:26):

Diagrammatic sets, instead, take an opposite "ideological commitment" that directed cells and spaces are primitive, so you build a self-contained theory of higher categories, that specialises to a theory of higher groupoids.

Ruby Khondaker (she/her) (Jul 11 2025 at 06:29):

Sure sure - so what I’m imagining is something like a presheaf of spaces on the category of regular directed complexes (maybe a simplicial presheaf?). The segal condition would tell you that the space of molecules of a given shape is weakly homotopy equivalent to an appropriate limit of spaces of atoms, I think. And then you can add a “completeness” condition?

I think I’m going off of what Mike said in another thread (why model categories, I believe), where one of his favourite classes of models involved having “space” as a primitive, and formulating omega-categories internal to this. Do let me know if this makes any sense!!

Mike Shulman (Jul 11 2025 at 15:13):

Amar Hadzihasanovic said:

One could do that, but Segal-type models have a kind of different, somewhat incompatible "ideology" behind

I don't agree. I mean, maybe your philosophy behind diagrammatic sets is different from the CSS philosophy, but that doesn't mean the two are mathematically incompatible in any way. Other shape categories like $\Delta$ and $\Theta$ can be used for both set-based definitions and space-based definitions, and it seems to me that so should atoms be usable.

Amar Hadzihasanovic (Jul 11 2025 at 15:15):

But I never said that it was mathematically incompatible... my first words were literally "one could do that".

Amar Hadzihasanovic (Jul 11 2025 at 15:20):

Maybe I shouldn't have used the word "incompatible" at all (:

Fernando Yamauti (Jul 11 2025 at 15:39):

Amar Hadzihasanovic said:

But I never said that it was mathematically incompatible... my first words were literally "one could do that".

Is that an easy thing to do? I mean: is the cat diagrammatic sets loc pres $\omega$-accessible? And, if so, is there a good description of the compacts (perhaps an explicit synctatic description of the underlying ess alg theory)?

Amar Hadzihasanovic (Jul 11 2025 at 15:53):

The category of atoms is like the simplex category (it has only co-face and co-degeneracy morphisms) and presumably could be used for a variant of the "n-fold complete Segal space" definition.
For something in the style of $\Theta$-spaces, maybe one should use something like the categories we consider here that also have subdivision/"co-composition" maps like $\Theta$. I have not thought much about this so I cannot tell how easy any of that is.

Fernando Yamauti (Jul 11 2025 at 16:15):

Amar Hadzihasanovic said:

The category of atoms is like the simplex category (it has only co-face and co-degeneracy morphisms) and presumably could be used for a variant of the "n-fold complete Segal space" definition.
For something in the style of $\Theta$-spaces, maybe one should use something like the categories we consider here that also have subdivision/"co-composition" maps like $\Theta$. I have not thought much about this so I cannot tell how easy any of that is.

Perhaps I didn't write properly. Is it true that diagrammatic $(\infty, n)$-cats (as a 1-cat) is loc pres fin accessible? Or, even better, is that a 1-topos?

If so, up to completeness (univalence), it shouldn't be difficult to give an explicit interpretation of internal diagrammatic $(\infty, n)$-cats inside any higher topoi.

Amar Hadzihasanovic (Jul 11 2025 at 16:23):

The category of diagrammatic sets is a presheaf topos.

Mike Shulman (Jul 11 2025 at 16:24):

Amar Hadzihasanovic said:

But I never said that it was mathematically incompatible... my first words were literally "one could do that".

Fair enough. But then you went on to imply that one shouldn't do that, and that's what I was objecting to. I don't see how there could be anything intrinsic to the category of atoms that's attached to the set-based philosophy rather than the space-based philosophy, even if that was your philosophy while developing it.

Fernando Yamauti (Jul 11 2025 at 16:30):

Amar Hadzihasanovic said:

The category of diagrammatic sets is a presheaf topos.

Yes. But I'm talking about the subcat of bifibrant ones.

Amar Hadzihasanovic (Jul 11 2025 at 17:08):

Ah, apologies. I don't know the answer to that.

Mike Shulman (Jul 11 2025 at 17:21):

The 1-category of bifibrant objects in a model category is almost never itself locally presentable. In fact it almost never even has limits and colimits.

Amar Hadzihasanovic (Jul 11 2025 at 17:41):

Mike Shulman said:

Fair enough. But then you went on to imply that one shouldn't do that, and that's what I was objecting to. I don't see how there could be anything intrinsic to the category of atoms that's attached to the set-based philosophy rather than the space-based philosophy, even if that was your philosophy while developing it.

Yes, sorry. Usually when I make statements like that, what I mean is that this is the reason why I personally have not looked into something nor have plans to do so, but I guess it comes across as discouraging others from doing otherwise. I will try to be more careful.

Ruby Khondaker (she/her) (Jul 11 2025 at 17:51):

Personally I feel like complete segal space type models might be my favourite now :D

Having a “space” of cells just makes a lot more sense to me, and it neatly respects the principle of equivalence, and it can be internalised to any notion of “space”, and it naturally has “uniqueness as contractibility” built-in, and it can be internalised to other type theories!!! I think this might be the “mental picture” I carry forward with me as I continue learning and using higher category theory, then! :)

Mike Shulman (Jul 11 2025 at 18:00):

Yay, I'm happy to have made a convert. (-:

Amar Hadzihasanovic (Jul 11 2025 at 18:03):

Mike Shulman said:

Yay, I'm happy to have made a convert. (-:

I didn't even notice you were trying!

Ruby Khondaker (she/her) (Jul 11 2025 at 18:03):

Mike Shulman said:

Yay, I'm happy to have made a convert. (-:

Of course!! I mean, I think I originally got inspired by Riehl's talk on them, based on joint work you two did :).

I remember her saying that she likes it a lot because the definition of $(\infty, 1)$-category is so simple - the space of binary composites is contractible, and the notion of "equivalence" is compatible with your ambient notion of space.

And then it feels like there's a very natural generalisation of this to $(\infty, \infty)$ categories - given a pasting diagram, if you impose boundary conditions, then the space of "solutions" (i.e. composites) is contractible; which feels like some kind of de Rham cohomology thing, if I continue the analogy of viewing a pasting diagram as a combinatorial differential equation. All you need is a "completeness" statement that says the notion of equivalence in your $\omega$-category agrees with the ambient notion of equivalence in your spaces. I guess that kinda sounds like univalence...?

Fernando Yamauti (Jul 11 2025 at 18:04):

Mike Shulman said:

The 1-category of bifibrant objects in a model category is almost never itself locally presentable. In fact it almost never even has limits and colimits.

Algebraic Kan complexes forms a loc pres cat. Internal complete Segal spaces (in, say, some topos) also give an accessible reflective localisation (and, therefore, loc pres) of simplicial objects.

I was hoping that the inclusion of bifibrants would be a reflective localisation, but maybe that's not true...

Ruby Khondaker (she/her) (Jul 11 2025 at 18:05):

Maybe I could combine that with the atom category thing too - so I have a continuous presheaf on the category of molecules which satisfies the segal conditions, and where the ambient notion of equivalence agrees with the $\omega$-categorical one

That way, I get to reap the added benefits of atoms being very expressive too, in terms of easily taking tensor products, joins and suspensions

Category theory is so so cool :D

Amar Hadzihasanovic (Jul 11 2025 at 18:07):

This work makes a formal connection between univalence and Rezk-completeness (I only have a shallow knowledge of it)

Mike Shulman (Jul 11 2025 at 18:07):

Fernando Yamauti said:

Algebraic Kan complexes forms a loc pres cat.

Algebraic Kan complexes are not the full subcategory of bifibrant objects in a model category.

Internal complete Segal spaces (in, say, some topos) also give an accessible reflective localisation (and, therefore, loc pres) of simplicial objects.

I'm not sure how to interpret this. Do you mean to have some $\infty$- prefixes on things? When talking about model categories we are usually talking only about 1-categories.

Mike Shulman (Jul 11 2025 at 18:11):

One general statement is that given a model category $C$ and a left Bousfield localization $LC$ of it, the $(\infty,1)$-category presented by $LC$ is a reflective localization of the $(\infty,1)$-category presented by $C$. For instance, if $C$ is bisimplicial sets, and $LC$ is the model structure for complete Segal spaces, we get that the $(\infty,1)$-category of $(\infty,1)$-categories (presented by $LC$) is a reflective localization of the $(\infty,1)$-category of simplicial $\infty$-groupoids -- and therefore, like the latter, a locally presentable $(\infty,1)$-category. But that's different from saying that the 1-category of complete Segal spaces is a locally presentable 1-category (it isn't).

Fernando Yamauti (Jul 11 2025 at 18:18):

Mike Shulman said:

I'm not sure how to interpret this. Do you mean to have some $\infty$- prefixes on things? When talking about model categories we are usually talking only about 1-categories.

I didn't know that algebraic Kan complexes were not the bifibrant.

Regarding the second paragraph, yes, everything is higher.

What I mean is that I was hoping diagrammatic $(\infty, n)$-cats would be the local objects for some set given by the fillers. But that fails for quasicats, so I guess what I was hoping is not reasonable.

Fernando Yamauti (Jul 11 2025 at 18:21):

Mike Shulman said:

But that's different from saying that the 1-category of complete Segal spaces is a locally presentable 1-category (it isn't).

Yes. The correct analogy here would be that Segal cats are reflective in simplicial sets.

Ruby Khondaker (she/her) (Jul 11 2025 at 18:21):

Oh, looking up the definition of local objects, maybe the boundary maps work for this? In terms of - if you want to compose a pasting diagram, you just need to choose a way to compose the boundary (which is kind of like specifying a map from the boundary of the n-cell to the boundary of the pasting diagram), and then there's a contractible space of ways to compose it. I don't quite know the details but it might be related to this.

Mike Shulman (Jul 11 2025 at 18:26):

Fernando Yamauti said:

Yes. The correct analogy here would be that Segal cats are reflective in simplicial sets.

That doesn't even typecheck: Segal categories are not simplicial sets.

Mike Shulman (Jul 11 2025 at 18:28):

Ruby Khondaker (she/her) said:

Oh, looking up the definition of local objects, maybe the boundary maps work for this? In terms of - if you want to compose a pasting diagram, you just need to choose a way to compose the boundary (which is kind of like specifying a map from the boundary of the n-cell to the boundary of the pasting diagram), and then there's a contractible space of ways to compose it.

This generally works for space-based definitions, where there is already an ambient notion of "contractible" and so the locality condition can say that "there is a contractible space of composites". It doesn't really work for set-based definitions like quasicategories and diagrammatic sets, where there isn't already a notion of "contractibility" except for one given by the fillers themselves. In other words, a model like complete Segal spaces are a localization of a model structure like simplicial spaces, but quasicategories are not the localization of any natural pre-existing model structure on simplicial sets.

Ruby Khondaker (she/her) (Jul 11 2025 at 18:29):

All the more reason for me to prefer space-based models!!

Fernando Yamauti (Jul 11 2025 at 18:30):

Mike Shulman said:

Fernando Yamauti said:

Yes. The correct analogy here would be that Segal cats are reflective in simplicial sets.

That doesn't even typecheck: Segal categories are not simplicial sets.

Ok. The naming was bad. I meant something complete Segal sets.

Fernando Yamauti (Jul 11 2025 at 18:34):

I haven't realised that until now. But, by the same reasoning, internal quasicats inside some higher topoi are not reflective inside simplicial objects. That's very frustrating because those are the usual choices when defining atlases of stacks.

Ruby Khondaker (she/her) (Jul 11 2025 at 18:36):

Actually, I had a naive question about space-based models - could you also have such a diagram within smooth spaces? Since homotopy types just correspond to topological spaces as far as I know... Maybe that way, you could a "smooth" notion of $\omega$-category too!

Or I guess more generally - what's the requirement on the notion of "space" that you have so that you can internalise space-based models to them?

Mike Shulman (Jul 11 2025 at 18:39):

Fernando Yamauti said:

I haven't realised that until now. But, by the same reasoning, internal quasicats inside some higher topoi are not reflective inside simplicial objects. That's very frustrating because those are the usual choices when defining atlases of stacks.

As I said in the other thread, the correct definition of an internal $(\infty,1)$-category in an $(\infty,1)$-topos should not be called an "internal quasicategory". It can be called an "internal complete Segal object", and these are indeed reflective in the $(\infty,1)$-category of simplicial objects.

Mike Shulman (Jul 11 2025 at 18:40):

Ruby Khondaker (she/her) said:

what's the requirement on the notion of "space" that you have so that you can internalise space-based models to them?

Basically they need to have a "homotopy theory". Best would be if they form a model category.

Beware, by the way, that when we say "space" in "space-based model" we mean an $\infty$-groupoid, not a topological space.

Fernando Yamauti (Jul 11 2025 at 18:40):

Ruby Khondaker (she/her) said:

Or I guess more generally - what's the requirement on the notion of "space" that you have so that you can internalise space-based models to them?

Being a theory given by an essentially algebraic theory is the usual good assumption. Then you can internalise inside any higher topoi and sheaves valued in models of the theory coincide with internal models.

Ruby Khondaker (she/her) (Jul 11 2025 at 18:41):

So cool :D

Fernando Yamauti (Jul 11 2025 at 18:43):

Mike Shulman said:

Fernando Yamauti said:

I haven't realised that until now. But, by the same reasoning, internal quasicats inside some higher topoi are not reflective inside simplicial objects. That's very frustrating because those are the usual choices when defining atlases of stacks.

As I said in the other thread, the correct definition of an internal $(\infty,1)$-category in an $(\infty,1)$-topos should not be called an "internal quasicategory". It can be called an "internal complete Segal object", and these are indeed reflective in the $(\infty,1)$-category of simplicial objects.

Agreed. Still people are using internal Kan complexes to model atlases. The lack of completeness is solved by taking Morita equivalences, but, yes, I agree that's not a good choice.

Mike Shulman (Jul 11 2025 at 18:46):

I certainly didn't mean that internal Kan complexes (with mere filler existence) aren't a thing, or that they don't have uses. It makes total sense to use them for atlases. I just meant that when you change from the existence of fillers to the uniqueness of fillers, you need to also change the name of the structure you're talking about from "Kan complex / quasicategory" to something like "Segal object".

Mike Shulman (Jul 11 2025 at 18:47):

Fernando Yamauti said:

Ruby Khondaker (she/her) said:

Or I guess more generally - what's the requirement on the notion of "space" that you have so that you can internalise space-based models to them?

Being a theory given by an essentially algebraic theory is the usual good assumption. Then you can internalise inside any higher topoi and sheaves valued in models of the theory coincide with internal models.

That's a condition on the kinds of theory you can internalize. I think Ruby was asking instead about the kinds of spaces you can internalize theories in.

Ruby Khondaker (she/her) (Jul 11 2025 at 18:48):

Yes, I think that's more accurate? So for example, you have smooth sets, which are sheaves of sets on an appropriate cartesian site. But perhaps one could instead consider "smooth spaces", which are sheaves of spaces on an appropriate cartesian site. Then maybe "smooth spaces" have a good enough homotopy theory that you can internalise space-based models of $\omega$-categories into?

Mike Shulman (Jul 11 2025 at 18:53):

Yes, that should work. But it's a bit confusing because "smooth space" is traditionally used to mean what you called a "smooth set". To be clear you could say "smooth $\infty$-groupoid".

Ruby Khondaker (she/her) (Jul 11 2025 at 18:53):

Thanks again so much for your help!

Fernando Yamauti (Jul 11 2025 at 18:58):

Mike Shulman said:

I certainly didn't mean that internal Kan complexes (with mere filler existence) aren't a thing, or that they don't have uses. It makes total sense to use them for atlases. I just meant that when you change from the existence of fillers to the uniqueness of fillers, you need to also change the name of the structure you're talking about from "Kan complex / quasicategory" to something like "Segal object".

Right. The Segal conditions are given by an iso , whereas the horn fillers are given by an effective epi. But, since people call those atlases higher groupoids, it would still be better to use complete Segal spaces since the underlying cat where simplicial objects are being considered is (sometimes) not given only by 0-truncated guys.

Mike Shulman (Jul 11 2025 at 19:13):

But an atlas is made of coverings, so I would expect that in the naturally-occurring examples the maps in question are effective epis and not isomorphisms.

Fernando Yamauti (Jul 11 2025 at 19:34):

Mike Shulman said:

But an atlas is made of coverings, so I would expect that in the naturally-occurring examples the maps in question are effective epis and not isomorphisms.

That’s a good point I was overseeing. Yes, the Kan fibrations are usually not of height 0. So I guess the solution is, then, to stop calling atlases of n-stacks n-groupoids and just call them internal Kan simplicial objects.

Mike Shulman (Jul 11 2025 at 19:39):

Another thing one could call them is something like "n-pregroupoids". Having surjections instead of equivalences is similar to failing a higher-dimensional version of the completeness/univalence condition, so what you have is like an n-groupoid but with a whole space of n-morphisms where, say, an isomorphism of n-morphisms isn't the same as an (n+1)-morphism.