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Stream: learning: questions

Topic: Horns filling up to some degree

Tim Hosgood (May 28 2023 at 00:06):

Say that $X_\bullet$ is a simplicial set such that some horn filling condition (e.g. inner horns fill, inner horns fill uniquely, all horns fill) holds for horns of dimension $\leq n$ for some $n\geq2$. Then what is $X_\bullet$? Can we say something about $\operatorname{sk}_n X_\bullet$ or $\operatorname{cosk}_n X_\bullet$?

Tim Hosgood (May 28 2023 at 00:07):

If $n=\infty$ then we get a quasi-category/nerve/Kan complex, but I'm interested in the case of finite $n$

Tim Hosgood (May 28 2023 at 00:08):

I wanted to say something like $\operatorname{cosk}_n X_\bullet$ is a quasi-category/nerve/Kan complex, but taking the coskeleton only ensures that boundaries fill uniquely above dimension $n$, not that horns do

Reid Barton (May 28 2023 at 00:22):

One thing I think you can say is that you can ensure that all (inner) hors fill (uniquely) (etc.) by running the small object argument with only horns of dimension $> n$, and in particular this will not change the (n-1) skeleton or so

Tim Hosgood (May 28 2023 at 15:15):

oh that's interesting!

Tim Hosgood (May 28 2023 at 15:22):

I think I'm looking for something in sort of the other direction though, two sort of related questions:

1. Is there some name for the structure of having e.g. inner horns filling up to dimension $n$? Naively I'd like to say that e.g. if only 2-horns fill then you have some sort of composition structure, but since 3-horns don't fill, this structure isn't associative.

2. If I have a simplicial set satisfying some horn filling condition up to dimension $n$, is there some canonical way of extending that condition to all dimensions? My guess would be that it involves the skeleton or coskeleton somehow, as a way to formalise "truncate at dimension $n$ and then just add in whatever simplices you need to ensure the filling condition is satisfied".

James Deikun (Jul 17 2023 at 06:25):

I think this might be an instantiation of the structure referred to as an $A_{n-1}$-category ... in the case of unique fillers it would be instantiated in something like a set or 1-categorial context, and for simple fillers it would be instantiated in a groupoidal or $(\infty,1)$-categorial context ...

Tim Hosgood (Jul 17 2023 at 09:34):

oh, that's interesting! do you have any references to suggest where I can read about this structure?

Tim Hosgood (Jul 17 2023 at 09:35):

(note: I also asked this question on MO but didn't get any answers there)

John Baez (Jul 17 2023 at 10:17):

When I hear "$A_n$" I think of Stasheff's pioneering work on $A_n$ spaces, which are spaces with a multiplication that's associative up to homotopy which obeys the pentagon identity up to homotopy which obeys the 3d associahedron identity up to homotopy... and so on, but only up to the nth stage.

John Baez (Jul 17 2023 at 10:19):

The original references to Stasheff are here but I don't think he was working simplicially.

Tim Hosgood (Jul 17 2023 at 12:30):

I've heard of $A_n$-spaces but only of $A_n$-categories for $n=\infty$

Tim Hosgood (Jul 17 2023 at 12:31):

(and "A_n-category" is one of those things that is really hard to google!)

Chris Grossack (they/them) (Jul 17 2023 at 15:45):

Tim Hosgood said:

(and "A_n-category" is one of those things that is really hard to google!)

You can definitely define an $A_n$-category for $n \neq \infty$. Though I'm not sure if there's a reason to! Here's the maneuver:

An operad gives you a notion of theory that you can interpret in your favorite symmetric monoidal category, in much the same way that an algebraic theory gives you a notion that you can interpret in your favorite finite product category. Now just like there's an algebraic theory called "groups", there's an operadic theory called "$A_n$" for each $n$ (including $\infty$), and we can interpret these in our favorite monoidal category.

For the purposes of this discussion (and the history of the field) it's helpful if your favorite symmetric monoidal category is the category of chain complexes. In this case, an $A_n$-algebra is a monoid (read: a dg-algebra) that's only associative up to homotopy, and these homotopies only cohere up to dimension $n$. So you should think about $A_\infty$ as being the "nicest possible" (things work in every dimension), and as you decrease $n$ you're increasing your tolerance for bad behavior (since you're allowing bad behavior to start sooner).

Now we categorify this whole story. We used to have a monoid that was only associative up to homotopy. But a monoid "is" a one-object category! So we can play this whole game again to get a category enriched in chain complexes, where multiplication (read: composition) is only associative up to homotopy! By far the most prevalent notion is an $A_\infty$ category, where we have homotopies in all dimensions, but you can see how to play this game to get an $A_n$-category, which is a category enriched in chain complexes where associativity homotopies only exist up to dimension $n$.

Tim Hosgood (Jul 17 2023 at 19:30):

that's a really nice explanation, thanks!

Tim Hosgood (Jul 17 2023 at 19:31):

but this seems to indicate to me that the structure I was originally describing maybe isn't an $A_n$-category then? since when you take $n=\infty$ you should recover quasi-categories/Kan complexes, and $A_\infty$-categories are really models for stable $(\infty,1)$-categories

Tim Hosgood (Jul 17 2023 at 19:32):

(it does so happen that this whole question came to me when working with dg-categories, so maybe there is something $A_n$-happening there, but I'm confused as to how it might apply to the general case)

Mike Shulman (Jul 17 2023 at 19:32):

Traditionally in some fields an "$A_\infty$-category" means one enriched over chain complexes, but it doesn't have to -- it could be defined using an $A_\infty$-operad in any enrichment context, such as spaces or simplicial sets.

Mike Shulman (Jul 17 2023 at 19:33):

The latter aren't then necessarily stable.

Tim Hosgood (Jul 17 2023 at 19:33):

something I realised is that, if all 2-horns fill, then you can still define the simplicial/combinatorial $\pi_0$ (and normally you can only define the simplicial/combinatorial homotopy groups for Kan complexes)

Tim Hosgood (Jul 17 2023 at 19:34):

Mike Shulman said:

Traditionally in some fields an "$A_\infty$-category" means one enriched over chain complexes, but it doesn't have to -- it could be defined using an $A_\infty$-operad in any enrichment context, such as spaces or simplicial sets.

oh ok, this makes sense then

John Baez (Jul 17 2023 at 19:46):

Yes, an $A_\infty$ operad describes things that are "associative up to coherent homotopy", nothing stable. If you want to study things that are stable, i.e. "associative and commutative up to coherent homotopy", you use an $E_\infty$ operad.

Mike Shulman (Jul 17 2023 at 20:00):

I think Tim meant "stable" in the sense of spectra or chain complexes that already have "additive" commutativity built in, while the extra "multiplicative" $A_\infty$ structure may not be commutative. Like an $A_\infty$ ring spectrum.

John Baez (Jul 17 2023 at 20:10):

Oh, okay. Like a ring spectrum is additively stable but not multiplicatively. But yeah, algebras of $A_\infty$ operads don't need to have any additive structure.