Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: theory: category theory

Topic: the (∞, 1)-category of strict ∞-groupoids

Amar Hadzihasanovic (Oct 22 2022 at 13:37):

Strict $n$-categories or $n$-groupoids only form a (large) strict $(n+1)$-category with “boring” higher cells, namely, “strict” natural transformations (made of strictly commutative squares), modifications, and so on; this is already visible when $n = 2$, in that (strict 2-categories, strict 2-functors, pseudonatural transformations, modifications) form a Gray-category and not a strict 3-category.

On the other hand, it should be expected that the $(n+1)$-categories with “interesting” higher cells are also somewhat strict, in that the hom-objects should have a structure of strict $n$-category, and weakness should only affect the horizontal composition of higher cells.

I had it in my mind that this question was addressed in some 90s papers, perhaps by Ronnie Brown, but I don't seem to find anything.
I will write some thoughts on it, in the case of strict $\infty$-groupoids even though much of it should apply to strict $\infty$-categories too.

With strict $\infty$-groupoids, higher homotopies can be represented using the closed monoidal structure given by the (Brown-Higgins) tensor product, so that, for $n > 0$, an $n$-cell corresponds to a morphism
$O^{n-1} \to \mathrm{Hom}(C, D)$
or equivalently
$O^{n-1} \otimes C \to D$
where the $\{O^k\}$ are classifying objects for $k$-cells (the groupoidal version of $k$-globes).

Since there are also classifying objects for compositions of cells, this also tells you how to compose $n$-cells in any direction except horizontally, and these compositions will satisfy associativity, interchange and unitality strictly.
It remains to define horizontal composition. But given an $n$-cell $x$ and $m$-cell $y$ that are horizontally composable, represented by morphisms
$x: O^{n-1} \otimes C \to D$,
$y: O^{m-1} \otimes D \to E$,
we do get a morphism
$y \circ (\mathrm{id} \otimes x): O^{m-1} \otimes O^{n-1} \otimes C \to E$,
and $O^{m-1} \otimes O^{n-1}$ is a composable pasting scheme of dimension $m+n-2$, which implies that this pulls back to an $(m+n-1)$ cell
$\chi_{x,y} : O^{n+m-2} \otimes C \to E$.

When $n = 1$ or $m = 1$ this has the right dimension for a composite of $x$ and $y$, and tells us how to “whisker” a higher cell with a 1-cell.
Otherwise, we can try splitting the composition into “whiskerings” + compositions along other directions. My intuition is that the $\chi_{x,y}$ should precisely be the coherences ensuring that the different ways of doing this are all equal up to higher homotopy.
For example, when $n = m = 2$, $\chi_{x,y}$ has exactly the type of a Gray-category interchanger.

So it seems to me that the $(\infty, 1)$-category of strict $\infty$-groupoids should naturally admit a semistrict structure that generalises that of a Gray-category, not in the sense of “weakening interchange wrt composition in all directions”, but only in the sense of having also higher coherences for interchange wrt horizontal composition.

Has such a notion of semistrict $(\infty, 1)$-category been defined somewhere?

Alexander Campbell (Oct 23 2022 at 09:19):

@Amar Hadzihasanovic

As you say, the category of strict ∞-groupoids is symmetric monoidal closed via the Gray tensor product. Are you after something other than categories enriched over this base?

Amar Hadzihasanovic (Oct 23 2022 at 09:55):

@Alexander Campbell Well, yes and no (and thank you for asking because my question needs clarifying).

“No” in the sense that the data that I'm describing does come from self-enrichment with the tensor product, and “a category enriched in strict $\infty$-groupoids with the tensor product” could eventually be a concise definition, in the same way as a Gray-category is concisely described as a category enriched in 2-categories with the “pseudo” Gray tensor product.

However, just as for Gray-categories, it is non-trivial that this data is equivalent to something that we want to call an algebraic higher category; I'm thinking, in this case, of something like “an algebra for a contractible globular operad” in the Batanin-Leinster framework. That is, that we can turn this data into globular composition operations, together with coherence data which ensures that composites of pasting diagrams are unique up to a contractible space of choices.
(It certainly is not the case for every self-enrichment; you can self-enrich strict $\omega$-categories with the lax Gray product and that will not give you anything we would like to call an $\omega$-category structure.)

So what I'm hoping for is an explicitation of this idea, i.e. a definition of a semistrict $(\infty, 1)$-category as “globular composition operations + a mixture of equations & coherence isomorphisms” which has this as a model...

John Baez (Oct 23 2022 at 12:09):

Amar Hadzihasanovic said:

On the other hand, it should be expected that the $(n+1)$-categories with “interesting” higher cells are also somewhat strict, in that the hom-objects should have a structure of strict $n$-category, and weakness should only affect the horizontal composition of higher cells.

Why should this be expected? Every weak 3-category is equivalent to one where the hom-2-categories are strict, but not every weak 4-category is equivalent to one where the hom-3-categories are strict.

I believe we can see this from the fact that not every weak 3-category is equivalent to a strict one.

Let C be a weak 3-category that's not equivalent to a strict one. Then we can create a 4-category with two objects $x,y$ and with $\mathrm{hom}(x,y)$ being the 3-category C, while $\mathrm{hom}(x,x)$ and $\mathrm{hom}(y,y)$ contain only identity cells and $\mathrm{hom}(y,x)$ is empty.

If every weak 4-category were equivalent to one where all the hom-3-categories are strict, I believe we could use this to find a strict 3-category equivalent to C. (I haven't checked this.)

Amar Hadzihasanovic (Oct 23 2022 at 12:11):

Oh I'm sorry, that sentence was ambiguous: I meant
“it should be expected that the $(n+1)$-categories of (small) strict $n$-categories with “interesting” higher cells are also somewhat strict”