The n-Category Café

Seeing that Voevodsky gained much by thinking directly about univalent bundles, defined in terms of what pullbacks it affords (nLab: univalence axiom), could one not use the opportunity of the nice geometric model of Ayala, Francis, and Rozenblyum to support intuitions?

They have an important striation sheaf called $\mathcal{B}un$, which plays a universal role. For instance, it classifies constructible bundles of stratified spaces. What would it need to be to act as a “directed” universe?

Anyway, the general question is whether some help on the geometric front could assist with a “natively” directed type theory, similarly to how it may be helpful to think in terms of points and paths in HoTT.

Well, to play devil’s advocate a bit, I often feel these days that the longstanding implicit identification of $\infty$-groupoids with topological spaces was a serious inhibitor of progress in higher category theory, and one of the more important recent steps has been to disentangle the notions. So it seems somewhat backwards to suggest that a similar identification of $(\infty,1)$-categories with certain topological spaces would be helpful in understanding the former.

There are already plenty of candidates for directed universes: the $(\infty,1)$-category of $\infty$-groupoids; the $(\infty,1)$-category of $(\infty,1)$-categories and functors; its opposite; the $(\infty,1)$-category of $(\infty,1)$-categories and profunctors; etc. Each of them classifies something different, and satisfies some appropriate kind of “univalence”.

I certainly see some philosophical commentators on HoTT being led astray by the ‘path’ language, and point out, to whoever listens, passages such as your

Classically, $\infty$-groupoids arose to prominence gradually, as repositories for the homotopy-theoretic information contained in a topological space… As we will see, however, the synthetic viewpoint emphasizes that this structure of a “homotopy space” is essentially orthogonal to other kinds of space structure, so that an object can be both “homotopical” and (for example) “topological” or “smooth” in unrelated ways. (Homotopy type theory: the logic of space)

Perhaps we need a ‘rational reconstruction’ of how one might have arrived at the univalence axiom through a line of reasoning which did not rely on topological understanding.

How might such a rational narrative look? Could the type theorists have done it by themselves? Would there have to have been some crucial category theoretic insight?

Is he using arbitrary (possibly even small) stable $(\infty,1)$-categories, or locally presentable ones? A locally presentable stable $(\infty,1)$-category bears a specific formal resemblance to an abelian group, or more precisely a module. Locally presentable $(\infty,1)$-categories admit a tensor product that represents two-variable functors preserving colimits in each variable (“bilinear maps”). The monoids for this monoidal structure (“rings”) are closed monoidal locally presentable $(\infty,1)$-categories, and the “modules” over such a monoid are enriched $(\infty,1)$-categories. Stable locally presentable $(\infty,1)$-categories are then exactly the modules over the monoidal $(\infty,1)$-category of spectra.

He’s using arbitrary stable $\infty$-categories – perhaps it’s best to think of them as small. Of course, by taking Ind-categories everywhere, this could easily be recast as a story in the presentable setting. And anyway, the tensor product of presentable $\infty$-categories restricts to a tensor product of compactly-generated $\infty$-categories, or equivalently a tensor product of idempotent-complete $\infty$-categories with finite colimits, right?

It seems that Dyckerhoff’s functor $N$ from chain complexes of stable categories to 2-simplicial stable categories is a sort of iterated $S_\bullet$-construction. In particular,

1. If $B_\bullet$ is a stable $\infty$-category $B$ regarded as a chain complex of stable $\infty$-categories concentrated in degree 0, then $N(B)$ is basically the nerve of $B$ – $N(B)_n$ is the category of $[n]$-chains of morphisms in $B$.

2. If $B[1]_\bullet$ is a stable $\infty$-category $B$ regarded as a chain complex of stable $\infty$-categories concentrated in degree 1, then $N(B)$ is the $S_\bullet$ construction applied to $B$.

The first bullet tells us the standard way to regard an $\infty$-category $B$ as a 2-simplicial $\infty$-category $B_\bullet$: let $B_n = B^{\Delta[n]}$. This is a familiar construction from ordinary 2-category theory: $B$ is an object in a finitely-complete 2-category, then we get a 2-simplicial object by taking the powers $B^{[n]}$. In $Cat$, for example, this means we have simplicial maps between between $B^{[1]}$ and $B$, given by $codomain: B^{[1]} \to B$, $identityarrow: B \to B^{[1]}$, and $domain: B^{[1]} \to B$, and the 2-cells of a 2-simplicial object say that we have adjunctions $codomain \dashv identityarrow \dashv domain$ (which are fun to check yourself!). It’s an interesting fact that if $B$ is stable, then this adjoint string continues infinitely in both directions – I don’t know if this is relevant to Dyckerhoff’s construction.

The second bullet tells us that “delooping” is given by the $S_\bullet$ construction. And it seems roughly accurate to say that in general Dyckerhoff’s functor is a sort of “iterated $S_\bullet$” functor.

Here’s a speculative take on how to extend the analogy, which actually circumvents trying to understand what the analogs of “loops” or “spaces” are.

Dyckerhoff is pretty clear that:

• The analog of an abelian group is a stable $\infty$-category.

• The analog of a topological abelian group is a 2-simplicial stable $\infty$-category, localized at levelwise equivalences.

• The analog of a connective chain complex of abelian groups is a connective chain complex of stable $\infty$-categories, localized at levelwise equivalences.

And Dyckerhoff’s Dold-Kan theorem says that the latter two are equivalent, as expected. From what I’ve seen in the paper, it seems that

• The analog of delooping an abelian group is the $S_\bullet$ construction.

This jives with other contexts where $S_\bullet$ is seen as some kind of “suspension”. Let me get increasingly speculative:

• Note that for a 1-simplicial object to be 2-simplicial is basically a property rather than a structure: it says that the simplicial structure maps form adjoint strings. These strings of adjoints might be analogous to the trivialization of Postnikov invariants that happens for a topological abelian group. So perhaps the analog of a general connective spectrum is simply a 1-simplicial stable $\infty$-category. Then, for example, a double $\infty$-category (which is a certain type of 1-simplicial $\infty$-category which is typically not 2-simplicial) satisfying certain stability conditions would be an example of a “generalized connective spectrum”. This would accord with the fact that the $S_\bullet$ construction can be applied to double $\infty$-categories.

• In the ordinary context, we can think of an $\Omega$-spectrum as a sequence of infinite loop spaces $X_0, X_1, \dots$ with 1-coconnected infinite loop maps $BX_n \to X_{n+1}$. Analogously here, we would have a sequence of simplicial stable $\infty$-categories $X_0, X_1, \dots$ with maps $S_\bullet X_n \to X_{n+1}$. One sort of truly nonconnective example could be obtained from a suitable double $\infty$-category by taking $X_{n+1}$ to be like $S_\bullet X_n$, except without the condition that “diagonal” objects in the $S_\bullet$ construction be zero.