The n-Category Café

But should these higher cells be invertible, at least in a sufficiently weak sense?

Yes.

If so, then to answer Jim’s question: a strong, not lax, sort of pseudo.

Ah. I have become used to the convention

“lax”:= up to directed cells,

“psedo”:=up to weak isos.

But I intentionally did not want to be specific about this weakining of the respect for pullbacks in $\infty$-stacks here. The question I ask above is really: can’t we just restrict to $\infty$-stacks where everything pulls back strictly.

A kind of $n$Lab:semi-strictification.

We expect for any notion of $\infty$-category an $\infty$-Yoneda lemma. Using this as described above would seem to provide an explicit way to rectify any $\infty$-stack.

(I should mention that this goes back to discussion I am having with Thomas Nikolaus.)

More concretely, one can read one of the main results of my work with Konrad Waldorf as saying that for instance

- the 1-stack of bundles with connections has a natural rectification; ()

- the 2-stack of 2-bundles with connection has a natural rectification

These statement follow a pattern which one clearly expects to continue to $\infty$:

$n$-bundles with connection are given by the $n$-stack which is the $\infty$-stackification of the (naturally recitified) prestack of globally smooth parallel transport functors:

$$G-TrivBund_\nabla(-) : U \mapsto hom(P_n(U), \mathbf{B}G) \,.$$

See for instance the first two pages of these notes.

Konrad and I naturally rectify (= make restriction maps be strictly preserved) the $\infty$-stackification of this by using “parallel transport $n$-functors”, namely by taking

$$G-Bund_\nabla(-)$$

to send $U$ to the $n$-category of “parallel transport” $n$-functors $$U \mapsto hom_{nCat(Sets)}(P_n(U), \mathbf{B}G) \cap \{has local smooth trivialization\} \,,$$ i.e. $n$-functors as before, but now allowed to not respect the ambient topological/smooth structure globally, but required to have the property of admitting a “smooth local trivialization”.

We prove that such “(locally smoothly trivializable) =: (parallel transport)-$n$-functors” are equivalent to differential nonabelian $n$-cocycles, hence to $n$-bundles with connection (as I said, done for $n \leq 2$, but the general construction is certainly expected to go through for all $n$). This means that the $n$-stack of parallel transport $n$-functors is a natural rectification of the $n$-stack pseudofunctor of $n$-bundles with connection.

The above Yoneda question is supposed to support the expectation that this goes through for $n \to \infty$ from a different angle.

In fact, there is a model for $\infty$-stacks which are fully rectified (i.e. are just plain contravariant functors on the underlying site) in the context of Segal categories, one model for $(\infty,1)$-categories:

this is recalled on the top of page 12 in Toën’s review:

it says there that, in this context, a model for $\infty$-stacks on a site $C$ is given by precisely those presheaves on $C$ with values in simplicial sets (“rectified pre-$\infty$-stacks”) which satisfy descent.

This is precisely the kind of statement I am referring to. Similarly we say, following Street, that an $\omega$-category valued (co)presheaf is an $\omega$-(co)stack if it satisfies $\omega$-(co)descent.