The n-Category Café

Some of what Tom describes is covered in Nick Gurski’s thesis “Algebraic tricategories”. He addresses the, as Tom puts it, “holistic” approach to coherence for bicategories.

A correction:

I described the result that the inclusion $\mathbf{Gray} \rightarrow \mathbf{Bicat}$ is not a triequivalence as “not new”. But actually, this seems to be its first time in print, even though the main ingredient in Steve’s proof appeared a few years ago (in another paper of Steve’s).

What happens when we restrict to the 2-groupoidal case?

In other words, we’ve got

BiGpd = [weak 2-groupoids, weak functors, weak transformations, modifications]

and

[strict 2-groupoids, strict functors, weak transformations, modifications]

Are they triequivalent?

I suspect they’re not.

BiGpd is important, because it should be triequivalent to the tricategory of homotopy 2-types. (Has someone shown this yet? I only know partial results along these lines.)

So, it would be nice to see how far we can go towards ‘strictifying’ the objects, morphisms, etc. in BiGpd while still getting a triequivalent tricategory. But, my hunch is that

[strict 2-groupoids, strict functors, weak transformations, modifications]

is not triequivalent to BiGpd.

I believe every weak 2-groupoid is equivalent to a strict one inside BiGpd. So, I guess that BiGpd is triequivalent to

[strict 2-groupoids, weak functors, weak transformations, modifications]

Similarly, I guess that BiCat is triequivalent to

[strict 2-categories, weak functors, weak transformations, modifications]

If any of you know results that turn my hunches into theorems, I’d love to hear about them!

Urs wrote:

Given a smooth space X, there are several flavors of strict 2-categories of 2-paths in X. I am working on understanding what strict smooth 2-functors with weak transformations and modifications between them on these strict 2-categories are like.

Needless to say, I’m fascinated by this too.

In particular, these 2-functors tend to be expressible in terms of $p$-form data on X, something that can be traced back to strictness of the domain 2-path 2-category.

I am wondering how much information is lost by restricting to strict 2-paths here.

From Tom’s summary, you can see the basic “problem” doesn’t come from weak 2-categories that aren’t equivalent to strict ones - they all are. It comes from weak functors between strict 2-categories that aren’t equivalent to strict functors.

So, instead of trying to replace the strict 2-category of 2-paths $P_2(M)$ in a smooth space $M$ by some cleverly defined weak 2-category of 2-paths, maybe what we should do is look at weak 2-functors

$hol: P_2(M) \to G$

and see if some fail to be equivalent to strict ones.

(Here $G$ can still be a strict 2-group - i.e. a strict 2-category with one object, with all morphisms and 2-morphism strictly invertible.)

I think in the case of 2-functors between strict 2-groupoids, there is a cohomological obstruction to replacing a weak 2-functor by an equivalent weak one. Understanding this might help us see what’s really going on here.

One can get a sense of this by recalling the cohomological description of the theory of 2-groups given by Joyal and Street and reviewed in HDA5. The idea is that you can use group cohomology to describe a strict 2-category equivalent to

[weak 2-groups, weak monoidal functors, monoidal natural transformations]

Using this, you can easily see the obstruction to finding a strict skeletal 2-group that’s equivalent to a given weak one. This obstruction comes from the associator

$$a_{x,y,z} : (x \otimes y) \otimes z \to (x \otimes y) \otimes z$$

Since $a_{x,y,z}$ eats three objects and spits out a morphism, this obstruction lies in H³(G,A), where G is the group of objects of our 2-group and A is the group of endomorphisms of the unit object.

(We assume without loss of generality that our 2-group is skeletal so its objects form a group.)

This obstruction is well-known - it’s called the Sinh invariant, named not after the hyperbolic sine function but after a student of Grothendieck.

But, this group cohomology stuff also lets you see the obstruction to finding a strict monoidal functor between 2-groups that’s isomorphic to a given weak one F: C → C’. The obstruction comes from the natural isomorphism

$$F_{x,y}: F(x) \otimes F(y) \cong F(x \otimes y)$$

Since $F_{x,y}$ eats two objects of C and spits out a morphism in C’, this obstruction lies in H²(G,A’), where now G is the group of objects of the 2-group C and A’ is the group of endomorphisms of the unit object of the 2-group C’.

(Again, we assume without loss of generality that C is skeletal.)

Of course, all this will get a bit more elaborate in the smooth context — and also more elaborate if we’re thinking about the weak 3-category of weak 2-groupoids, instead of the strict 2-category of weak 2-groups.

But, it should be a lot of fun to think about.