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December 13, 2006

Bicat is Not Triequivalent to Gray

Posted by David Corfield

Stephen Lack has a paper with that title out today. Abstract:

Bicat is the tricategory of bicategories, homomorphisms, pseudonatural transformations, and modifications. Gray is the subtricategory of 2-categories, 2-functors, pseudonatural transformations, and modifications. We show that these two tricategories are not triequivalent.

Posted at December 13, 2006 8:43 AM UTC

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Re: Bicat is Not Triequivalent to Gray

Since Toby did me the favour of summarizing my paper here, I will do Steve the favour (?) of summarizing his paper here.

As it says in his abstract (in old-fashioned terminology), he compares the tricategories

Bicat = weak 2-categories + weak functors + weak transformations + modifications

(everything weak) and

Gray = strict 2-categories + strict functors + weak transformations + modifications

(a mixture of strengths).

Steve proves that Bicat and Gray are not triequivalent. (I’ll drop the ‘tri’.) And Steve, being a precise person, really means that. In other words, he doesn’t just prove that the canonical inclusion GrayBicat\mathbf{Gray} \rightarrow \mathbf{Bicat} is not an equivalence; he proves that there is no equivalence in the world.

Actually, the first step is to prove that this inclusion isn’t an equivalence. The proof boils down to exhibiting a weak functor between strict 2-categories that is not equivalent to any strict functor between them. This is not new (though may be due to Steve originally).

The second step is to show that if there were any equivalence between Bicat and Gray, then the inclusion would be an equivalence. But we’ve just seen that it isn’t: QED. This is the new part.

Steve then makes some comments about coherence for bicategories. As I understand it, the thought lurking in the background is the following: if you cite the coherence theorem as ‘every weak 2-category is equivalent to a strict one’ then you’re ignoring functors, transformations etc between 2-categories. To be more holistic, we should be looking for a statement of the coherence theorem that takes account of them. Steve’s result has a bearing on this.

I’ll probably say something about this in my Fields talk next month, though presumably Steve will present his result in his own talk.

Posted by: Tom Leinster on December 13, 2006 6:07 PM | Permalink | Reply to this

Re: Bicat is Not Triequivalent to Gray

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.

Posted by: Eugenia Cheng on December 15, 2006 1:48 AM | Permalink | Reply to this

Re: Bicat is Not Triequivalent to Gray

A correction:

I described the result that the inclusion GrayBicat\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).

Posted by: Tom Leinster on December 15, 2006 11:56 PM | Permalink | Reply to this

Re: Bicat is Not Triequivalent to Gray

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!

Posted by: John Baez on December 16, 2006 12:53 AM | Permalink | Reply to this

Re: Bicat is Not Triequivalent to Gray

Is there a useful way to characterize the “complement” of Gray\mathbf{Gray} in Bicat\mathbf{Bicat}, up to equivalence?

For instance: what would be the largest sub-tricategory of Bicat\mathbf{Bicat} that is still (tri-)equivalent to Gray\mathbf{Gray}?

I have a concrete application in mind when asking this question. I am wondering about the following:

Given a smooth space XX, there are several flavors of strict 2-categories of 2-paths in XX. I am working on understanding what strict smooth 2-functors with weak transformations and modifications between them on these strict 2-categories are like. In particular, these 2-functors tend to be expressible in terms of pp-form data on XX, 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. Is there an entire world of interesing 2-bundles-with-connection structures missed by that assumption, or are the examples missed maybe just “uninteresting” generalizations, in some sense.

Posted by: urs on December 13, 2006 10:00 PM | Permalink | Reply to this

Re: Bicat is Not Triequivalent to Gray

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 pp-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)P_2(M) in a smooth space MM 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)Ghol: P_2(M) \to G

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

(Here GG 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:(xy)z(xy)za_{x,y,z} : (x \otimes y) \otimes z \to (x \otimes y) \otimes z

Since a x,y,za_{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)F(y)F(xy)F_{x,y}: F(x) \otimes F(y) \cong F(x \otimes y)

Since F x,yF_{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.

Posted by: John Baez on December 16, 2006 2:30 AM | Permalink | Reply to this

Re: Bicat is Not Triequivalent to Gray

the basic “problem” […] comes from weak functors between strict 2-categories that aren’t equivalent to strict functors.

Right, okay.

maybe what we should do is look at weak 2-functors […] and see if some fail to be equivalent to strict ones.

I see.

this obstruction lies in H 2(G,A)H_2(G,A'),

All right. I once played around with this “by hand”, trying to figure out what a weak 2-functor from 2-paths to a 2-group would be like.

But before anything came out of this playing around I got more interested in looking at pseudo-2-functors into a 3-group (namely inner automorphisms of the original 2-group) that are “as strict as possible”.

While there seemed to be something interesting coming out of this, clearly the full picture here - more along general lines as you sketched - has not emerged yet (not to my mind at least).

Posted by: urs on December 17, 2006 2:08 PM | Permalink | Reply to this

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