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September 26, 2006

Our Raison D’être

Posted by David Corfield

Marni Sheppeard reports from the AustMS2006 conference, which, as anyone who knows about Australian mathematics might expect, is holding a category theory session. Dominic Verity is giving one of the talks, in which he considers the raison d’être for higher category theory, and so by extension that of the Café. Of course, we also come here for the coffee.

Posted at September 26, 2006 1:51 PM UTC

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Re: Our Raison D’être

Dominic Verity wrote:

Non-abelian cohomology as the raison d’être for higher category theory

What would you all understand under “non-abelian”?

Am I mislead if I feel that if “non-abelian” cohomology is the raison d’être of anything, then of higher groupoid theory (only)?

I have never seen a general definition: “non-abelian cohomology is…”, but all the examples I know are of the form:

let CC be a 1-groupoid and TT be an nn-groupoid, then an object in the nn-groupoid of pseudofunctors

(1)[C,T] [C,T]

is a non-abelian cocycle with values in TT, a morphism between two of these is a coboundary, and so on. The cohomology is hence the (set, usually) [C,T] [C,T]_\sim of equivalence classes in [C,T][C,T].

By removing the requirement that CC be a 1- and TT be an nn-groupoid (in some weak sense), we can get something even more general, which I wouldn’t in general compare to “abelian” anymore.

Also, is the idea really that the concept [C,T] [C,T]_\sim is the raison d’être of category theory? Or is my concept of non-abelian cohomology above too narrow-minded?

Posted by: urs on September 26, 2006 3:07 PM | Permalink | Reply to this

Re: Our Raison D’être

Hi all.

I’m at the conference too, so I’ll put in a few words of what I think this might mean (and say more after the fact).

So John Roberts in the late 70’s realised that n-dimensional cohomology should have an n-category as the coefficient object. For the usual abelian group-valued cohomology, this is just the groupoid with only its n-morphisms non-trivial, consisting of the group AA (written as, say Σ nA\Sigma^nA). Ross Street went further and wrote a paper saying cohomology should have as `base space’ an omega-category and should have as coefficients an ω\omega-category. We all know ω\omega-groupoids are secretly topological spaces, so we’ve been doing some of this part already.

Explicitly (this is talked about in John and Mike’s cohomology notes) this is all just a big, big generalisation of Schreier theory and Galois theory - realised by Grothendieck. So in dimension one we have that opfibrations over a category XX are equivalent to [X,Cat][X,\mathbf{Cat}] (at least, that’s what I recall and I’m pressed for time - please someone tell me if I have egg on my face).

It’s just that we are used to dealing with spaces = homotopy types = groupoids. There is work on `directed homotopy theory’, in which paths are thought of (I gather) as paths in categories, not groupoids. There of course, things are not necessarily invertible.

Posted by: David Roberts on September 26, 2006 11:54 PM | Permalink | Reply to this

Re: Our Raison D’être

Explicitly (this is talked about in John and Mike’s cohomology notes) this is all just a big, big generalisation of Schreier theory and Galois theory - realised by Grothendieck.

And knowing that Klein’s Erlanger Program was part of the same story got me started on our Klein 2-geometry quest. What I find amazing is that more work isn’t being done on similarly concrete lines, such as we are trying to carry out on specific Lie 2-groups. Surely enough is now known about higher category theory that if applied in fairly concrete situations either there will be a rich pay-off or else we should be worried. Why is there not more being done on categorifying basic undergraduate Galois theory? A step further, should we not expect treasures to be found categorifying differential galois theory (or here)? I can understand the thrill of attempting to sort out the definitions of ω-categories, and I also know that categorification can be hard work, but if one wanted to run the higher categories program rationally would one not direct more resources to such concrete questions as ‘What is projective 2-space?’?

Posted by: David Corfield on September 27, 2006 10:58 AM | Permalink | Reply to this

Re: Our Raison D’être

I hadn’t read the next post yet, so my comment above seems dated. My question is:

Given a directed space XX and a category CC, do we have an adjunction

Hom Cat(Π 1 cX,C)Hom dTop(X,||NC||)Hom_{Cat}(\Pi_1^cX,C) \simeq Hom_{dTop}(X,||NC||)

where Π 1 c\Pi_1^c is the fundamental category? (cc is for category - I’ll append this whenever necessary to differentiate it from the groupoid case) Or put another way, are categories equivalent to directed homotopy types?

This would, I imagine, require delicate thought about what `vanishing homotopy’ is, because the fundamental monoid π 1 c\pi_1^c at a basepoint xx (equal to the endomorphisms of xx in the fundamental category) contains less information about Π 1 c\Pi_1^c than π 1\pi_1 does about Π 1\Pi_1. Clearly this is because we can’t set up an isomorphism between the fundamental monoids at different basepoints as in the groupoid case, and we can have different π 1 c\pi_1^c. But the required notion of `homotopically trivial’ is already done for us in Baez-Shulman; jj-homotopy is trivial if there is a (directed) j+1j+1-path between parallel (directed) jj-paths. I suppose it’s best to think of this is a lax version of homotopic (recall: pseudo- means up to invertible morphism one dimension higher, lax- means up to any old morphism one dimension higher).

an object in the n-groupoid of pseudofunctors [C,T] is a non-abelian cocycle with values in T

So what you were after for a cocycle category was the n-category of lax functors [X,T][X,T] between a category XX and an n-category TT (preferably semi-strict - what Street would call an n-file). The last section of Street’s `Orientals’ paper discusses all this.

Posted by: David Roberts on September 27, 2006 1:57 AM | Permalink | Reply to this

Re: Our Raison D’être

The last section of Street’s ‘Orientals’ paper discusses all this.

I am experiencing difficulties with retrieving this document.

Instead I have taken a look at Street’s descent theory paper.

The descent 2-category spelled out on pp. 2-3 there is exactly the type of 2-category which I encounter as the transition 2-category of 2-transport (my definition 5 here, can be regarded as the descent 2-category of a 2-stack of 2-transport functors).

If asked to identify the general nature of this 2-category, I would have said that it is an example of a 2-category of lax functors from a certain 1-category to a certain 2-category.

For instance the triangle on p.2 of Street’s paper above would be the compositor of these lax functors, the tetrahedron on the bottom of the page the coherence for these compositors, the square on top of p. 3 would be one face involved in peusdonatural transformations, the next diagram is the naturality condition for these pseudonatural transformations. Finally, the last diagram on p. 3 of Street’s “descent theory” is the equation for a modification of pseudonatural transformations.

Posted by: urs on September 28, 2006 5:05 PM | Permalink | Reply to this

Re: Our Raison D’être

It’s much too late to say that nonabelian cohomology is the raison d’etre for n-categories, since we’ve learned by now that n-categories are great for lots of things. I’d rather say nonabelian cohomology is a raison d’etre.

But one can argue that nonabelian cohomology was the raison d’etre, since we can see it looming over Grothendieck’s letter Pursuing Stacks, and also John Roberts’ paper that introduced n-categories in mathematical physics:

  • John E. Roberts, Mathematical aspects of local cohomology, in Algèbres d’Opérateurs et Leurs Applications en Physique Mathématique, CNRS, Paris, 1979, pp. 321–332.

And, it’s still fun to see nonabelian cohomology as the raison d’etre for n-categories - it’s a useful viewpoint.

As David Roberts points out, the idea is to subsume many different kinds of cohomology into one big generalization: the cohomology

H(X,Y)H(X,Y)

of a weak ω\omega-category XX with coefficients in a weak ω\omega-category YY. You can think of this as the set

[X,Y][X,Y]

of weak natural isomorphism classes of weak ω\omega-functors f:XYf : X \to Y. Or, even better, you can think of it as the weak ω\omega-category

hom(X,Y)hom(X,Y)

The point is that the set [X,Y][X,Y] tends to classify interesting `things over X’ up to equivalence, while hom(X,Y)hom(X,Y) describes the ω\omega-category of such things.

Topologists are most familiar with the example where XX is a space and Y=BGY = B G is the classifying space for GG-bundles, where GG is a topological group. We can think of XX, GG and BGB G as ω\omega-groupoids. Then [X,Y][X,Y] is the set of isomorphism classes of principal GG-bundles over XX, while hom(X,Y)hom(X,Y) is the ω\omega-groupoid of such bundles.

This example shows that nonabelian cohomology with coefficients in an ω\omega-groupoid is nothing new if you’re a topologist. The real fun starts when we hit ω\omega-categories.

But, Grothendieck was really interested in a big further generalization. In algebraic geometry, cohomology with coefficients in a group is regarded as a pathetically dull special case of cohomology with coefficients in a sheaf of groups! So, when we replace groups by ω\omega-categories, we should also replace sheaves by their ω\omega-categorified versions, which Grothendieck called “stacks”.

All this seems quite intimidating the first time you meet it, and also the second time, and the third time. That’s why I explained it very gently in my paper with Mike Shulman. We don’t go much beyond the ω\omega-groupoid case, but I think we explain some aspects of this business more thoroughly than anyone ever had.

Posted by: John Baez on September 29, 2006 5:58 PM | Permalink | Reply to this

morphisms of omega-groupoids

Above I admitted # that when thinking about nonabelian cohomology I am thinking about categories of functors

(1)[C,T] [C,T]

from a 1-groupoid to an nn-groupoid (for any nn, so probably I could just say ω\omega-groupoid).

If I understand correctly, John says in

the idea is to subsume many different kinds of cohomology into one big generalization: the cohomology

(2)H(X,Y) H(X,Y)

of a weak ω\omega-category XX with coefficients in a weak ω\omega-category YY.

that all I need to change in my point of view in order to obtain the established general point of view is that I allow the domain category to be not just a 1-category, but an ω\omega-category itself.

As an example, he offers the classical

example where XX is a space and Y=BGY=B G is the classifying space for GG-bundles, where GG is a topological group. We can think of XX, GG and BGB G as ω\omega-groupoids. Then [X,Y][X,Y] is the set of isomorphism classes of principal GG-bundles over XX, while hom(X,Y)\mathrm{hom}(X,Y) is the ω\omega-groupoid of such bundles.

Now, this classical example can just as well be described using just a domain 1-category.

I would let XX be the Čech-groupoid of any good covering of XX and let Y=Σ(G)Y = \Sigma(G) be the group GG regarded as a category with a single object.

Or to generalize, let Y=Σ(HG)Y = \Sigma(H \to G) be the suspension of any strict 2-group and [X,Y][X,Y] classifies (HG)(H\to G)-2-bundles (1-gerbes) on XX - for instance.

So now I am wondering: can I maybe always restrict to XX being just a 1-groupoid when talking about cohomology?

If not, what would be a counter example?

Posted by: urs on October 1, 2006 1:55 PM | Permalink | Reply to this

Re: morphisms of omega-groupoids

Here is a better link to the review of Street’s paper: The algebra of oriented simplexes

urs wrote:

So now I am wondering: can I maybe always restrict to X being just a 1-groupoid when talking about cohomology?

If not, what would be a counter example?

I would guess: let XX be the stack of G 2G_2-2-bundles, for G 2G_2 a topological/smooth 2-group. It’s cohomology a la Chern-Weil should give us a lot. I say `stack’, but we can present this by a bigroupoid, coming from G 2G_2 itself, namely ΣG 2\Sigma G_2.

Yes, I did let slip the s-word (`set’) there. Of course cohomology comes with coboundaries, being just (n1)(n - 1)-morphisms between nn-morphisms in some nn-functor groupoid. Since people don’t seem to have been grinding away at the specifics of non-abelian cohomology (with diagrams, say) for nn above 1, they haven’t needed to worry too much about coboundaries between coboundaries, or haven’t wanted to. However, there is a paper by Dedecker

Dedecker, Paul
Three dimensional non-abelian cohomology for groups.
1969 Category Theory, Homology Theory and their Applications, II (Battelle Institute Conference, Seattle, Wash., 1968, Vol. Two) pp. 32–64 Springer, Berlin

in which he treats the 3-cohomology of groups, with values in a `hypercrossed complex’==3-group, without unnecessarily dividing out by coboundaries - this paper needs a diagrammatic and n-categorical workout, as it all seems very mysterious without these two friends of ours.

Posted by: David Roberts on October 3, 2006 8:27 AM | Permalink | Reply to this

Re: morphisms of omega-groupoids

Here is a better link to the review of Street’s paper:

Thanks!

If not, what would be a counter example?

I would guess: let XX be the stack of G 2G_2-2-bundles, for G 2G_2 a topological/smooth 2-group. It’s cohomology a la Chern-Weil should give us a lot.

Which functor category [S,T][S,T] do you have in mind, where SS cannot be taken to be a 1-groupoid?

Posted by: urs on October 5, 2006 8:04 PM | Permalink | Reply to this

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