## May 27, 2008

### Double Categories — Warm and Cuddly?

#### Posted by John Baez

This abstract of a forthcoming talk by Ross Street is sort of interesting… and not just because it’s the first abstract of a math talk I’ve seen that contains the word ‘cuddly’.

• Ross Street, Double categories for better or worse, Australian Category Seminar, Macquarie University, Wednesday May 28, 2008.

Abstract: I suspect no-one actually thinks of double categories as warm and cuddly, or wants to study them for their own sakes. Usually they are studied as a means to an end. Ehresmann defined them but I suspect they were incidental to his interest in internal categories and groupoids for differential geometry. He also defined 2-categories as vertically discrete double categories. Much interest in double categories has been to use them to study 2-categories. I thought at one point I could use higher multiple categories to easily obtain a simplicial nerve for an n-category; but that turned out of not much help. Of course, a double category has an obvious nerve as a double simplicial set, and this has had important ramifications for weak n-category theory. Double categories in the category of groups are Loday’s algebraic model for connected homotopy 3-types; but I mostly prefer the Joyal-Tierney model. Having said all that, there is a definite appeal to trying to develop category theory in a category E replacing Set and then trying to apply it to the case E = Cat… and so on. For this, we cannot develop the theory merely for a topos E since Cat is not a topos. Then there are the questions of size, weak equivalence, and so on; how do they iterate? But the goal is a short talk.
Posted at May 27, 2008 12:38 AM UTC

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### Re: Double Categories — Warm and Cuddly?

Double and higher groupoids are of interest in their own right. They
arise in nature from Poisson Lie groups and from constructions in theoretical mechanics.

The concept of Poisson Lie group is not categorical’ by which I mean that although the multiplication is a Poisson map the other maps (inversion, inclusion of the identity, translations) are not Poisson (though inversion is antiPoisson). But the group structure induces a group_oid_ structure on the cotangent bundle and the Poisson structure induces a Lie algebroid structure ditto, and all the groupoid structure maps are then Lie algebroid morphisms. So the cotangent of a Poisson Lie group is an LA-groupoid’ (groupoid object in the category of Lie algebroids) and may (under weak onditions) be integrated to a double Lie groupoid and differentiated to a double Lie algebroid.

If a double groupoid satisfies double versions of transitivity it is
determined by what Ronnie Brown and I called its `core diagram’; for double groupoids which have rotational symmetry this reduces to the equivalence between crossed modules and cat^1 groups in the form given by Loday (1982).

Categories and groupoids are not the same, of course, and I think that the ifferences should be emphasized rather than overlooked. But those who damn category theory usually damn groupoids too.

So are double groupoids warm and cuddly? I would say they’re prickly but interesting.

Thanks to Bruce Bartlett for pointing me at Ross’ post.

Posted by: Kirill Mackenzie on June 27, 2008 12:01 PM | Permalink | Reply to this

### Re: Double Categories — Warm and Cuddly?

So the cotangent of a Poisson Lie group is an ‘LA-groupoid’ (groupoid object in the category of Lie algebroids) and may (under weak onditions) be integrated to a double Lie groupoid and differentiated to a double Lie algebroid.

At the conference in Sheffield (or Bakewell, rather) I was wondering how double Lie algebroids (which I haven’t really studied yet much, I have to admit) relate to categories/groupoids internal to the category of Lie algebroids. Maybe I asked a question about that, but I am not sure what the answer is.

Can you say more about this? Are double Lie algebroids equivalent to groupoids internal to Lie algebroids? If not, how can one understand double Lie algebroids naturally (I mean, apart from giving their detailed definition)? Do you obtain their definition directly from starting with double Lie groupoids? (I suppose the answer to this might involve an answer to a similar question about double vector bundles (?))

Recently I have come across the thesis (I think) by Rajan Amit Mehta: Supergroupoids, double structures, and equivariant cohomology (which I also haven’t fully studied yet, due to lack of time) but which looks at groupoids internal to $\infty$-algebroids in the form of differential $\mathbb{N}$-graded commutative algebras.

There are several further structures here which naturally suggest themselves, and of which it would be good to know how they all hang together:

either of the following should make sense and should be equivalent to something like strict $\infty$ Lie algebroids, I suppose:

strict $\infty$-groupoids internal to Lie algebroids; simplicial Lie algebroids; Kan complexes internal to Lie algebroids.

I really have to try and work through that thesis of Rajan Mehta - it seems like it is a nice introduction to $Q$-manifolds and so on. Rajan and Alfonso Gracia-Saz were actually in Barcelona for some of the talks at the categorical groups conference.