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 noone 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 2categories as vertically discrete double categories. Much interest in double categories has been to use them to study 2categories. I thought at one point I could use higher multiple categories to easily obtain a simplicial nerve for an ncategory; 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 ncategory theory. Double categories in the category of groups are Loday’s algebraic model for connected homotopy 3types; but I mostly prefer the JoyalTierney 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.
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 `LAgroupoid’ (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.