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There is a lot of work of course on convolution algebras of measured groupoids, and this gives "Noncommutative geometry". However there is a lot of interest in algebraically structured groupoids, for example groupoids internal to the categories of say groups, or groupoids, or Lie algebras. Thus double groupoids, i.e. groupoids internal to groupoids, can be seen as "more noncommutative" than groupoids. They are also quite difficult to understand in general, though special cases have been studied extensively, e.g. 2-groupoids, and what are called 2-groups (groupoids internal to groups). They all have relations with crossed modules.

How does one find then a jacking up of Noncommutative Geometry to take into account these algebraically structured groupoids?

The usual formula for a convolution of functions $f,g$ on a finite groupoid $G$ given by

$$(f*g)(z) = \sum _{xy=z} f(x)g(y)$$

can be extended to a kind (or rather many kinds!) of matrix convolution looking at all decompositions of $z$ as a matrix composition (I find it difficult to write this down in this system!) but it all depends on the size $m \times n$ of the matrix, and because one needs the interchange law it all gets complicated and not reducible to the individual compositions in the double groupoid.

If this could be done, it might open up new worlds!

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Pedro Resende understands this well. The interchange law in a double algebra (defined by Resende) is not satisfied by a double groupoid convolution algebra but I think that doesn't necessarily mean that a category fibred over the double groupoid is not a double category. So you can drop the interchange law, well at least that is what we considered doing. In the end it seemed that the idea of a weak Hopf algebra by Natale and Andruskiewitsch was the best approach! (So there is already a counterpart of a Hopf algebra for a group coming from coproduct if not a counterpart of a group convolution algebra coming from product.)

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  • $\begingroup$ Again, Pedro Resende understands this far better....if such an algebra were to satisfy the interchange law (in the sense of a unital double algebra, i.e. as a condition on the elements of an algebra with two mult laws) then it would collapse to just an ordinary algebra by the same calculations as in the Eckman-Hilton argument. (A double algebra turns out to be an algebra.) $\endgroup$ – Rachel Dec 17 '12 at 16:27

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