# Convolution algebras for double groupoids?

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!