# On Yetter's Invariant and an Extension of the Dijkgraaf-Witten Invariant to Categorical Groups

## Joao Faria Martins and Timothy Porter

We give an interpretation of Yetter's Invariant of manifolds M in terms of the homotopy type of the function space TOP(M,B(\cal G))\$, where \cal G is a crossed module and B(\cal G) is its classifying space. From this formulation, there follows that Yetter's invariant depends only on the homotopy type of M, and the weak homotopy type of the crossed module \cal G. We use this interpretation to define a twisting of Yetter's Invariant by cohomology classes of crossed modules, defined as cohomology classes of their classifying spaces, in the form of a state sum invariant. In particular, we obtain an extension of the Dijkgraaf-Witten Invariant of manifolds to categorical groups. The straightforward extension to crossed complexes is also considered.

Keywords: Categorical Groups; Crossed Modules; Cohomology of Crossed Modules; State Sum Invariants of Manifolds; Dijkgraaf-Witten Invariant; Yetter's Invariant

2000 MSC: 18F99; 55P99; 57M27; 57R56; 81T45

Theory and Applications of Categories, Vol. 18, 2007, No. 4, pp 118-150.

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