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

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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.

http://www.tac.mta.ca/tac/volumes/18/4/18-04.dvi

http://www.tac.mta.ca/tac/volumes/18/4/18-04.ps

http://www.tac.mta.ca/tac/volumes/18/4/18-04.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/18/4/18-04.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/18/4/18-04.ps

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