Crossed module

Abstract: We define the thin fundamental Gray 3-groupoid S3(M) of a smooth manifold M and define (by using differential geometric data) 3-dimensional holonomies, to be smooth strict Gray 3-groupoid maps S3(M) ! C(H), where H is a 2-crossed module of Lie groups and C(H) is the Gray 3groupoid naturally constructed from H. As an application, we define Wilson 3-sphere observables.

Abstract: We define the thin fundamental categorical group P2(M,�) of a based smooth manifold (M,�) as the categorical group whose objects are rank-1 homotopy classes of based loops on M, and whose morphisms are rank2 homotopy classes of homotopies between based loops on M. Here two maps are rank-n homotopic, when the rank of the differential of the homotopy between them equals n. Let C(G) be a Lie categorical group coming from a Lie crossed module G = (∂: E ! G, ⊲). We construct categorical holonomies, defined to be smooth morphisms P2(M,�) ! C(G), by using a notion of categorical connections, being a pair (ω, m), where ω is a connection 1-form on P, a principal G bundle over M, and m is a 2-form on P with values in the Lie algebra of E, with the pair (ω, m) satisfying suitable conditions. As a further result, we are able to define Wilson spheres in this context.

Abstract: We define the thin fundamental categorical group ${\mathcal P}_2(M,*)$ of a based smooth manifold $(M,*)$ as the categorical group whose objects are rank-1 homotopy classes of based loops on $M$, and whose morphisms are rank-2 homotopy classes of homotopies between based loops on $M$. Here two maps are rank-$n$ homotopic, when the rank of the differential of the homotopy between them equals $n$. Let $\C(\Gc)$ be a Lie categorical group coming from a Lie crossed module ${\Gc= (\d\colon E \to G,\tr)}$. We construct categorical holonomies, defined to be smooth morphisms ${\mathcal P}_2(M,*) \to \C(\Gc)$, by using a notion of categorical connections, being a pair $(\w,m)$, where $\w$ is a connection 1-form on $P$, a principal $G$ bundle over $M$, and $m$ is a 2-form on $P$ with values in the Lie algebra of $E$, with the pair $(\w,m)$ satisfying suitable conditions. As a further result, we are able to define Wilson spheres in this context.

Abstract: We give an interpretation of Yetter's Invariant of manifolds M in terms of the homotopy type of the function space TOP(M,B(G)), where G is a crossed module and B(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 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.