Interpretations of Yetter's Notion of G-coloring : Simplicial Bre Bundles and Non-abelian Cohomology

In 20], Yetter makes the following deenition: Fix a nite group, G. For any space X and a triangulation, T, a G-coloring of T is a map : T (1) ! G such that given any 2 T (2) , (e 1) " 1 (e 2) " 2 (e 3) " 3 = 1, whenever @ = e " 1 1 e " 2 2 e " 3 3 , for " i = 1 denoting in the rst expression non-inversion or inversion in the group G and in the second preservation or reversal of orientation. We denote the set of all G-colorings of T by G (T). Yetter then deenes Z G (X; T) to be the vector space having G (T) as basis. Restricting to the case where X is a surface, he shows that if T 0 is a triangulation obtained from T by iterated subdivision of edges, then there is a well deened map res T 0 ;T : G (T 0) ! G (T) which induces a map res T 0 ;T on the corresponding vector spaces. The Z G (X; T)s and res T 0 ;T deene a diagram of vector spaces and he takes Z G (X) to be the colimit of this diagram. It is then shown that 1. this deenes a (2 + 1)-dimensional topological quantum eld theory in the sense of Atiyah 1]; 2. the vector space Z G (X) is isomorphic to the vector space whose basis is the set of conjugacy classes of representations from (X) to G (i.e. of natural isomorphism classes of functors from (X) to G, regarded as a groupoid with one object). Yetter then extends the construction to take coeecients in a crossed G-set and in a second paper, 21], shows how to adapt the method to handling coeecients in an algebraic model of a homotopy 2-type. In both cases the theory gives a TQFT and there are hints at an interpretation in terms analogous to 2. above. Here we will provide alternative proofs of some of Yetter's results. This gives an interpretation in terms of simplicial bre bundles and of 2-descent data or non-abelian cocycles.