  A group-category is an additively semisimple category with a monoidal product
structure in which the simple objects are invertible. For example in the
category of representations of a group, 1-dimensional representations are the
invertible simple objects. This paper gives a detailed exploration of
"topological quantum field theories" for group-categories, in hopes of finding
clues to a better understanding of the general situation. Group-categories are
classified in several ways extending results of Froelich and Kerler.
Topological field theories based on homology and cohomology are constructed,
and these are shown to include theories obtained from group-categories by
Reshetikhin-Turaev constructions. Braided-commutative categories most naturally
give theories on 4-manifold thickenings of 2-complexes; the usual 3-manifold
theories are obtained from these by normalizing them (using results of Kirby)
to depend mostly on the boundary of the thickening. This is worked out for
group-categories, and in particular we determine when the normalization is
possible and when it is not.

Geometric Topology;Quantum Algebra