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 Frölich 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.
Dedicated to Rob Kirby, on the
occasion of his 60th birthday.
Keywords
topological quantum field theory, braided
category
