#### Geometry & Topology Monographs 2 (1999),
Proceedings of the Kirbyfest,
paper no. 21, pages 407-453.

## Group categories and their field theories

### Frank Quinn

**Abstract**.
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.
**Keywords**.
Topological quantum field theory, braided category

**AMS subject classification**.
Primary: 18D10.
Secondary: 81R50, 55B20.

**E-print:** `arXiv:math.GT/9811047`

Submitted: 9 November 1998.
(Revised: 27 January 1999.)
Published: 21 November 1999.

Notes on file formats
Frank Quinn

Department of Mathematics, Virginia Tech

Blacksburg VA 24061-0123, USA

Email: quinn@math.vt.edu

GT home page

## Archival Version

**These pages are not updated anymore.
They reflect the state of
.
For the current production of this journal, please refer to
http://msp.warwick.ac.uk/.
**