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Duality and traces for indexed monoidal categories

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Kate Ponto and Michael Shulman

By the Lefschetz fixed point theorem, if an endomorphism of a
topological space is fixed-point-free, then its Lefschetz number
vanishes. This necessary condition is not usually sufficient,
however; for that we need a refinement of the Lefschetz number called
the Reidemeister trace. Abstractly, the Lefschetz number is a trace
in a symmetric monoidal category, while the Reidemeister trace is a
trace in a bicategory; in this paper we relate these contexts using
indexed symmetric monoidal categories.

In particular, we will show that for any symmetric monoidal category
with an associated indexed symmetric monoidal category, there is an
associated bicategory which produces refinements of trace analogous to
the Reidemeister trace. This bicategory also produces a new notion of
trace for parametrized spaces with dualizable fibers, which refines
the obvious ``fiberwise'' traces by incorporating the action of the
fundamental group of the base space. We also advance the basic theory
of indexed monoidal categories, including introducing a string diagram
calculus which makes calculations much more tractable. This abstract
framework lays the foundation for generalizations of these ideas to
other contexts.

Keywords:
duality, trace, monoidal category, indexed category, fiberwise duality

2010 MSC:
18D10, 18D30

*Theory and Applications of Categories,*
Vol. 26, 2012,
No. 23, pp 582-659.

Published 2012-11-05.

http://www.tac.mta.ca/tac/volumes/26/23/26-23.pdf

A version with diagrams coded for black and white printers is at:

http://www.tac.mta.ca/tac/volumes/26/23/26-23bw.pdf

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