Duality and traces for indexed monoidal categories

TL;DR

This paper explores the relationship between Lefschetz and Reidemeister traces within indexed symmetric monoidal categories, introducing a new bicategorical trace that refines fixed-point invariants and advances the theoretical framework.

Contribution

It establishes a connection between symmetric monoidal categories and bicategories via indexed categories, introducing a new trace for parametrized spaces with dualizable fibers.

Findings

Refined trace analogous to Reidemeister trace in bicategories

New notion of trace for parametrized spaces with fundamental group action

String diagram calculus simplifies calculations in indexed monoidal categories

Abstract

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.