The linearity of traces in monoidal categories and bicategories

TL;DR

This paper demonstrates that in symmetric monoidal categories, absolute weights for colimits preserve dualizability and allow trace calculations to be expressed as linear combinations, with broad generalizations to homotopical and bicategorical contexts.

Contribution

It establishes a general framework for trace linearity in monoidal categories and extends these results to homotopical and bicategorical settings, unifying various classical theorems.

Findings

Dualizability is preserved under certain colimits in symmetric monoidal categories.

Trace of colimit endomorphisms can be computed as linear combinations of object traces.

Generalizations include additivity of Euler characteristic, Lefschetz number, and orbit-counting theorems.

Abstract

We show that in any symmetric monoidal category, if a weight for colimits is absolute, then the resulting colimit of any diagram of dualizable objects is again dualizable. Moreover, in this case, if an endomorphism of the colimit is induced by an endomorphism of the diagram, then its trace can be calculated as a linear combination of traces on the objects in the diagram. The formal nature of this result makes it easy to generalize to traces in homotopical contexts (using derivators) and traces in bicategories. These generalizations include the familiar additivity of the Euler characteristic and Lefschetz number along cofiber sequences, as well as an analogous result for the Reidemeister trace, but also the orbit-counting theorem for sets with a group action, and a general formula for homotopy colimits over EI-categories.