Traces in monoidal derivators, and homotopy colimits

TL;DR

This paper extends the concept of trace to closed monoidal derivators, providing a functorial framework and an explicit formula for homotopy colimits, generalizing classical trace additivity in a homotopical setting.

Contribution

It introduces a trace in monoidal derivators applicable to fiberwise dualizable objects and derives a formula for the trace of homotopy colimits over finite EI-categories.

Findings

Established functoriality of the trace in derivators.

Derived an explicit formula for traces of homotopy colimits.

Generalized additivity of traces to a homotopical context.

Abstract

A variant of the trace in a monoidal category is given in the setting of closed monoidal derivators, which is applicable to endomorphisms of fiberwise dualizable objects. Functoriality of this trace is established. As an application, an explicit formula for the trace of the homotopy colimit of endomorphisms over finite EI-categories is deduced. This result can be seen as a generalization of the additivity of traces in monoidal categories with a compatible triangulation.