The linearity of traces in monoidal categories and bicategories

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.

References in corpus (6)

• Homotopy limits and colimits and enriched homotopy theory
• Duality and traces for indexed monoidal categories
• The theory and practice of Reedy categories
• The multiplicativity of fixed point invariants
• Traces in monoidal derivators, and homotopy colimits
• Coincidence invariants and higher Reidemeister traces

Cited by in corpus (16)

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• $t$-Structures on stable derivators and Grothendieck hearts
• Abstract representation theory of Dynkin quivers of type A
• Tilting theory for trees via stable homotopy theory
• Chromatic Cyclotomic Extensions
• Iterated chromatic localisation
• Characterizations of abstract stable homotopy theories
• Universal property of triangulated derivators via Keller's towers
• Abstract cubical homotopy theory
• Generalized stability for abstract homotopy theories
• Derivator Six-Functor-Formalisms -- Construction II
• Characters and transfer maps via categorified traces
• The bivariant parasimplicial ${\mathsf{S}}_{\bullet }$-construction
• Exactness of Bousfield localizations of simplicial presheaves and local lifting
• Euler characteristic and homotopy cardinality
• Higher symmetries in abstract stable homotopy theories