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We show that in any symmetric monoidal category, if a weight for colimits isabsolute, then the resulting colimit of any diagram of dualizable objects isagain dualizable. Moreover, in this case, if an endomorphism of the colimit isinduced by an endomorphism of the diagram, then its trace can be calculated asa linear combination of traces on the objects in the diagram. The formal natureof this result makes it easy to generalize to traces in homotopical contexts(using derivators) and traces in bicategories. These generalizations includethe familiar additivity of the Euler characteristic and Lefschetz number alongcofiber 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 generalformula for homotopy colimits over EI-categories.

The linearity of traces in monoidal categories and bicategories