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The linearity of traces in monoidal categories and bicategories

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Kate Ponto and Michael Shulman

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.

Keywords:
duality, trace, derivator, absolute colimit

2010 MSC:
18D05, 18D20, 55U30

*Theory and Applications of Categories,*
Vol. 31, 2016,
No. 23, pp 594-689.

Published 2016-06-29.

http://www.tac.mta.ca/tac/volumes/31/23/31-23.pdf

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