Authors define and analyze a monoid of endofunctors on the stable homotopy category that includes chromatic localizations, develop an axiomatic framework usable in equivariant settings, and formalize combinatorial parts in Lean.
The linearity of traces in monoidal categories and bicategories
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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.
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math.AT 1years
2019 1verdicts
UNVERDICTED 1representative citing papers
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Iterated chromatic localisation
Authors define and analyze a monoid of endofunctors on the stable homotopy category that includes chromatic localizations, develop an axiomatic framework usable in equivariant settings, and formalize combinatorial parts in Lean.