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- W4313854816 abstract "Given a braided pivotal category $mathcal C$ and a pivotal module tensor category $mathcal M$, we define a functor $mathrm{Tr}_{mathcal C}:mathcal M to mathcal C$, called the associated categorified trace. By a result of Bezrukavnikov, Finkelberg and Ostrik, the functor $mathrm{Tr}_{mathcal C}$ comes equipped with natural isomorphisms $tau_{x,y}:mathrm{Tr}_{mathcal C}(x otimes y) to mathrm{Tr}_{mathcal C}(y otimes x)$, which we call the traciators. This situation lends itself to a diagramatic calculus of `strings on cylinders', where the traciator corresponds to wrapping a string around the back of a cylinder. We show that $mathrm{Tr}_{mathcal C}$ in fact has a much richer graphical calculus in which the tubes are allowed to branch and braid. Given algebra objects $A$ and $B$, we prove that $mathrm{Tr}_{mathcal C}(A)$ and $mathrm{Tr}_{mathcal C}(A otimes B)$ are again algebra objects. Moreover, provided certain mild assumptions are satisfied, $mathrm{Tr}_{mathcal C}(A)$ and $mathrm{Tr}_{mathcal C}(A otimes B)$ are semisimple whenever $A$ and $B$ are semisimple." @default.
- W4313854816 created "2023-01-10" @default.
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- W4313854816 date "2015-09-09" @default.
- W4313854816 modified "2023-10-16" @default.
- W4313854816 title "Categorified trace for module tensor categories over braided tensor categories" @default.
- W4313854816 doi "https://doi.org/10.48550/arxiv.1509.02937" @default.
- W4313854816 hasPublicationYear "2015" @default.
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