Geodesic
Categorified trace for module tensor categories over braided tensor categories
Documenta mathematica, Tome 21 (2016), pp. 1089-1149
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Given a braided pivotal category C and a pivotal module tensor category M, we define a functor TrC​:M→C, called the associated categorified trace. By a result of Bezrukavnikov, Finkelberg and Ostrik, the functor TrC​ comes equipped with natural isomorphisms τx,y​:TrC​(x⊗y)→TrC​(y⊗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 TrC​ 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 TrC​(A) and TrC​(A⊗B) are again algebra objects. Moreover, provided certain mild assumptions are satisfied, TrC​(A) and TrC​(A⊗B) are semisimple whenever A and B are semisimple. 
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     author = {James Tener and Andr\'e Henriques and David Penneys},
     title = {Categorified trace for module tensor categories over braided tensor categories},
     journal = {Documenta mathematica},
     pages = {1089--1149},
     year = {2016},
     volume = {21},
     doi = {10.4171/dm/553},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/553/}
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James Tener; André Henriques; David Penneys. Categorified trace for module tensor categories over braided tensor categories. Documenta mathematica, Tome 21 (2016), pp. 1089-1149. doi: 10.4171/dm/553

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