Shadows and traces in bicategories

By Kathleen Ponto, Michael Shulman. 2009

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Traces in symmetric monoidal categories are well-known and have many applications; for instance, their functoriality directly implies the Lefschetz fixed point theorem. However, for some applications, such as generalizations of the Lefschetz theorem, one needs “noncommutative” traces, such as the Hattori-Stallings trace for modules over noncommutative rings. In this paper we study a generalization of the symmetric monoidal trace which applies to noncommutative situations; its context is a bicategory equipped with an extra structure called a “shadow”. In particular, we prove its functoriality and 2-functoriality, which are essential to its applications in fixed-point theory. Throughout we make use of an appropriate “cylindrical” type of string diagram, which we justify formally in an appendix.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b4923f2fa0430b6c1d15c6825c5d4a82771a007b

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Citations in the corpus

Bicommutant categories from fusion categories (2015, methodology): “n we may dene the cyclic fusion in terms of the operation described in [BDH14a, Appendix A] as h H i 1+1 ::: H i 2 R i2 H i 2+1 ::: H i 3 R i3 :::::: R ik H i k+1 ::: H i 1 R i1 i cyclic Inspired by [PS13], we propose the following graphical calculus for morphisms between cyclic fusions. The Hilbert”
Categorified trace for module tensor categories over braided tensor categories (2015, methodology): “raciator. We point out that ˝ does not in general satisfy equation (4), but instead a mirrored version of that axiom. 13 Remark 3.2. A slightly less general notion of categoried trace was studied in [PS13] under the name \shadow". It diers from Denition3.1in that the authors also included the a”
Quantum link homology via trace functor I (2016, background): “ts the tube into a square (3.28). The string can be deformed (compare with (3.29)), but its endpoints must remain ﬁxed. In particular, performing a twist on the tube changes the morphism, contrary to [PS13]. The symmetric horizontaltraceissometimes called theannularization functor [MW10]. Its genera”