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Shadows and traces in bicategories
 J. Homotopy Relat. Struct
"... Abstract. Traces in symmetric monoidal categories are wellknown 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, s ..."
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Abstract. Traces in symmetric monoidal categories are wellknown 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 HattoriStallings 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 2functoriality, which are essential to its applications in fixedpoint theory. Throughout we make use of an appropriate “cylindrical ” type of string
DUALITY AND TRACES FOR INDEXED MONOIDAL CATEGORIES
, 2012
"... By the Lefschetz fixed point theorem, if an endomorphism of a topological space is fixedpointfree, then its Lefschetz number vanishes. This necessary condition is not usually sufficient, however; for that we need a refinement of the Lefschetz number called the Reidemeister trace. Abstractly, the ..."
Abstract

Cited by 9 (7 self)
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By the Lefschetz fixed point theorem, if an endomorphism of a topological space is fixedpointfree, then its Lefschetz number vanishes. This necessary condition is not usually sufficient, however; for that we need a refinement of the Lefschetz number called the Reidemeister trace. Abstractly, the Lefschetz number is a trace in a symmetric monoidal category, while the Reidemeister trace is a trace in a bicategory; in this paper we relate these contexts using indexed symmetric monoidal categories. In particular, we will show that for any symmetric monoidal category with an associated indexed symmetric monoidal category, there is an associated bicategory which produces refinements of trace analogous to the Reidemeister trace. This bicategory also produces a new notion of trace for parametrized spaces with dualizable fibers, which refines the obvious “fiberwise ” traces by incorporating the action of the fundamental group of the base space. We also advance the basic theory of indexed monoidal categories, including introducing a string diagram calculus which makes calculations much more tractable. This abstract framework lays the foundation for generalizations of these ideas to other