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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
ENRICHED INDEXED CATEGORIES
"... Abstract. We develop a theory of categories which are simultaneously (1) indexed over a base category S with finite products, and (2) enriched over an Sindexed monoidal category V. This includes classical enriched categories, indexed and fibered categories, and internal categories as special cases. ..."
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Abstract. We develop a theory of categories which are simultaneously (1) indexed over a base category S with finite products, and (2) enriched over an Sindexed monoidal category V. This includes classical enriched categories, indexed and fibered categories, and internal categories as special cases. We then describe the appropriate notion of “limit ” for such enriched indexed categories, and show that they admit “free cocompletions” constructed as usual with a Yoneda embedding.
The additivity of traces in monoidal derivators
 the Journal of Ktheory. arXiv:1212.3277
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