Shadows and traces in bicategories

@article{Ponto2009ShadowsAT,
  title={Shadows and traces in bicategories},
  author={Kate Ponto and Michael Shulman},
  journal={Journal of Homotopy and Related Structures},
  year={2009},
  volume={8},
  pages={151-200}
}
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… 

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References

SHOWING 1-10 OF 60 REFERENCES

Fixed Point Theory and Trace for Bicategories

The Lefschetz fixed point theorem follows easily from the identification of the Lefschetz number with the fixed point index. This identification is a consequence of the functoriality of the trace in

DUALITY AND TRACES FOR INDEXED MONOIDAL CATEGORIES

By the Lefschetz fixed point theorem, if an endomorphism of a topological space is fixed-point-free, then its Lefschetz number vanishes. This necessary condition is not usually sufficient, however;

Higher Dimensional Algebra: I. Braided Monoidal 2-Categories

Abstract We begin with a brief sketch of what is known and conjectured concerning braided monoidal 2-categories and their relevance to 4d TQFTs and 2-tangles. Then we give concise definitions of

Relative fixed point theory

The Lefschetz fixed point theorem and its converse have many generalizations. One of these generalizations is to endomorphisms of a space relative to a fixed subspace. In this paper we define

Monoidal Bicategories and Hopf Algebroids

Why are groupoids such special categories? The obvious answer is because all arrows have inverses. Yet this is precisely what is needed mathematically to model symmetry in nature. The relation

Rank Element of a Projective Module

  • A. Hattori
  • Mathematics
    Nagoya Mathematical Journal
  • 1965
In § 1 of this note we first define the trace of an endomorphism of a projective module P over a non-commutative ring A. Then we call the trace of the identity the rank element r(P) of P, which we

A Survey of Graphical Languages for Monoidal Categories

This article is intended as a reference guide to various notions of monoidal categories and their associated string diagrams. It is hoped that this will be useful not just to mathematicians, but also

Framed bicategories and monoidal fibrations

In some bicategories, the 1-cells are `morphisms' between the 0-cells, such as functors between categories, but in others they are `objects' over the 0-cells, such as bimodules, spans, distributors,

The multiplicativity of fixed point invariants

We prove two general factorization theorems for fixed-point invariants of fibrations: one for the Lefschetz number and one for the Reidemeister trace. These theorems imply the familiar
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