nLab bicategorical trace

Idea

A generalization of the notion of trace in a monoidal category to a bicategory context, based on equipping the bicategory with a “shadow” or categorified trace (including the trace of a category).

Examples

The following traditional notions of “traces” turn out to be special cases of bicategorical traces

• All symmetric monoidal traces, such as the Lefschetz number.

• The Reidemeister trace.

Related concepts

• trace

• higher trace

• span trace

References

• Kate Ponto, Michael Shulman, Shadows and traces in bicategories (arXiv:0910.1306)

A survey is in

• Mike Shulman, Traces in monoidal categories and bicategories (2008) (pdf slides)

Relating the bicategorical trace to the Dennis trace on algebraic K-theory:

• Jonathan Campbell, John Lind, Cary Malkiewich, Kate Ponto, Inna Zakharevich, K-theory of endomorphisms, the TR-trace, and zeta functions [arXiv:2005.04334]

For the concept of a universal shadow of a bicategory and an extension of shadows to (∞,2)-categories, see

• Kathryn Hess, Nima Rasekh, Shadows are Bicategorical Traces [arXiv:2109.02144]