Iterated traces in 2-categories and Lefschetz theorems

Abstract

While not obvious from its initial motivation in linear algebra, there are many context where iterated traces can be defined. In this paper we prove a very general theorem about iterated 2-categorical traces. We show that many Lefschetz-type theorems in the literature are consequences of this result and the new perspective we provide allows for immediate spectral generalizations. We also prove a novel theorem about ncharacters.

Riemann-Roch theorems in monoidal 2-categories

“…Copairing and pairing maps Lemma 6.6 is a fundamental observation from [CP]. Most of this result appeared in the proof of [CP,Theorem 6.18], but since the definition of 1-dualizability is slightly different in this paper (Remark 5.5) we will formally state and prove the result here. Lemma 6.6.…”

“…This shadow is the familiar one in many examples, including those in Example 2.2. Proposition 5.7 (Goal 3.1, [BZN], [CP,Prop. 6.9]).…”

“…Definition 6.1. [CP,6.15,6.16] A 1-dualizable 0-cell A is 2-dualizable if the witnessing 1-cells ( A , A ) satisfy any of the following equivalent conditions: i. A and A are left dualizable ii.…”