Exchange theorems and coherent duality in six functors

Abstract

We define the notion of an exchange theorem and show that any two functors satisfying an exchange theorem are canonically related via twisted norm maps. This is done by identifying the universal category receiving a pair of functors satisfying an exchange theorem. Additionally, we show that the twists occurring are $K$-theoretic in nature, parametrized by a categorified analogue of virtual vector bundles. As an application, we show that every 3-functor formalism has a canonical extension which encodes Poincar\'e duality and Thom twists internal to the formalism. This gives a 1-categorical realization of the "coherent six operations" outlined by Hoyois. In the process of proving universality, techniques for computing categories associated to bi- and $n$-simplicial spaces are developed. Many of the results in this direction may be viewed as model-independent rederivations of work of Liu--Zheng.