The goal of this paper is to construct trace maps for the six functor formalism of motivic cohomology after Voevodsky, Ayoub, and Cisinski-Déglise. We also construct an $\infty$-enhancement of such a trace formalism. In the course of the $\infty$-enhancement, we need to reinterpret the trace formalism in a more functorial manner. This is done by using Suslin-Voevodsky's relative cycle groups.

• 14F42 - Motivic cohomology; motivic homotopy theory
• 18N60 - \((\infty,1)\)-categories (quasi-categories, Segal spaces, etc.); \(\infty\)-topoi, stable \(\infty\)-categories