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

A bstract In this note, we investigate the implications of classical soft theorems for formalism developed by Kosower, Maybee and O’Connell (KMOC) to derive observables in gauge theory gravity from scattering amplitudes. particular, show that radiative electro-magnetic field at leading order expansion imposes an infinite hierarchy constraints on expectation value family generated monomials line...

We prove that Morel-Voevodsky's stable $\mathbb{A}^1$-homotopy theory affords the universal coefficient system, giving rise to Grothendieck's six operations.