K-theory and matrix transfers

Abstract

We introduce and study matrix transfers to achieve elementary models for bivariant $K$-theory. They share lots of common properties with Voevodsky's framed correspondences and lead to symmetric matrix motives of algebraic varieties introduced in this paper. Symmetric matrix motives recover $K$-motives and fit in a closed symmetric monoidal triangulated category of symmetric matrix motives constructed in this paper by using methods of enriched motivic homotopy theory.

The paper has been written on the occasion of the conference "Recent Developments in Algebraic K-theory", Warwick, UK (April, 2025)