Algebraic Kasparov K-theory. II

TL;DR

This paper develops a motivic stable homotopy theory for algebras, constructing explicit fibrant replacements and embedding various bivariant theories into triangulated categories, including the category of $K$-motives.

Contribution

It introduces a new motivic homotopy framework for algebras, constructs fibrant replacements, and identifies the $kk$-category with $K$-motives, advancing algebraic $K$-theory and homotopy theory.

Findings

Constructed explicit fibrant replacements for spectra of algebras.

Embedded universal bivariant theories into triangulated categories.

Identified the $kk$-category with the category of $K$-motives.

Abstract

A kind of motivic stable homotopy theory of algebras is developed. Explicit fibrant replacements for the $S^1$-spectrum and $(S^1,\mathbb G)$-bispectrum of an algebra are constructed. As an application, unstable, Morita stable and stable universal bivariant theories are recovered. These are shown to be embedded by means of contravariant equivalences as full triangulated subcategories of compact generators of some compactly generated triangulated categories. Another application is the introduction and study of the symmetric monoidal compactly generated triangulated category of $K$-motives. It is established that the triangulated category $kk$ of Corti\~{n}as--Thom can be identified with the $K$-motives of algebras. It is proved that the triangulated category of $K$-motives is a localization of the triangulated category of $(S^1,\mathbb G)$-bispectra. Also, explicit fibrant $(S^1,\mathbb G)$-bispectra representing stable algebraic Kasparov $K$-theory and algebraic homotopy $K$-theory are constructed.