Homotopy structures realizing algebraic kk-theory

TL;DR

This paper constructs a homotopy-theoretic framework for algebraic kk-theory, showing it can be realized as a stable category of fibrant objects and as a stable infinity category, thus connecting algebraic and homotopical structures.

Contribution

It demonstrates that algebraic kk-theory can be modeled as a stable category of fibrant objects and as a stable infinity category, providing a new homotopical perspective.

Findings

kk-theory forms a stable category of fibrant objects

The homotopy category of this model is kk

The Dwyer-Kan localization yields a stable infinity category

Abstract

Algebraic $kk$-theory, introduced by Corti\~nas and Thom, is a bivariant $K$-theory defined on the category $\mathrm{Alg}$ of algebras over a commutative unital ring $\ell$. It consists of a triangulated category $kk$ endowed with a functor from $\mathrm{Alg}$ to $kk$ that is the universal excisive, homotopy invariant and matrix-stable homology theory. Moreover, one can recover Weibel's homotopy $K$-theory $\mathrm{KH}$ from $kk$ since we have $kk(\ell,A)=\mathrm{KH}(A)$ for any algebra $A$. We prove that $\mathrm{Alg}$ with the split surjections as fibrations and the $kk$-equivalences as weak equivalences is a stable category of fibrant objects, whose homotopy category is $kk$. As a consecuence of this, we prove that the Dwyer-Kan localization $kk_\infty$ of the $\infty$-category of algebras at the set of $kk$-equivalences is a stable infinity category whose homotopy category is $kk$.