Homotopy structures realizing algebraic kk-theory

Algebraic $kk$-theory, introduced by Cortiñas and Thom, is a bivariant $K$-theory defined on the category $\mathrm{A}\mathrm{l}\mathrm{g}$ of algebras over a commutative unital ring $\mathrm{\ell }$. It consists of a triangulated category $kk$ endowed with a functor from $\mathrm{A}\mathrm{l}\mathrm{g}$ to $kk$ that is the universal excisive, homotopy invariant and matrix-stable homology theory. Moreover, one can recover Weibel’s homotopy $K$-theory $\mathrm{K}\mathrm{H}$ from $kk$ since we have $kk(\mathrm{\ell },A)=\mathrm{K}\mathrm{H}(A)$ for any algebra $A$. We prove that $\mathrm{A}\mathrm{l}\mathrm{g}$ 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 $k{k}_{\mathrm{\infty }}$ of the $\mathrm{\infty }$-category of algebras at the set of $kk$-equivalences is a stable infinity category whose homotopy category is $kk$.