Homotopy structures realizing algebraic kk-theory

Algebraic $kk$-theory, introduced by Cortiñas 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 $kk$.

💡 Research Summary

This paper provides a homotopy-theoretic foundation for algebraic kk-theory by constructing explicit stable infinity-categorical and homotopical models whose homotopy categories recover the triangulated category kk. Algebraic kk-theory, introduced by Cortiñas and Thom, is a bivariant K-theory defined for algebras over a commutative ring ℓ. It is characterized as the universal excisive, homotopy invariant, and matrix-stable homology theory on the category Alg, and it recovers Weibel’s homotopy K-theory KH.

The authors achieve this through two principal constructions. First, they equip the category Alg itself with a homotopical structure. They define a class of weak equivalences (W_kk) consisting of maps that become isomorphisms in kk, and a class of fibrations consisting of split surjections. They prove that with this structure, Alg forms a stable category of fibrant objects, a notion weaker than a model category but sufficient for homotopy theory. A key theorem establishes that the homotopy category of this category of fibrant objects is equivalent to the triangulated category kk. This shows that kk arises naturally from inverting a specific set of maps within Alg under a controlled homotopical framework.

Second, and as a main corollary, they consider the higher categorical perspective. They form the Dwyer-Kan localization kk∞ of the infinity category of algebras at the set W_kk of kk-equivalences. Their central result is that kk∞ is a stable infinity category, and its homotopy category (obtained by taking connected components of mapping spaces) is triangle equivalent to kk. This embeds algebraic kk-theory firmly into the realm of modern stable homotopy theory, implying it can be naturally enriched over spectra.

To prove these results, the paper develops several technical tools. It refines the universal property of the functor j: Alg → kk, showing it is initial among finite product-preserving functors that send three specific families of maps to isomorphisms: polynomial homotopy equivalences, the upper-left corner inclusions A → M∞A (for matrix stability), and the classifying maps of extensions with contractible middle term or which are infinite-sum rings (related to excision). This offers a concise categorical characterization of kk.

An important observation highlighted in the paper is a fundamental discrepancy between the algebraic and topological (C*-algebraic) settings. While for C*-algebras, a similar category of fibrant objects structure exists with continuous homotopy equivalences as weak equivalences, the authors argue that such a structure likely does not exist for Alg with polynomial homotopy equivalences as weak equivalences. This indicates that the methods used to study topological KK-theory cannot be directly transplanted, underscoring the need for distinct techniques in algebraic kk-theory.

In summary, this work successfully realizes algebraic kk-theory as the homotopy category of both a concrete category of fibrant objects and a stable infinity category, thereby clarifying its homotopical nature and connecting it to contemporary higher categorical structures.