The loop-stable homotopy category of algebras

Abstract

Let l be a commutative ring with unit. Garkusha constructed a functor from the category of l-algebras into a triangulated category D, that is a universal excisive and homotopy invariant homology theory. Later on, he provided different descriptions of D, as an application of his motivic homotopy theory of algebras. Using these, it can be shown that D is triangulated equivalent to a category, denote it by k, whose objects are pairs (A,m) with A an l-algebra and m an integer, and whose Hom-sets can be described in terms of homotopy classes of morphisms. All these computations, however, require a heavy machinery of homotopy theory. In this paper, we give a more explicit construction of the triangulated category k and prove its universal property, avoiding the homotopy-theoretic methods and using instead the ones developed by Corti\~nas-Thom for defining kk-theory. Moreover, we give a new description of the composition law in k, mimicking the one in the suspension-stable homotopy category of bornological algebras defined by Cuntz-Meyer-Rosenberg. We also prove that the triangulated structure in k can be defined using either extension or mapping path triangles.