# The loop-stable homotopy category of algebras

## TL;DR

This paper constructs an explicit, homotopy-theoretic machinery-free model of the loop-stable homotopy category of algebras, providing a universal, triangulated category with a new composition law and multiple descriptions of its structure.

## Contribution

It offers a new explicit construction of the triangulated category of algebras, avoiding heavy homotopy theory, and introduces a novel composition law inspired by bornological algebra homotopy theory.

## Key findings

- Explicit construction of the triangulated category k
- Universal property of the category k
- New description of the composition law

## 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.