A Cartan-Eilenberg approach to Homotopical Algebra

TL;DR

This paper introduces a new categorical framework for homotopical algebra using strong and weak equivalences, extending classical derived functor theory to non-additive contexts with applications to minimal models and acyclic models theorems.

Contribution

It develops the concept of Cartan-Eilenberg categories, establishing an equivalence between localizations that generalizes classical homotopical algebra to non-additive settings.

Findings

Defined Cartan-Eilenberg categories with strong and weak equivalences.

Extended derived functor theory to non-additive categories.

Proved existence of filtered minimal models for cdg algebras.

Abstract

In this paper we propose an approach to homotopical algebra where the basic ingredient is a category with two classes of distinguished morphisms: strong and weak equivalences. These data determine the cofibrant objects by an extension property analogous to the classical lifting property of projective modules. We define a Cartan-Eilenberg category as a category with strong and weak equivalences such that there is an equivalence between its localization with respect to weak equivalences and the localised category of cofibrant objets with respect to strong equivalences. This equivalence allows us to extend the classical theory of derived additive functors to this non additive setting. The main examples include Quillen model categories and functor categories with a triple, in the last case we find examples in which the class of strong equivalences is not determined by a homotopy relation. Among other applications, we prove the existence of filtered minimal models for \emph{cdg} algebras over a zero-characteristic field and we formulate an acyclic models theorem for non additive functors.