This paper proposes an approach to homotopical algebra in which there are two classes \({\mathcal S}\) and \({\mathcal W}\) of distinguished morphisms called the strong and the weak equivalences. An object of the category \({\mathcal C}\) is said to be cofibrant if any morphism from it to the codomain of a weak equivalence lifts uniquely to one from it to the domain in the category \({\mathcal C}[{\mathcal S}^{-1}]\), obtained by inverting the strong equivalences. This allows a neat development of a new form of homotopical algebra.
A triple \(({\mathcal {C,S,W}})\) is called a (left) Cartan-Eilenberg category if every object has a cofibrant left model in a fairly obvious sense. These Cartan-Eilenberg categories are characterised by the property that \({\mathcal C}[{\mathcal W}^{-1}]\) is equivalent to a relative localisation of the category of cofibrant objects with respect to strong equivalences.
The paper develops analogues of aspects of the standard theory in this new context, including derived functors. In section 4, the authors describe how Cartan-Eilenberg categories relate to some of the other axiomatisations of homotopy theory, including that of Quillen model categories. They also give an abstract version of aspects of Sullivan’s theory of minimal models. In the final section, they discuss how Cartan-Eilenberg categories may be defined on a functor category using a comonad/cotriple and then develop a general acyclic model theorem in this setting.