nLab Cartan-Eilenberg category

Idea

The notion of Cartan-Eilenberg categories is a tool in homotopical algebra.

The structure of a Cartan-Eilenberg category (SAPR 10) on a category is the structure of a category with weak equivalences equipped with the further data of “strong” equivalences inducing a notion of cofibrant objects, such that the localization of the category at the weak equivalences is equivalently that of the full subcategory of cofibrant objects at just the strong equivalences.

The motivation for this axiomatics in (SAPR 10) is that it is the general context in which the original construction in (Cartan-Eilenberg 56) of derived functors of additive functors between categories of modules generalizes.

Every model category is in particular a Cartan-Eilenberg category, the strong equivalences in this case are the left homotopy equivalences. But the notion of CE-categories is weaker and exists in situations where a full model category structure is not available. In this way the notion of Cartan-Eilenberg category structures is analogous to other such structures in between categories with weak equivalences and fully-fledged model categories, such as categories of fibrant objects, Waldhausen categories etc.

In this approach to homotopical algebra, there is a ambient category, $\mathcal{C}$, in which there are two classes $\mathcal{S}$ and $\mathcal{W}$ of distinguished morphisms called the strong and the weak equivalences and with $S\subseteq W$. 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 within the category $\mathcal{C}[\mathcal{S}^{-1}]$, obtained by inverting the strong equivalences. (Note as the context is slightly different cofibrant means something slightly different here.)

Definition

A triple $(\mathcal{C},\mathcal{S},\mathcal{W})$ is called a (left) Cartan-Eilenberg category if every object has a cofibrant left model that is, for each object, $X$, of $\mathcal{C}$, there is a cofibrant object, $M$, and a morphism $q: M\to X$ in $\mathcal{C}[\mathcal{S}^{-1}]$, which is an isomorphism in $\mathcal{C}[\mathcal{W}^{-1}]$.

Properties

These Cartan-Eilenberg categories have 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.

References

The original constructions from which “Cartan-Eilenberg categories” got their name are due to

• Henri Cartan, Samuel Eilenberg, Homological Algebra. Princeton University Press, 1956

The abstraction of Cartan-Eilenberg categories then appears in

• F. Guillén Santos, Vicente Navarro Aznar, Pere Pascual, Agusti Roig, A Cartan-Eilenberg approach to Homotopical Algebra, arxiv/0707.3704 and J. Pure Applied. Alg. 214 (2010) 140 - 164.

Further discussion is in

• Pere Pascual, Some remarks on Cartan-Eilenberg categories, pdf and Collect. Math. 63, No. 2, 203-216 (2012);

• Pere Pascual, Cartan-Eilenberg categories and descent categories, pdf

• Joana Cirici, Francisco Guillén, Homotopy theory of mixed Hodge complexes, arxiv/1304.6236