In this paper we propose an approach to homotopical algebra w
here the basic ingredient
is a category with two classes of distinguished morphisms: s
trong and weak equivalences. These data
determine the cofibrant objects by an extension property ana
logous to the classical lifting property
of projective modules. We define a Cartan-Eilenberg categor
y as a category with strong and weak
equivalences such that there is an equivalence of categorie
s between its localisation with respect to
weak equi...
In this paper we propose an approach to homotopical algebra w
here the basic ingredient
is a category with two classes of distinguished morphisms: s
trong and weak equivalences. These data
determine the cofibrant objects by an extension property ana
logous to the classical lifting property
of projective modules. We define a Cartan-Eilenberg categor
y as a category with strong and weak
equivalences such that there is an equivalence of categorie
s between its localisation with respect to
weak equivalences and the relative localisation of the subc
ategory of cofibrant objets with respect to
strong equivalences. This equivalence of categories allow
s us to extend the classical theory of derived
additive functors to this non additive setting. The main exa
mples include Quillen model categories
and categories of functors defined on a category endowed with
a cotriple (comonad) and taking values
on a category of complexes of an abelian category. In the latt
er case there are examples in which the
class of strong equivalences is not determined by a homotopy
relation. Among other applications of
our theory, we establish a very general acyclic models theor
em
Citation
Guillén, F. [et al.]. A Cartan-Eilenberg approach to homotopical algebra. "Journal of pure and applied algebra", 2010, vol. 214, p. 140-164.