Title:
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A Cartan-Eilenberg approach to homotopical algebra
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Author:
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Guillén Santos, Francisco; Navarro Aznar, Vicente; Pascual Gainza, Pere; Roig Martí, Agustín
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Other authors:
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Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada I; Universitat Politècnica de Catalunya. EGSA - Equacions Diferencials, Geometria, Sistemes Dinàmics i de Control, i Aplicacions |
Abstract:
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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 |
Abstract:
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Peer Reviewed |
Subject(s):
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-Àrees temàtiques de la UPC::Matemàtiques i estadística -Algebra, Homological -Àlgebra homològica |
Rights:
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Attribution-NonCommercial-NoDerivs 3.0 Spain
http://creativecommons.org/licenses/by-nc-nd/3.0/es/ |
Document type:
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Article - Published version Article |
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