Title:
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A Cartan-Eilenberg approach to homotopical algebra
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Author:
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Pascual Gainza, Pere; Roig Martí, Agustín; Guillén Santos, Francisco; Navarro Aznar, Vicente
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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 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 of categories between its localisation with respect to
weak equivalences and the relative localisation of the subcategory of cofibrant objets with respect to
strong equivalences. This equivalence of categories allows us to extend the classical theory of derived
additive functors to this non additive setting. The main examples 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 latter 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 theorem. |
Subject(s):
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-Category theory (Mathematics) -Homological algebra -Derived functors, -Minimal models -Acyclic models -Quillen model category -Models of a functor -Cofibrant object -Relative localisation -Àlgebra homològica -Categories (Matemàtica) -Classificació AMS::55 Algebraic topology::55U Applied homological algebra and category theory |
Rights:
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Attribution-NonCommercial-NoDerivs 2.5 Spain
http://creativecommons.org/licenses/by-nc-nd/2.5/es/ |
Document type:
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Article |
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