Homological epimorphisms, homotopy epimorphisms and acyclic maps

Chuang, Joseph and Lazarev, Andrey (2020) Homological epimorphisms, homotopy epimorphisms and acyclic maps. Forum Mathematicum, 32 (6). 1395–1406. ISSN 0933-7741

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Abstract

We show that the notions of homotopy epimorphism and homological epimorphism in the category of differential graded algebras are equivalent. As an application we obtain a characterization of acyclic maps of topological spaces in terms of induced maps of their chain algebras of based loop spaces. In the case of a universal acyclic map we obtain, for a wide class of spaces, an explicit algebraic description for these induced maps in terms of derived localization.