Algebraic & Geometric Topology

Homotopy theory of unital algebras

Brice Le Grignou

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We provide an extensive study of the homotopy theory of types of algebras with units, for instance unital associative algebras or unital commutative algebras. To this purpose, we endow the Koszul dual category of curved coalgebras, where the notion of quasi-isomorphism barely makes sense, with a model category structure Quillen equivalent to that of unital algebras. To prove such a result, we use recent methods based on presentable categories. This allows us to describe the homotopy properties of unital algebras in a simpler and richer way. Moreover, we endow the various model categories with several enrichments which induce suitable models for the mapping spaces and describe the formal deformations of morphisms of algebras.

Article information

Algebr. Geom. Topol., Volume 19, Number 3 (2019), 1541-1618.

Received: 25 May 2018
Revised: 25 September 2018
Accepted: 17 October 2018
First available in Project Euclid: 29 May 2019

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Mathematical Reviews number (MathSciNet)

Primary: 18D50: Operads [See also 55P48] 18G30: Simplicial sets, simplicial objects (in a category) [See also 55U10] 18G55: Homotopical algebra 55U15: Chain complexes 55U40: Topological categories, foundations of homotopy theory

operads Koszul duality bar and cobar constructions


Le Grignou, Brice. Homotopy theory of unital algebras. Algebr. Geom. Topol. 19 (2019), no. 3, 1541--1618. doi:10.2140/agt.2019.19.1541.

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