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Commutative homotopical algebra embeds into non-commutative homotopical algebra

Abstract : Over a field of characteristic zero, we show that the forgetful functor from the homotopy category of commutative dg algebras to the homotopy category of dg associative algebras is faithful. In fact, the induced map of derived mapping spaces gives an injection on all homotopy groups at any basepoint. We prove similar results both for unital and non-unital algebras, and also Koszul dually for the universal enveloping algebra functor from dg Lie algebras to dg associative algebras. An important ingredient is a natural model for these derived mapping spaces as Maurer-Cartan spaces of complete filtered dg Lie algebras (or curved Lie algebras, in the unital case).
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Preprints, Working Papers, ...
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https://hal.archives-ouvertes.fr/hal-03855171
Contributor : Ricardo Campos Connect in order to contact the contributor
Submitted on : Wednesday, November 16, 2022 - 11:18:28 AM
Last modification on : Friday, November 18, 2022 - 3:38:33 AM

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  • HAL Id : hal-03855171, version 1
  • ARXIV : 2211.02387

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Ricardo Campos, Dan Petersen, Daniel Robert-Nicoud, Felix Wierstra. Commutative homotopical algebra embeds into non-commutative homotopical algebra. 2022. ⟨hal-03855171⟩

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