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Preprints, Working Papers, ... Year : 2022

Commutative homotopical algebra embeds into non-commutative homotopical algebra

Dan Petersen
  • Function : Author
Daniel Robert-Nicoud
  • Function : Author
Felix Wierstra
  • Function : Author

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).

Dates and versions

hal-03855171 , version 1 (16-11-2022)

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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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