Commutative homotopical algebra embeds into non-commutative homotopical algebra

TL;DR

This paper demonstrates that the homotopy category of commutative dg algebras can be faithfully embedded into the homotopy category of dg associative algebras, revealing deep structural connections via derived mapping spaces.

Contribution

It establishes faithfulness of the forgetful functor from commutative to associative dg algebras in the homotopy category, including unital and non-unital cases, and explores Koszul duality with dg Lie algebras.

Findings

Faithful embedding of commutative dg algebras into associative dg algebras.

Derived mapping spaces modeled as Maurer-Cartan spaces of filtered dg Lie algebras.

Injective maps on all homotopy groups at any basepoint.

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).