An elementary approach to the model structure on DG-Lie algebras

TL;DR

This paper provides an elementary proof for the classical model structure on unbounded DG-Lie algebras over a field of characteristic zero, highlighting properties of free and semifree extensions and their cofibrancy.

Contribution

It offers a new elementary proof of the model structure and explores properties of free, semifree, and non-cofibrant DG-Lie algebras, with specific examples.

Findings

Cobar construction of a cocommutative coalgebra is semifree DG-Lie algebra.

Existence of non-cofibrant DG-Lie algebra with free underlying graded Lie algebra.

In the bounded above case, such DG-Lie algebras are always cofibrant.

Abstract

This paper contains an elementary proof of the existence of the classical model structure on the category of unbounded DG-Lie algebras over a field of characteristic zero, with an emphasis on the properties of free and semifree extensions, which are particularly nice cofibrations. The cobar construction of a locally conilpotent cocommutative coalgebra is shown to be an example of semifree DG-Lie algebra. We also give an example of a non-cofibrant DG-Lie algebra whose underlying graded Lie algebra is free; this cannot occur in the bounded above case, where DG-Lie algebras of this form are always cofibrant.