Homotopy theory of unital algebras

TL;DR

This paper develops a homotopy theory framework for unital algebras by establishing a model category structure on curved coalgebras, enabling a deeper understanding of their homotopy and deformation properties.

Contribution

It introduces a model category structure on the Koszul dual of curved coalgebras, providing a new approach to study homotopy and deformations of unital algebras.

Findings

Model category structure on curved coalgebras is Quillen equivalent to unital algebras.

Homotopy properties of unital algebras are described in a simpler, richer framework.

Enrichments induce models for mapping spaces and formal deformations.

Abstract

This paper provides an extensive study of the homotopy theory of types of algebras with units, like unital associative algebras or unital commutative algebras for instance. 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.