Homotopy unital A_infinity-algebras

TL;DR

This paper constructs a cofibrant replacement for the operad of unital associative differential graded algebras, enabling a homotopy theory framework for unital A_infinity-algebras and their morphisms.

Contribution

It introduces a new cofibrant replacement operad for unital associative dg-algebras and establishes the homotopy properties of unital A_infinity-algebras and their morphisms.

Findings

Constructed a cofibrant replacement operad for unital associative dg-algebras.

Proved that the operad bimodule of A_infinity-morphisms is a cofibrant replacement.

Established the cofibrant replacement for the bimodule of unital A_infinity-morphisms.

Abstract

It is well known that the differential graded operad of A_infinity-algebras is a cofibrant replacement (a dg-resolution) of the operad of associative differential graded algebras without units. In this article we find a cofibrant replacement of the operad of associative differential graded algebras with units. Algebras over it are called homotopy unital A_infinity-algebras. We prove that the operad bimodule of A_infinity-morphisms is a cofibrant replacement of the operad bimodule of morphisms of dg-algebras without units. Similarly we show that the operad bimodule of homotopy unital A_infinity-morphisms is a cofibrant replacement of the operad bimodule of morphisms of dg-algebras with units.