Homotopy units in A-infinity algebras

TL;DR

This paper demonstrates that in many symmetric monoidal model categories, the canonical map from associative to unital associative operads is a homotopy epimorphism, linking unital and non-unital algebra structures.

Contribution

It establishes the homotopy epimorphism property of the canonical map in a broad class of categories, clarifying the relationship between unital and non-unital associative algebras.

Findings

The map from associative to unital associative operad is a homotopy epimorphism.

Unital algebra structures form a subset of non-unital structures up to homotopy.

Provides a framework for understanding unital structures in homotopical algebra.

Abstract

We show that the canonical map from the associative operad to the unital associative operad is a homotopy epimorphism for a wide class of symmetric monoidal model categories. As a consequence, the space of unital associative algebra structures on a given object is up to homotopy a subset of connected components of the space of non-unital associative algebra structures.