Compactified configuration space of points on a line and homotopies of $A_\infty$ morphisms

TL;DR

This paper constructs a geometric model for a 2-colored operad that encodes two $A_ abla$-algebras, their morphisms, and a homotopy between these morphisms, linking geometric configuration spaces with algebraic structures.

Contribution

It introduces a configuration space model for a 2-colored differential graded operad capturing $A_ abla$-algebras, morphisms, and homotopies, and identifies its cohomology with a known operad.

Findings

The constructed operad models the structure of two $A_ abla$-algebras and their morphisms.

The cohomology of the operad corresponds to the operad for associative algebras and morphisms.

The model provides a geometric perspective on homotopies of $A_ abla$-morphisms.

Abstract

We construct a configuration space model for a particular 2-colored differential graded operad encoding the structure of two $A_\infty$ algebras with two $A_\infty$ morphisms and a homotopy between the morphisms. The cohomology of this operad is shown to be the well-known 2-colored operad encoding the structure of two associative algebras and of an associative algebra morphism between them.