Action of derived automorphisms on infinity-morphisms

TL;DR

This paper explores how to modify homotopy algebras and their infinity morphisms simultaneously in a homotopically unique manner using operads, specifically through derivations of the cooperad.

Contribution

It introduces a method to change homotopy algebras and infinity morphisms together via derivations, ensuring homotopical uniqueness, using operadic structures like Cyl(C).

Findings

Constructs new al{C}-algebras and morphisms from derivations

Ensures modifications are unique up to homotopy

Operads, especially Cyl(C), govern the structure of these transformations

Abstract

In this paper we investigate how to simultaneously change homotopy algebras of a certain type and a corresponding infinity morphism between them, and show that this can be done in a homotopically unique way. More precisely, for a reduced cooperad C, given \Omega(C)-algebras V and W and an infinity-morphism U from V to W, for any derivation \phi of \Omega(C) we produce new \Omega(C)-algebras V' and W' and a new infinity-morphism U' between them, that are unique up to homotopy. Operads play the central role in answering this question, in particular a 2-colored operad Cyl(C) that governs pairs of homotopy algebras and infinity-morphisms between them.