Homotopy Diagrams of Algebras

TL;DR

This paper provides explicit models for diagrams of homomorphisms of strongly homotopy algebras, enhancing understanding of their homotopy invariance and categorical structure, with applications to homological perturbation theory.

Contribution

It explicitly describes algebraic models for colored operads of diagrams of homomorphisms, advancing the algebraic understanding of homotopy invariance in strongly homotopy algebras.

Findings

Explicit models for diagrams of homomorphisms are constructed.

Strongly homotopy algebras form a genuine category, not just a weak Kan category.

Provides a conceptual framework for homotopies through homomorphisms.

Abstract

In [math.AT/9907138] we proved that strongly homotopy algebras are homotopy invariant concepts in the category of chain complexes. Our arguments were based on the fact that strongly homotopy algebras are algebras over minimal cofibrant operads and on the principle that algebras over cofibrant operads are homotopy invariant. In our approach, algebraic models for colored operads describing diagrams of homomorphisms played an important role. The aim of this paper is to give an explicit description of these models. A possible application is an appropriate formulation of the `ideal' homological perturbation lemma for chain complexes with algebraic structures. Our results also provide a conceptual approach to `homotopies through homomorphism' for strongly homotopy algebras. We also argue that strongly homotopy algebras form a honest (not only weak Kan) category. The paper is a continuation of our program to translate the famous book "M. Boardman, R. Vogt: Homotopy Invariant Algebraic Structures on Topological Spaces" to algebra.