Homotopy Algebras for Operads

TL;DR

This paper defines homotopy algebras for operads within monoidal categories, explores their properties through examples like loop spaces, and discusses their relation to other algebraic structures and potential infinity-categorical extensions.

Contribution

It introduces a new definition of homotopy algebra for operads in monoidal categories and demonstrates its applications to loop spaces and classifying spaces.

Findings

Any loop space is a homotopy monoid.

n-fold loop spaces are n-fold homotopy monoids.

Classifying space of a homotopy monoidal category is a homotopy topological monoid.

Abstract

We present a definition of homotopy algebra for an operad, and explore its consequences. The paper should be accessible to topologists, category theorists, and anyone acquainted with operads. After a review of operads and monoidal categories, the definition of homotopy algebra is given. Specifically, suppose that M is a monoidal category in which it makes sense to talk about algebras for some operad P. Then our definition says what a homotopy P-algebra in M is, provided only that some of the morphisms in M have been marked out as `homotopy equivalences'. The bulk of the paper consists of examples of homotopy algebras. We show that any loop space is a homotopy monoid, and, in fact, that any n-fold loop space is an n-fold homotopy monoid in an appropriate sense. We try to compare weakened algebraic structures such as A_infinity-spaces, A_infinity-algebras and non-strict monoidal categories to our homotopy algebras, with varying degrees of success. We also prove results on `change of base', e.g. that the classifying space of a homotopy monoidal category is a homotopy topological monoid. Finally, we reflect on the advantages and disadvantages of our definition, and on how the definition really ought to be replaced by a more subtle infinity-categorical version.