Homotopy Algebras for Operads

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 inM 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∞-spaces, A∞-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 ∞-categorical version. This paper is long (100 pages), but a taste of it can be got from the introductory paper [Lei4] (8 pages), which does not use operads.