  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.
