The following two papers on homotopy-algebraic structures - or "weakened" algebraic structures, if you prefer - are now available. The first one, "Up-to-Homotopy Monoids", is 8 pages long and is essentially a set of notes for the talk I gave at the Louvain-la-Neuve PSSL in October. It can serve as an introduction to the second one, "Homotopy Algebras for Operads" (100 pages, but don't let that scare you: it should be easy for category theorists). Abstracts are below. Tom Leinster * * * "Up-to-Homotopy Monoids" Informally, a homotopy monoid is a monoid-like structure in which properties such as associativity only hold `up to homotopy' in some consistent way. This short paper comprises a rigorous definition of homotopy monoid and a brief analysis of some examples. It is a much-abbreviated version of the paper `Homotopy Algebras for Operads', and does not assume any knowledge of operads. Available on the mathematics archive: http://xxx.lanl.gov/abs/math.QA/9912084 * * * "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 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. Available on the mathematics archive: http://xxx.lanl.gov/abs/math.QA/0002180