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• W1644606943 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." @default.
• W1644606943 created "2016-06-24" @default.
• W1644606943 creator A5047971691 @default.
• W1644606943 date "2000-02-22" @default.
• W1644606943 modified "2023-09-27" @default.
• W1644606943 title "Homotopy Algebras for Operads" @default.
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