Homotopy Algebras for Operads

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.

100 pages. An introductory paper is available, 8 pages and operad-free: math.QA/9912084

References in corpus (12)

• Operads and Motives in Deformation Quantization
• On operad structures of moduli spaces and string theory
• Derived categories for the working mathematician
• General Operads and Multicategories
• Structures in higher-dimensional category theory
• Homotopy Algebras via Resolutions of Operads
• Generalized Enrichment for Categories and Multicategories
• Up-to-Homotopy Monoids
• Homotopy Algebras are Homotopy Algebras
• Iterated Monoidal Categories
• From Coherent Structures to Universal Properties
• Deformations of homotopy algebras

Cited by in corpus (14)

• Operads in Higher-Dimensional Category Theory
• Homotopical and Higher Categorical Structures in Algebraic Geometry
• Up-to-Homotopy Monoids
• Algebraic Geometry over model categories (a general approach to derived algebraic geometry)
• K-theory for 2-categories
• Segal Enriched Categories I
• Action of derived automorphisms on infinity-morphisms
• Twisting structures and strongly homotopy morphisms
• Coherence for Categorified Operadic Theories
• The homotopy theory of strong homotopy algebras and bialgebras
• Simplicial localization of monoidal structures, and a non-linear version of Deligne's conjecture
• Colax Reedy diagrams
• Curved homotopy coalgebras
• Toward a strictification theorem for co-Segal categories