Homology, Homotopy and Applications

Volume 12 (2010)

Number 2

The homotopy theory of strong homotopy algebras and bialgebras

Pages: 39 – 108

DOI: http://dx.doi.org/10.4310/HHA.2010.v12.n2.a3

Author

J. P. Pridham (Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, University of Cambridge, United Kingdom)

Abstract

Lada introduced strong homotopy algebras to describe the structures on a deformation retract of an algebra in topological spaces. However, there is no satisfactory general definition of a morphism of strong homotopy (s.h.) algebras. Given a monad ⊤ on a simplicial category $\mathcal{C}$, we instead show how s.h. ⊤-algebras over $\mathcal{C}$ naturally form a Segal space. Given a distributive monad-comonad pair (⊤, ⊥), the same is true for s.h. (⊤,⊥)-bialgebras over $\mathcal{C}$; in particular this yields the homotopy theory of s.h. sheaves of s.h. rings. There are similar statements for quasi-monads and quasi-comonads. We also show how the structures arising are related to derived connections on bundles.

Keywords

algebraic theories; simplicial categories; Segal spaces

2010 Mathematics Subject Classification

18C15, 18D20, 18G30, 55U40

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