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# Homology, Homotopy and Applications

## Volume 12 (2010)

### Number 2

### The homotopy theory of strong homotopy algebras and bialgebras

Pages: 39 – 108

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

#### Author

#### 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

Published 1 January 2010