The homotopy theory of strong homotopy algebras and bialgebras

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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Published 1 January 2010