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 C, we instead show how s.h. ⊤-algebras over C naturally form a Segal space. Given a distributive monad-comonad pair (⊤, ⊥), the same is true for s.h. (⊤, ⊥)-bialgebras over 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.
Homology, Homotopy and Applications, Vol. 12 (2010), No. 2, pp.39-108.
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