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 C, we instead show how s.h. >-algebras over C naturally form a Segal space. Given a distributive monadcomonad pair (>,{box drawings light up and horizontal}), the same is true for s.h. (>,{box drawings light up and horizontal})-bialgebras over C; in particular this yields the homotopy theory of s.h. sheaves of s.h. rings. There are similar statements for quasimonads and quasi-comonads. We also show how the structures arising are related to derived connections on bundles.