The homotopy theory of strong homotopy algebras and bialgebras

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

2009-08-02

arxiv.org/abs/0908.0116v2

Mathematics

Category Theory

58 pages; v2 final version, to appear in HHA.

Scientific paper

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 T on a simplicial category C, we instead show how s.h. T-algebras over C naturally form a Segal space. Given a distributive monad-comonad pair (T, S), the same is true for s.h. (T, S)-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.

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