TY - JOUR

T1 - The homotopy theory of strong homotopy algebras and bialgebras

AU - Pridham, J.P.

PY - 2010/1/1

Y1 - 2010/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?partnerID=yv4JPVwI&eid=2-s2.0-77956307152&md5=c470179fb3023ca535c4f70c348bedcc

U2 - http://dx.doi.org/10.4310/HHA.2010.v12.n2.a3

DO - http://dx.doi.org/10.4310/HHA.2010.v12.n2.a3

M3 - Article

AN - SCOPUS:77956307152

VL - 12

SP - 39

EP - 108

JO - Homology, Homotopy and Applications

JF - Homology, Homotopy and Applications

SN - 1532-0073

IS - 2

ER -