## Homology, Homotopy and Applications

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

J. P. Pridham

#### Abstract

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 $\top$ on a simplicial category $\mathcal{C}$, we instead show how s.h. $\top$-algebras over $\mathcal{C}$ naturally form a Segal space. Given a distributive monad-comonad pair ($\top, \bot$), the same is true for s.h. ($\top, \bot$)-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.

#### Article information

Source
Homology Homotopy Appl., Volume 12, Number 2 (2010), 39-108.

Dates
First available in Project Euclid: 28 January 2011

Permanent link to this document
https://projecteuclid.org/euclid.hha/1296223878

Mathematical Reviews number (MathSciNet)
MR2721031

Zentralblatt MATH identifier
1236.18016

#### Citation

Pridham, J. P. The homotopy theory of strong homotopy algebras and bialgebras. Homology Homotopy Appl. 12 (2010), no. 2, 39--108. https://projecteuclid.org/euclid.hha/1296223878