## Homology, Homotopy and Applications

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

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

#### 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

**Subjects**

Primary: 18C15: Triples (= standard construction, monad or triad), algebras for a triple, homology and derived functors for triples [See also 18Gxx] 55U40: Topological categories, foundations of homotopy theory 18D20: Enriched categories (over closed or monoidal categories) 18G30: Simplicial sets, simplicial objects (in a category) [See also 55U10]

**Keywords**

Algebraic theories simplicial categories Segal spaces

#### 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