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

Derived deformations of Artin stacks

“…Given a monoidal category C and a set O, recall from [Pri3] that a C-valued quasi-descent datum X on objects O consists of:…”

Constructing derived moduli stacks

“…Although cosimplicial groups can be used to construct derived moduli in all characteristics for many problems, they are insufficiently flexible to arise in the generality we need. Instead, we use the quasi-comonoids introduced in [Pri4] 5.1. Quasi-comonoids.…”

“…Then [Pri4] Lemma 3.2 gives a model structure on QM * (S) in which a morphism E → F is a (trivial) fibration whenever the canonical maps Proof. This is a direct consequence of [Pri4] Corollary 3.12, which shows that MC is a right Quillen functor.…”

“…Proposition 4.38 gives a cosimplicial group governing derived moduli of torsors, a problem not easily accessible via DGLAs.Cosimplicial groups cannot handle all moduli problems, so Section 5 begins by recalling the quasi-comonoids of [Pri4], and the derived Deligne groupoid Del(E) of a simplicial quasi-comonoid E. Corollary 5.43 then shows that quasi-comonoid-valued functors E give rise to well-behaved derived moduli functors Del(E). In §5.2.1, we recall basic properties of monads, together with results from [Pri4] showing how these give rise to quasi-comonoids. Monads are ubiquitous, arising whenever there is some kind of algebraic structure.…”

Curved Homotopy Coalgebras

“…It is a particular case of homotopy comonoid defined by Leinster [Lei99, Definition 2.2]. This concept under the name of quasi-comonoid is important also for Pridham [Pri10,Definition 1.4].…”