A first goal of this paper is to precisely relate the homotopy theories of bialgebras and E 2 -algebras. For this, we construct a conservative and fully faithful ∞-functor from pointed conilpotent homotopy bialgebras to augmented E 2 -algebras which consists in an appropriate "cobar" construction. Then we prove that the (derived) formal moduli problem of homotopy bialgebras structures on a bialgebra is equivalent to the (derived) formal moduli problem of E 2 -algebra structures on this "cobar" construction. We show consequently that the E 3 -algebra structure on the higher Hochschild complex of this cobar construction, given by the solution to the higher Deligne conjecture, controls the deformation theory of this bialgebra. This implies the existence of an E 3 -structure on the deformation complex of a dg bialgebra, solving a long-standing conjecture of Gerstenhaber-Schack. On this basis we solve a long-standing conjecture of Kontsevich, by proving the E 3 -formality of the deformation complex of the symmetric bialgebra. This provides as a corollary a new proof of Etingof-Kazdhan deformation quantization of Lie bialgebras which extends to homotopy dg Lie bialgebras and is independent from the choice of an associator. Along the way, we establish new general results of independent interest about the deformation theory of algebraic structures, which shed a new light on various deformation complexes and cohomology theories studied in the literature.
Abstract. We connect the homotopy type of simplicial moduli spaces of algebraic structures to the cohomology of their deformation complexes. Then we prove that under several assumptions, mapping spaces of algebras over a monad in an appropriate diagram category form affine stacks in the sense of Toen-Vezzosi's homotopical algebraic geometry. This includes simplicial moduli spaces of algebraic structures over a given object (for instance a cochain complex). When these algebraic structures are parametrised by properads, the tangent complexes give the known cohomology theory for such structures and there is an associated obstruction theory for infinitesimal, higher order and formal deformations. The methods are general enough to be adapted for more general kinds of algebraic structures.
We prove that a weak equivalence between two cofibrant (colored) props in chain complexes induces a Dwyer-Kan equivalence between the simplicial localizations of the associated categories of algebras. This homotopy invariance under base change implies that the homotopy category of homotopy algebras over a prop P does not depend on the choice of a cofibrant resolution of P , and gives thus a coherence to the notion of algebra up to homotopy in this setting. The result is established more generally for algebras in combinatorial monoidal dg categories.
We study several homotopical and geometric properties of Maurer-Cartan spaces for L ∞ -algebras which are not nilpotent, but only filtered in a suitable way. Such algebras play a key role especially in the deformation theory of algebraic structures. In particular, we prove that the Maurer-Cartan simplicial set preserves fibrations and quasi-isomorphisms. Then we present an algebraic geometry viewpoint on MaurerCartan moduli sets, and we compute the tangent complex of the associated algebraic stack.
We prove that a weak equivalence between cofibrant props induces a weak
equivalence between the associated classifying spaces of algebras. This
statement generalizes to the prop setting a homotopy invariance result which is
well known in the case of algebras over operads. The absence of model category
structure on algebras over a prop leads us to introduce new methods to overcome
this difficulty. We also explain how our result can be extended to algebras
over colored props in any symmetric monoidal model category tensored over chain
complexes.Comment: Final version, to appear in Algebraic \& Geometric Topolog
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