Deformation theory of representations of prop(erad)s I

@inproceedings{Merkulov2009DeformationTO,
  title={Deformation theory of representations of prop(erad)s I},
  author={S. Merkulov and B. Vallette},
  year={2009}
}
Abstract In this paper and its follow-up [Merkulov and Vallette, J. reine angew. Math.], we study the deformation theory of morphisms of properads and props thereby extending Quillen's deformation theory for commutative rings to a non-linear framework. The associated chain complex is endowed with an L ∞-algebra structure. Its Maurer-Cartan elements correspond to deformed structures, which allows us to give a geometric interpretation of these results. To do so, we endow the category of prop(erad… Expand

Figures from this paper

Derived deformation theory of algebraic structures
The main purpose of this article is to develop an explicit derived deformation theory of algebraic structures at a high level of generality, encompassing in a common framework various kinds ofExpand
Props in model categories and homotopy invariance of structures
Abstract We prove that any category of props in a symmetric monoidal model category inherits a model structure. We devote an appendix, about half the size of the paper, to the proof of the modelExpand
Deformation theory of Cohomological Field Theories
We develop the deformation theory of cohomological field theories (CohFTs), which is done as a special case of a general deformation theory of morphisms of modular operads. This leads us to introduceExpand
Moduli Spaces of (Bi)algebra Structures in Topology and Geometry
After introducing some motivations for this survey, we describe a formalism to parametrize a wide class of algebraic structures occurring naturally in various problems of topology, geometry andExpand
Deformation theory of bialgebras, higher Hochschild cohomology and Formality
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 conilpotentExpand
Higher Lie theory
We present a novel approach to the problem of integrating homotopy Lie algebras by representing the Maurer-Cartan space functor with a universal cosimplicial object. This recovers Getzler's originalExpand
Resolutions of operads arising from Koszul (bi)algebras
We introduce a construction that produces from each bialgebra H an operad AssH controlling associative algebras in the monoidal category of H-modules or, briefly, H-algebras. When the underlyingExpand
Kontsevich's graph complex, GRT, and the deformation complex of the sheaf of polyvector fields
We generalize Kontsevich's construction of L-infinity derivations of polyvector fields from the affine space to an arbitrary smooth algebraic variety. More precisely, we construct a map (in theExpand
Curved Koszul duality theory
We extend the bar–cobar adjunction to operads and properads, not necessarily augmented. Due to the default of augmentation, the objects of the dual category are endowed with a curvature. As usual,Expand
Notes on Algebraic Operads, Graph Complexes, and Willwacher's Construction
We give a detailed proof of T. Willwacher's theorem arXiv:1009.1654 which links the cohomology of the full graph complex fGC to the cohomology of the deformation complex of the operad GER, governingExpand
...
1
2
3
4
5
...