In this paper and its follow-up [32], 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 y -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)s with a model category structure. We provide a complete study of models for prop(erad)s. A new e¤ective method to make minimal models explicit, that extends the Koszul duality theory, is introduced and the associated notion is called homotopy Koszul.As a corollary, we obtain the (co)homology theories of (al)gebras over a prop(erad) and of homotopy (al)gebras as well. Their underlying chain complex is endowed with an L y -algebra structure in general and a Lie algebra structure only in the Koszul case. In particular, we make the deformation complex of morphisms from the properad of associative bialgebras explicit. For any minimal model of this properad, the boundary map of this chain complex is shown to be the one defined by Gerstenhaber and Schack. As a corollary, this paper provides a complete proof of the existence of an L y -algebra structure on the Gerstenhaber-Schack bicomplex associated to the deformations of associative bialgebras. IntroductionThe theory of props and properads, which generalizes the theory of operads, provides us with a universal language to describe many algebraic, topological and di¤erential geometric structures. Our main purpose in this paper is to establish a new and surprisingly strong link between the theory of prop(erad)s and the theory of L y -algebras.We introduce several families of L y -algebras canonically associated with prop(erad)s, moreover, we develop new methods which explicitly compute the associated L y -brackets in terms of prop(erad)ic compositions and di¤erentials. Many important dg Lie algebras in algebra and geometry (such as Hochschild, Poisson, Schouten, Frö licher-Nijenhuis and many others) are proven to be of this prop(erad)ic origin.In the appendix of [32], we endow the category of dg prop(erad)s with a model category structure which is used throughout the text.The paper is organized as follows. In §1 we remind key facts about properads and props and we define the notion of non-symmetric prop(erad). In §2 we introduce and study the convolution prop(erad) canonically associated with a pair, ðC; PÞ, consisting of an arbitrary coprop(erad) C and an arbitrary prop(erad) P; our main result is the construction 53 Merkulov and Vallette, Deformation theory of representations of prop(erad )s I Brought to you by | MPI fuer Mathematik Authenticated | 192.68.254.219 Download Date | 9/19/13 7:27 PMof a Lie algebra structure on this convolution properad, as well as higher operations. In §3 we discuss bar and cobar constructions for (co)prop(erad)s. We introduce the notion of twisting morphism (cochain) for prop(erads) and prove Theor...

Abstract. We introduce and study wheeled PROPs, an extension of the theory of PROPs which can treat traces and, in particular, solutions to the master equations which involve divergence operators. We construct a dg free wheeled PROP whose representations are in one-to-one correspondence with formal germs of SP-manifolds, key geometric objects in the theory of BatalinVilkovisky quantization. We also construct minimal wheeled resolutions of classical operads Com and Ass as rather non-obvious extensions of Com ∞ and Ass ∞ , involving, e.g., a mysterious mixture of associahedra with cyclohedra. Finally, we apply the above results to a computation of cohomology of a directed version of Kontsevich's complex of ribbon graphs.

Abstract. 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)s with a model category structure. We provide a complete study of models for prop(erad)s. A new effective method to make minimal models explicit, that extends the Koszul duality theory, is introduced and the associated notion is called homotopy Koszul.As a corollary, we obtain the (co)homology theories of (al)gebras over a prop(erad) and of homotopy (al)gebras as well. Their underlying chain complex is endowed an L∞-algebra structure in general and a Lie algebra structure only in the Koszul case. In particular, we make the deformation complex of morphisms from the properad of associative bialgebras explicit . For any minimal model of this properad, the boundary map of this chain complex is shown to be the one defined by Gerstenhaber and Schack. As a corollary, this paper provides a complete proof of the existence of an L∞-algebra structure on the Gerstenhaber-Schack bicomplex associated to the deformations of associative bialgebras.

We introduce and study wheeled PROPs, an extension of the theory of PROPs which can treat traces and, in particular, solutions to the master equations which involve divergence operators. We construct a dg free wheeled PROP whose representations are in one-to-one correspondence with formal germs of SP-manifolds, key geometric objects in the theory of Batalin-Vilkovisky quantization. We also construct minimal wheeled resolutions of classical operads Com and Ass as rather non-obvious extensions of Com_infty and Ass_infty, involving, e.g., a mysterious mixture of associahedra with cyclohedra. Finally, we apply the above results to a computation of cohomology of a directed version of Kontsevich's complex of ribbon graphs.Comment: LaTeX2e, 63 pages; Theorem 4.2.5 on bar-cobar construction is strengthene