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

One of the core issues of nanotechnology involves masking the foreignness of nanomaterials to enable in vivo longevity and long-term immune evasion. Dextran-coated superparamagnetic iron oxide nanoparticles are very effective magnetic resonance imaging (MRI) contrast agents, and strategies to prevent immune recognition are critical for their clinical translation. Here we prepared 20 kDa dextran-coated SPIO nanoworms (NWs) of 250 nm diameter and a high molar transverse relaxivity rate R2 (~400 mM−1 s−1) to study the effect of cross-linking-hydrogelation with 1-chloro-2,3-epoxypropane (epichlorohydrin) on the immune evasion both in vitro and in vivo. Cross-linking was performed in the presence of different concentrations of NaOH (0.5 to 10 N) and different temperatures (23 and 37 °C). Increasing NaOH concentration and temperature significantly decrease the binding of anti-dextran antibody and dextran-binding lectin conconavalin A to the NWs. The decrease in dextran immunoreactivity correlated with the decrease in opsonization by complement component 3 (C3) and with the decrease in the binding of the lectin pathway factor MASP-2 in mouse serum, suggesting that cross-linking blocks the lectin pathway of complement. The decrease in C3 opsonization correlated with the decrease in NW uptake by murine peritoneal macrophages. Optimized NWs demonstrated up to 10 h circulation half-life in mice and minimal uptake by the liver, while maintaining the large 250 nm size in the blood. We demonstrate that immune recognition of large iron oxide nanoparticles can be efficiently blocked by chemical cross-linking-hydrogelation, which is a promising strategy to improve safety and bioinertness of MRI contrast agents.

We construct a functor from the derived category of homotopy Gerstenhaber algebras, g, with finite-dimensional cohomology to the purely geometric category of so-called F ∞ -manifolds. The latter contains Frobenius manifolds as a subcategory (so that a pointed Frobenius manifold is itself a homotopy Gerstenhaber algebra). If g happens to be formal as a L ∞ -algebra, then its F ∞ -manifold comes equipped with the Gauss-Manin connection. Mirror Symmetry implications are discussed.

The genetic code appears to be universal; . . . " Britannica.0. Introduction. The first instances of algebraic and topological strongly homotopy, or infinity, structures have been discovered by Stasheff [St] long ago. Since that time infinities have acquired a prominent role in algebraic topology and homological algebra. We argue in this paper that some classical local geometries are of infinity origin, i.e. their smooth formal germs are (homotopy) representations of cofibrant PROPs P ∞ in spaces concentrated in degree zero; in particular, they admit natural infinity generalizations when one considers homotopy representations of P ∞ in generic differential graded (dg) spaces. The simplest manifestation of this phenomenon is provided by the Poisson geometry (or even by smooth germs of tensor fields!) and is the main theme of the present paper. Another example is discussed in [Mer2]. The PROPs P ∞ are minimal resolutions of PROPs P which are graph spaces built from very few basic elements, genes, subject to simple engineering rules. Thus to a local geometric structure one can associate a kind of a code, genome, which specifies it uniquely and opens a new window of opportunities of attacking differential geometric problems with the powerful tools of homological algebra. In particular, the genetic code of Poisson geometry discovered in this paper has been used in [Mer3] to give a new short proof of Kontsevich's [Ko1] deformation quantization theorem.Formal germs of geometric structures discussed in this paper are pointed in the sense that they vanish at the distinguished point. This is the usual price one pays for working with (di)operads without "zero terms" (as is often done in the literature). As structural equations behind the particular geometries we study in this paper are homogeneous, this restriction poses no problem: say, a generic non-pointed Poisson structure, ν, in R n can be identified with the pointed one, ν, in R n+1 , being the extra coordinate.We introduce in this paper a dg free dioperad whose generic representations in a graded vector space V can be identified with pointed solutions of the Maurer-Cartan equations in the Lie algebra of polyvector fields on the formal manifold associated with V . The cohomology of this dioperad can not be computed directly. Instead one has to rely on some fine mathematics such as Koszulness [GiKa, G] and distributive laws [Mar1, G]. One of the main results of this paper is a proof of Theorem 3.2 which identifies the cohomology of that dg free dioperad with a surprisingly small dioperad, Lie 1 Bi, of Lie 1bialgebras, which are almost identical to the dioperad, LieBi, of usual Lie bialgebras except that degree of generating Lie and coLie operations differ by 1 (compare with Gerstenhaber versus Poisson algebras). The dioperad Lie 1 Bi is proven to be Koszul. We use the resulting geometric interpretation of Lie 1 Bi ∞ algebras to give their homotopy classification (see Theorem 3.4.5) which is an extension of Kontsevich's homotopy classification [Ko1] of L ∞ algebras.As...

We compute the homotopy derivations of the properads governing even and odd Lie bialgebras as well as involutive Lie bialgebras. The answer may be expressed in terms of the Kontsevich graph complexes. In particular, this shows that the Grothendieck-Teichmüller group acts faithfully (and essentially transitively) on the completions of the properads governing even Lie bialgebras and involutive Lie bialgebras, up to homotopy. This shows also that by contrast to the even case the properad governing odd Lie bialgebras admits precisely one non-trivial automorphism -the standard rescaling automorphism, and that it has precisely one non-trivial deformation which we describe explicitly.