Deformation theory of representations of prop(erad)s II

Merkulov, Sergeï; Vallette, Bruno

doi:10.1515/crelle.2009.084

Cited by 69 publications

(106 citation statements)

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“…Traditionally, the deformation complex is given by the canonical cofibrant resolution of a properad. Since a general cofibrant resolution is often quite large, Merkulov and Vallette construct minimal models using "homotopy" Koszul duality of properads [MV09a,MV09b]. A homotopy Koszul properad has a space of generators equal to the Koszul dual of a quadratic properad associated to it.…”

Section: Deformation Theorymentioning

confidence: 99%

Infinity Properads and Infinity Wheeled Properads

Hackney

Robertson

Yau

2015

Lecture Notes in Mathematics

180

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PrefaceThis monograph fits in the intersection of two long and intertwined stories. The first part of our story starts in the mid-twentieth century, when it became clear that a new conceptual framework was necessary for the study of higher homotopical structures arising in algebraic topology. Some better known examples of these higher homotopical structures appear in work of J. F. Adams and S. Mac Lane [Mac65] on the coproduct in the bar construction and work of J. Stasheff [Sta63], J. M. Boardman and R. M. Vogt [BV68,BV73], and J. P. May [May72] on recognition principles for (ordinary, n-fold, or infinite) loop spaces. The notions of 'operad' and 'prop' were precisely formulated for the purpose of this work; the former is suitable for modeling algebraic or coalgebraic structures, while the latter is also capable of modeling bialgebraic structures, such as Hopf algebras. Operads came to prominence in other areas of mathematics beginning in the 1990s (but see, e.g. This renaissance in the world of operads [Lod96, LSV97] and the popularity of quantum groups [Dri83,Dri87] in the 1990s lead to a resurgence of interest in props, which had long been in the shadow of their little singleoutput nephew. Properads were invented around this time, during an effort of B. Vallette to formulate a Koszul duality for props [Val07]. Properads and props both model algebraic structures with several inputs and outputs, but properads govern a smaller class of such structures, those whose generating operations and relations among operations can be taken to be connected. This class includes most types of bialgebras that arise in nature, such as biassociative bialgebras, (co)module bialgebras, Lie bialgebras, and Hopf algebras. The use of 'colored' or 'multisorted' variants of operads or props [Bri00, BV73], where composition is only partially defined, allows one to address many other situations of interest. It allows one, for instance, to model morphisms of algebras associated to a given operad. There is a two-colored operad which encodes the data of two associative algebras as well as a map from one to the other; a resolution of this operad precisely gives the correct notion of morphism of A ∞ -algebras [Mar04,Mar02]. It also provides a unified way to treat operads, cyclic operads, modular operads, properads, and so on: for each there is a colored operad which controls the structure in question.The second part of our story is an extension of the theory of categories. Categories are pervasive in pure mathematics and, for our purposes, can be loosely described as tools for studying collections of objects up to isomorphism and comparisons between collections of objects up to isomorphism. When the objects we want to study have a homotopy theory, we need to generalize category theory to identify two objects which are, while possibly not isomorphic in the categorical sense, equivalent up to homotopy. For example, when discussing topological spaces we might replace 'homeomorphism' with 'homotopy equivalence'. This leads to the theory of ∞-ca...

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Section: Deformation Theorymentioning

confidence: 99%

Infinity Properads and Infinity Wheeled Properads

Hackney

Robertson

Yau

2015

Lecture Notes in Mathematics

180

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“…Remark 2.2. According to [68], the similar free-forgetful adjunction between Σbiojects and dg properads equips dg properads with a cofibrantly generated model category structure with componentwise fibrations and weak equivalences.…”

Section: Homotopy Theory Of (Bi)algebrasmentioning

confidence: 99%

“…The twisting of the complete L ∞ -algebra Hom Σ (C, End X ) by a properad morphism ϕ : P ∞ → End X is the deformation complex of ϕ, and we have an isomorphism g ϕ P,X = Hom Σ (C, End X ) ϕ ∼ = Der ϕ (Ω(C), End X ) where the right-hand term is the complex of derivations with respect to ϕ [68,Theorem 12], whose L ∞ -structure induced by the twisting of the left-hand side is equivalent to the one of […”

Section: 3mentioning

confidence: 99%

Moduli Spaces of (Bi)algebra Structures in Topology and Geometry

Yalin

2018

MATRIX Book Series

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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 and mathematical physics. This allows us to define an "up to homotopy version" of algebraic structures which is coherent (in the sense of ∞-category theory) at a high level of generality. To understand the classification and deformation theory of these structures on a given object, a relevant idea inspired by geometry is to gather them in a moduli space with nice homotopical and geometric properties. Derived geometry provides the appropriate framework to describe moduli spaces classifying objects up to weak equivalences and encoding in a geometrically meaningful way their deformation and obstruction theory. As an instance of the power of such methods, I will describe several results of a joint work with Gregory Ginot related to longstanding conjectures in deformation theory of bialgebras, En-algebras and quantum group theory.

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“…It is easy to check that the constructed map F k has degree 1 − k. It is also a straightforward untwisting of the definitions of differentials d in Defq and δ in LieB ∞ to show that equations (17) follow directly from the basic property, δ • F = F • d, of the morphism F (cf. again [Ma3,MeVa]). (18) i :…”

Section: 3mentioning

confidence: 99%