Simplicial localisation of homotopy algebras over a prop

Yalin, Sinan

doi:10.1017/s0305004114000437

Cited by 7 publications

(10 citation statements)

References 23 publications

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“…The goal of this section is to prove that the classifying space of weak equivalences of P -algebras is weakly equivalent to the classifying space of acyclic fibrations of P -algebras: Remark 3.2. Actually, the methods of [30] can be transposed in our setting to prove the following much stronger statement. We refer the reader to the seminal papers [8], [9] and [10] for the notions of simplicial localization, hammock localization and Dwyer-Kan equivalences of simplicial categories.…”

Section: The Subcategory Of Acyclic Fibrationsmentioning

confidence: 98%

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Function spaces and classifying spaces of algebras over a prop

Yalin

2016

Algebr. Geom. Topol.

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The goal of this paper is to prove that the classifying spaces of categories of algebras governed by a prop can be determined by using function spaces on the category of props. We first consider a function space of props to define the moduli space of algebra structures over this prop on an object of the base category. Then we mainly prove that this moduli space is the homotopy fiber of a forgetful map of classifying spaces, generalizing to the prop setting a theorem of Rezk. The crux of our proof lies in the construction of certain universal diagrams in categories of algebras over a prop. We introduce a general method to carry out such constructions in a functorial way.Comment: 28 pages, modifications mainly in section 2 (more details in some proofs and additional explanations), typo corrections. Final version, to appear in Algebr. Geom. Topo

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Section: The Subcategory Of Acyclic Fibrationsmentioning

confidence: 98%

“…Remark 3.2. Actually, the methods of [30] can be transposed in our setting to prove the following much stronger statement. We refer the reader to the seminal papers [8], [9] and [10] for the notions of simplicial localization, hammock localization and Dwyer-Kan equivalences of simplicial categories.…”

Section: For Any Objectmentioning

confidence: 98%

Function spaces and classifying spaces of algebras over a prop

Yalin

2016

Algebr. Geom. Topol.

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“…By [94,Section 3.3], this means that F induces an equivalence of weak presheaves of ∞-categories as in Theorem 3.25. Then, we can mimick the proof of Theorem 3.25 as follows: we replace the presheaves of ∞-categories by these presheaves of classification spaces, t ake based loop spaces which gives back the homotopy automorphisms as well, and apply Theorem 3.24.…”

Section: Theorem 324mentioning

confidence: 99%

Derived deformation theory of algebraic structures

Ginot¹,

Yalin²

2019

Preprint

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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 of algebras (associative, commutative, Poisson...) or bialgebras (associative and coassociative, Lie, Frobenius...), that is algebraic structures parametrized by props.A central aspect is that we define and study moduli spaces of deformations of algebraic structures up to quasi-isomorphisms (and not only up to isomorphims or ∞-isotopies). To do so, we implement methods coming from derived algebraic geometry, by encapsulating these deformation theories as classifying (pre)stacks with good infinitesimal properties and derived formal groups. In particular, we prove that the Lie algebra describing the deformation theory of an object in a given ∞-category of dg algebras can be obtained equivalently as the tangent complex of loops on a derived quotient of this moduli space by the homotopy automorphims of this object.Moreover, we provide explicit formulae for such derived deformation problems of algebraic structures up to quasi-isomorphisms and relate them in a precise way to other standard deformation problems of algebraic structures. This relation is given by a fiber sequence of the associated dg-Lie algebras of their deformation complexes. Our results provide simultaneously a vast generalization of standard deformation theory of algebraic structures which is suitable (and needed) to set up algebraic deformation theory both at the ∞categorical level and at a higher level of generality than algebras over operads.In addition, we study a general criterion to compare formal moduli problems of different algebraic structures and apply our formalism to En-algebras and bialgebras.

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“…To overcome these difficulties, one has to go through a completely new method based on the construction of a functorial path object of P -algebras and a corresponding equivalence of classification spaces proved in [96], then an argument using the equivalences of several models of (∞, 1)-categories [97]. The equivalence of Theorem 2.4 is stated and proved in [97] as an equivalence of hammock localizations in the sense of Dwyer-Kan [16]. Theorem 2.4 means that the notion of algebraic structure up to homotopy is coherent in a very general context, and in particular that transfer and realization problems make sense also for various kinds of bialgebras.…”

Section: Homotopy Theory Of (Bi)algebrasmentioning

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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Simplicial localisation of homotopy algebras over a prop

Cited by 7 publications

References 23 publications

Function spaces and classifying spaces of algebras over a prop

Function spaces and classifying spaces of algebras over a prop

Derived deformation theory of algebraic structures

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

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