Props in model categories and homotopy invariance of structures

Fresse, Benoît

doi:10.1515/gmj.2010.007

Cited by 21 publications

(38 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(cf. [32,Theorem 5.5]) The category of dg props P rop equipped with the classes of componentwise weak equivalences and componentwise fibrations forms a cofibrantly generated model category.…”

Section: Morphisms Of Props Are Equivariant Morphisms Of Collections ...mentioning

confidence: 99%

Derived deformation theory of algebraic structures

Ginot¹,

Yalin²

2019

Preprint

View full text Add to dashboard Cite

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.

show abstract

Section: Morphisms Of Props Are Equivariant Morphisms Of Collections ...mentioning

confidence: 99%

Derived deformation theory of algebraic structures

Ginot¹,

Yalin²

2019

Preprint

View full text Add to dashboard Cite

show abstract

“…Even though this is one of the most important and motivational aspects of the theory, we will not be concerned with algebras over PROPs in this paper; we refer the reader to [18] and [24] for more details on the theory of algebras and their homotopy theory.…”

Section: Enriched Coloured Propsmentioning

confidence: 99%

“…Unravelling definition 2.3 we get the following equivalent one (see [18]): called input-permutation and output-permutation respectively. These data have to satisfy the following conditions:…”

Section: Enriched Coloured Propsmentioning

confidence: 99%

“…As for operads, PROPs can be enriched in any symmetric monoidal category V. If V has a (sufficiently nice) model structure, the category of V-enriched C-coloured PROPs (i.e. with a fixed set of colours C) inherits a (semi-)model structure (see [18]).…”

Section: Introductionmentioning

confidence: 99%

“…The homotopy invariance of algebras over enriched (coloured) PROPs in the context of model categories was studied by Fresse in [18] and Johnson and Yau in [24]. The higher degree of complexity of the algebraic structures presented by PROPs makes their study more difficult with respect to operadic theories; for example, there is not always a free construction for this kind of algebras, thus there is no way to transfer the model structure from the ground category to the category of algebras in general.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Dwyer-Kan model structure for enriched coloured PROPs

Caviglia

2015

Preprint

View full text Add to dashboard Cite

Coloured PROPs are a generalisation of coloured operads. In this article, we prove the existence of a Dwyer-Kan model structure on the category of small coloured PROPs enriched in a (sufficiently nice) monoidal model category V. This model structure extends the known Dwyer-Kan model structures on the categories of (small) V-enriched categories and V-enriched coloured operads. Contents 1. Introduction 1 2. Enriched coloured PROPs 4 3. Operadic bifibrations 10 4. Homotopy theory of the fibers 16 5. The Dwyer-Kan model structure on enriched PROPs 18 6. The operads for PROPs 23 7. Push-out of a PROP along a morphism of operads 34 8. Locally presentable operadic families 37 Appendix A. Operadic morphisms 39 Appendix B. Graphs 50 References 61

show abstract

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

Yalin

2018

MATRIX Book Series

View full text Add to dashboard Cite

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.

show abstract

Props in model categories and homotopy invariance of structures

Cited by 21 publications

References 16 publications

Derived deformation theory of algebraic structures

Derived deformation theory of algebraic structures

The Dwyer-Kan model structure for enriched coloured PROPs

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

Contact Info

Product

Resources

About