Generators in formal deformations of categories

Blanc, Anthony; Katzarkov, Ludmil; Pandit, Pranav

doi:10.1112/s0010437x18007303

Cited by 9 publications

(9 citation statements)

References 8 publications

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“…This equivalence is an analogue to the classical correspondence between formal/algebraic/Lie groups and Lie algebras. This equivalence holds true not only in the commutative case but also for iterated loop spaces and noncommutative moduli problems, see [8,Proposition 2.15]. Remark 3.6.…”

Section: Derived Formal Groups Of Algebraic Structures and Associated...mentioning

confidence: 89%

“…( 2), the functor Ω * has a left adjoint (3.3) B f mp : M on gp E1 (F M P K ) −→ F M P K . The functor B f mp is obtained as a generalized bar construction given by the realization of a simplicial object in derived formal moduli problems, hence a homotopy colimit corresponding to a classifying space ∞-functor for derived formal groups (see [8,Lemma 2.16] and [67, Remark 5.2.2.8]). Composing equivalence (2) with Lurie's equivalence theorem [64] result into the equivalence…”

Section: Derived Formal Groups Of Algebraic Structures and Associated...mentioning

confidence: 99%

“…• We denote haut(X) the derived prestack group of homotopy automorphisms of the underlying complex X taken in the model category of chain complexes 8 . It is defined by…”

Section: Prestacks Of Algebras and Derived Groups Of Homotopy Automor...mentioning

confidence: 99%

“…The latter is a 1-proximate formal moduli problem in the sense of [64, Section 5.1] with associated formal moduli problem denoted L(ObjDef o X ). By [8,Lemma 2.11], there is an equivalence of formal moduli problems…”

Section: Proof Of Propoposition 48 First We Construct the Commutative...mentioning

confidence: 99%

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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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Section: Derived Formal Groups Of Algebraic Structures and Associated...mentioning

confidence: 89%

Section: Derived Formal Groups Of Algebraic Structures and Associated...mentioning

confidence: 99%

“…• We denote haut(X) the derived prestack group of homotopy automorphisms of the underlying complex X taken in the model category of chain complexes 8 . It is defined by…”

Section: Prestacks Of Algebras and Derived Groups Of Homotopy Automor...mentioning

confidence: 99%

Section: Proof Of Propoposition 48 First We Construct the Commutative...mentioning

confidence: 99%

See 2 more Smart Citations

Derived deformation theory of algebraic structures

Ginot¹,

Yalin²

2019

Preprint

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“…For example, even if A 0 is uncurved, there might exist curved deformations A for which A bc is empty (i.e., all objects might be obstructed). See [BKP17] for a more detailed analysis.…”

Section: Curvedmentioning

confidence: 99%

Versality in mirror symmetry

Sheridan

2017

Current Developments in Mathematics

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One of the attractions of homological mirror symmetry is that it not only implies the previous predictions of mirror symmetry (e.g., curve counts on the quintic), but it should in some sense be 'less of a coincidence' than they are and therefore easier to prove. In this survey we explain how Seidel's approach to mirror symmetry via versality at the large volume/large complex structure limit makes this idea precise.

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Deformation quantisation for unshifted symplectic structures on derived Artin stacks

Pridham¹

2018

Sel. Math. New Ser.

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We prove that every 0-shifted symplectic structure on a derived Artin n-stack admits a curved A∞ deformation quantisation. The classical method of quantising smooth varieties via quantisations of affine space does not apply in this setting, so we develop a new approach. We construct a map from DQ algebroid quantisations of unshifted symplectic structures on a derived Artin n-stack to power series in de Rham cohomology, depending only on a choice of Drinfeld associator. This gives an equivalence between even power series and certain involutive quantisations, which yield anti-involutive curved A∞ deformations of the dg category of perfect complexes. In particular, there is a canonical quantisation associated to every symplectic structure on such a stack, which agrees for smooth varieties with the Kontsevich-Tamarkin quantisation for even associators. The quasi-isomorphism gives a k -linear P 2,∞ -algebra quasi-isomorphism Pol(A, 0) ≃ Q Pol w (A, 0) sending the filtration { i F p−i i } p on the left to {F p } p , and inclusion of Pol(A, 0) on the left then gives rise to the quantisation map φ w : P(A, 0) → QP w (A, 0) on Maurer-Cartan spaces. This allows us to make a comparison with our constructions: Lemma 2.24. For A smooth over a field k ⊃ Q and w ∈ Levi GT (k), the map φ w : P(A, 0) → QP w (A, 0) from Poisson structures to quantisations, following [VdB, Remark 8.2.1 and Theorem 9.5.1], extends to map Comp(A, 0) → QComp w (A, 0),from compatible pairs to w-compatible pairs.Proof. Functoriality of µ implies that µ w (ω, φ w (π)) = φ w (µ(ω, π)), so φ w (π) is wcompatible with a pre-symplectic form ω whenever π is compatible with ω.When w comes from an even associator, we then have:

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Generators in formal deformations of categories

Cited by 9 publications

References 8 publications

Derived deformation theory of algebraic structures

Derived deformation theory of algebraic structures

Versality in mirror symmetry

Deformation quantisation for unshifted symplectic structures on derived Artin stacks

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