Homotopical Algebra for Lie Algebroids

Nuiten, Joost

doi:10.1007/s10485-019-09563-z

Cited by 13 publications

(31 citation statements)

References 34 publications

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“…Proof. For ordinary L ∞ -algebroids, without filtrations or curvature, this is proven in [Nui19a]. The proofs of loc.…”

Section: Homotopy Theory Of Curved Lie Algebroidsmentioning

confidence: 92%

“…Notice that Lie algebroids over a fixed base are not algebras over an operad, so that the usual methods of constructing a model structure on them do not quite work. In [Nui19a], the third author showed that Lie (or equivalently L ∞ ) algebroids carry a semimodel structure for which the weak equivalences are quasi-isomorphisms.…”

Section: Mod Grmentioning

confidence: 99%

“…In fact, we can go further than [Nui19a] and study Lie algebroids which are themselves curved. Such type of objects have been considered for instance in [Baa].…”

Section: Mod Grmentioning

confidence: 99%

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Lie algebroids are curved Lie algebras

Calaque,

Campos,

Nuiten

2021

Preprint

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“…Proof. For ordinary L ∞ -algebroids, without filtrations or curvature, this is proven in [Nui19a]. The proofs of loc.…”

Section: Homotopy Theory Of Curved Lie Algebroidsmentioning

confidence: 92%

Section: Mod Grmentioning

confidence: 99%

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Lie algebroids are curved Lie algebras

Calaque,

Campos,

Nuiten

2021

Preprint

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“…The following is the main result of [18]: Using the simplicial structure from [24] and the fact that every object in g/LieAlgd dg A is fibrant, one finds that g/LieAlgd dg A is a genuine (combinatorial) model category. Definition 2.6.…”

Section: Introductionmentioning

confidence: 98%

“…], so the result follows from (a shift of) Lemma 3.10.3.2.Tame dg-Lie algebroids. The tame model structure on dg-A-modules can be transferred to a semi-model structure on dg-Lie algebroids, by[18, Remark 4.25]: Proposition 3.12. The category of dg-Lie algebroids over A carries the tame semimodel structure, in which a map is a weak equivalence (fibration) if and only if it is a tame weak equivalence (fibration).…”

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confidence: 99%

Koszul duality for Lie algebroids

Nuiten

2019

Advances in Mathematics

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This paper studies the role of dg-Lie algebroids in derived deformation theory. More precisely, we provide an equivalence between the homotopy theories of formal moduli problems and dg-Lie algebroids over a commutative dg-algebra of characteristic zero. At the level of linear objects, we show that the category of representations of a dg-Lie algebroid is an extension of the category of quasi-coherent sheaves on the corresponding formal moduli problem. We describe this extension geometrically in terms of pro-coherent sheaves.between pro-coherent sheaves on X and pro-coherent T A/X -representations.Outline. The paper is outlined as follows. In Section 2, we recall the basic homotopy theory of dg-Lie algebroids over a commutative dg-algebra A, in which the weak equivalences are the quasi-isomorphisms. In Section 3 we provide model categorical descriptions of the homotopy theories of pro-coherent sheaves and procoherent Lie algebroids over A. Theorem 1.3 and its analogue in the coherent case are proven in Section 6, based on results about Lie algebroid cohomology that are discussed in Section 4 and 5. Section 7 is devoted to a proof of Theorem 1.4. As an application, we also show how Theorem 1.4 can be used to provide a simple point-set model for the Lie algebroid classifying the deformations of a (connective) commutative algebra over A (Proposition 7.18). Finally, in Section 8 we prove the equivalence of procoherent sheaves on formal moduli problems and representations of their associated Lie algebroid.Conventions. Throughout, let Q ⊆ k be a fixed connective commutative dgalgebra of characteristic zero and let A be a connective cdga over k. All differentialgraded objects are homologically graded, so that connective objects are concentrated in non-negative degrees. Given a chain complex V , we denote its suspension and cone by V [1] and V [0, 1]. Acknowledgement. The author was supported by NWO. Recollections on DG-Lie algebroidsIn this section we recall the homotopy theory of dg-Lie algebroids over a commutative dg-algebra, based on the discussion in [18].2.1. DG-Lie algebroids. Recall that the tangent module of a commutative dg-kalgebra A is the dg-A-module of k-linear derivations of A

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Algebraic foliations and derived geometry: the Riemann–Hilbert correspondence

Toën

Vezzosi

2022

Sel. Math. New Ser.

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This is the first in a series of papers about foliations in derived geometry. After introducing derived foliations on arbitrary derived stacks, we concentrate on quasi-smooth and rigid derived foliations on smooth complex algebraic varieties and on their associated formal and analytic versions. Their truncations are classical singular foliations defined in terms of differential ideals in the algebra of forms. We prove that a quasi-smooth rigid derived foliation on a smooth complex variety X is formally integrable at any point, and, if we suppose that its singular locus has codimension $$\ge 2$$ ≥ 2 , its analytification is a locally integrable singular foliation on the associated complex manifold $$X^h$$ X h . We then introduce the derived category of perfect crystals on a quasi-smooth rigid derived foliation on X, and prove a Riemann-Hilbert correspondence for them when X is proper. We discuss several examples and applications.

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Homotopical Algebra for Lie Algebroids

Cited by 13 publications

References 34 publications

Lie algebroids are curved Lie algebras

Lie algebroids are curved Lie algebras

Koszul duality for Lie algebroids

Algebraic foliations and derived geometry: the Riemann–Hilbert correspondence

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