Ginzburg algebras of triangulated surfaces and perverse schobers

Christ, Merlin

doi:10.1017/fms.2022.1

Cited by 4 publications

(3 citation statements)

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“…The vector spaces Ψ i , with 1 ď i ď n, are all equivalent and may be chosen for concreteness as the stalks of the n-th roots of unity. An ad-hoc categorification of this local description is described in [Chr22a], based on Waldhausen's relative S ‚ -construction. Using these local descriptions, we can describe perverse sheaves or perverse schobers on any surface S with 0-dimensional strata P and non-empty boundary by choosing a spanning graph G Ă S. The inclusion of G into S is required to be a homotopy equivalence and each stratum p P P is required to be the image of a vertex of G. A perverse sheaf or perverse schober on S can be encoded as a constructible sheaf and cosheaf on G, which restricts at each vertex of G to a diagram as in (5.2.4) or its categorification.…”

Section: Barbacovi's Theorem and Its Geometric Interpretationmentioning

confidence: 99%

“…Theorem 5.2.3 is a special case of a more general phenomenon exhibited by perverse schobers on a stratified surface S with non-empty boundary. We again choose a spanning graph G of S. By a G-parametrized perverse schober, we mean a constructible sheaf of stable 8-categories on G encoding a perverse schober on S, as explained above and defined in [Chr22a]. Given an edge e of G, we denote by Fpeq the stalk of F at any point on that edge.…”

Section: Barbacovi's Theorem and Its Geometric Interpretationmentioning

confidence: 99%

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Complexes of stable $\infty$-categories

Christ¹,

Dyckerhoff²,

Walde³

2023

Preprint

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We study complexes of stable 8-categories, referred to as categorical complexes. As we demonstrate, examples of such complexes arise in a variety of subjects including representation theory, algebraic geometry, symplectic geometry, and differential topology. One of the key techniques we introduce is a totalization construction for categorical cubes which is particularly well-behaved in the presence of Beck-Chevalley conditions. As a direct application we establish a categorical Koszul duality result which generalizes previously known derived Morita equivalences among higher Auslander algebras and puts them into a conceptual context. We explain how spherical categorical complexes can be interpreted as higher-dimensional perverse schobers, and introduce Calabi-Yau structures on categorical complexes to capture noncommutative orientation data. A variant of homological mirror symmetry for categorical complexes is proposed and verified for CP 2 . Finally, we develop the concept of a lax additive p8, 2q-category and propose it as a suitable framework to formulate further aspects of categorified homological algebra.

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Section: Barbacovi's Theorem and Its Geometric Interpretationmentioning

confidence: 99%

Section: Barbacovi's Theorem and Its Geometric Interpretationmentioning

confidence: 99%

Complexes of stable $\infty$-categories

Christ¹,

Dyckerhoff²,

Walde³

2023

Preprint

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“…Remark 5.5. For an oriented marked surface with ideal triangulation (S, T ), we denote by G T the associated relative Ginzburg algebra (see [8]). Let S be an oriented marked surface with two ideal triangulation T , T ′ related by a flip of an edge e of T .…”

Section: 1mentioning

confidence: 99%

Categorification of ice quiver mutation

Wu¹

2021

Preprint

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In 2009, Keller and Yang categorified quiver mutation by interpreting it in terms of equivalences between derived categories. Their approach was based on Ginzburg's Calabi-Yau algebras and on Derksen-Weyman-Zelevinsky's mutation of quivers with potential. Recently, Matthew Pressland has generalized mutation of quivers with potential to that of ice quivers with potential. In this paper, we show that his rule yields derived equivalences between the associated relative Ginzburg algebras, which are special cases of Yeung's deformed relative Calabi-Yau completions arising in the theory of relative Calabi-Yau structures due to Toën and Brav-Dyckerhoff. We illustrate our results on examples arising in the work of Baur-King-Marsh on dimer models and cluster categories of Grassmannians. We also give a categorification of mutation at frozen vertices as it appears in recent work of Fraser-Sherman-Bennett on positroid cluster structures. 2.1. Derived categories of dg algebras 2.2. Derived categories of pseudocompact dg algebras 2.3. Hochschild/cyclic homology in the pseudocompact setting 3. Relative Calabi-Yau completions in the pseudocompact setting 3.1. Relative deformed Calabi-Yau completions 4. Ice quiver mutations and complete relative Ginzburg algebras 4.1. Ice quivers 4.2. Combinatorial mutations 4.3. Ice quivers with potential 4.4. Algebraic mutations 4.5. The complete relative Ginzburg algebra and the Ginzburg functor 4.6. Cofibrant resolutions of simples over a tensor algebra 5. Main results 5.1. Compatibility with Morita functors and localizations 5.2. Derived equivalences 5.3. Stability under mutation of relative Ginzburg algebras concentrated in degree 0 6. Mutation at frozen vertices 6.1. Combinatorial mutations 6.2. Algebraic mutations 6.3. Categorical mutations References

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Categorification of ice quiver mutation

2023

Math. Z.

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Ginzburg algebras of triangulated surfaces and perverse schobers

Cited by 4 publications

References 50 publications

Complexes of stable $\infty$-categories

Complexes of stable $\infty$-categories

Categorification of ice quiver mutation

Categorification of ice quiver mutation

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