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| Block | Content | Key References | |-------|---------|----------------| | | What “perverse family’’ means; perverse t‑structures; “free‑category’’ (dg‑ or A∞‑categories generated freely by a quiver or by a stratification). | BBD (1982); Kashiwara–Schapira (1990); Lurie (2014‑2022); Gaitsgory–Rozenblyum (2017). | | 1. Classical perverse sheaves & families | Classical definition (constructible complexes on a stratified space), families over a base, variation of perverse sheaves, perverse local systems. | Goresky–MacPherson (1980, 1988); Beilinson‑Bernstein‑Deligne (1982); de Cataldo‑Migliorini (2009). | | 2. Perverse‑coherent sheaves & “free’’ categorical models | Perverse‑coherent t‑structures on derived categories of coherent sheaves, and their description via free dg‑categories (e.g. quiver algebras with relations). | Bezrukavnikov (2003); Arinkin‑Bezrukavnikov (2010); Riche (2016). | | 3. Parametrising families without a fixed ambient category | Stacks of perverse objects; derived algebraic stacks; moduli of perverse sheaves as free‑generated Artin/derived stacks. | Simpson (1994‑2001); Toen‑Vezzosi (2008‑2013); Porta‑Yu (2020). | | 4. Computational/search tools in “free’’ settings | Algorithms for constructing perverse families via quiver representations, cellular sheaves, and software (e.g. PerverseSheaves.jl , SageMath packages). | Curry–Robinson (2021); Baranovsky‑Pym (2022). | | 5. Recent trends: perverse schobers & categorified families | Perverse schobers (categorified perverse sheaves) live naturally in free‑generated triangulated/∞‑categories; they provide a higher‑dimensional analogue of perverse families. | Kapranov‑Schechtman (2015‑2023); Dyckerhoff‑Le (2021‑2024). | | 6. Open problems & “search strategies’’ | How to detect a perverse family inside a free dg‑category, criteria based on Ext‑vanishing, co‑t‑structures, and microlocal support. | Ongoing work by Nadler‑Shende (2022‑2025); forthcoming preprints by Jiang‑Zhang (2024). | To understand why this specific type of phrase

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"perverse family" refers to specific pathological structures within family systems theory, most notably the perverse triangle If you encounter the keyword "searching for perverse

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