导出的光滑与Banach高阶群胚：可表示性与下降
Derived Smooth and Banach Higher Groupoids: Representability and Descent
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中文总结 AI 辅助
本文研究特定几何中高阶群胚的同伦理论，通过区分可表示性与实现，建立了光滑与Banach情形的Brown结构，并推广至导出模型，涵盖有限与超完备情形。
中文摘要 AI 辅助
我们研究在特定几何中的高阶群胚的同伦理论，区分几何可表示性与实现及下降。普通的光滑和Banach开层允许完整的纤维对象的Brown范畴。对于普通的几何群胚，一个交叉轴障碍促使引入一个空兼容的不完整Brown结构。其分裂Banach版本需要一个固定的图表兼容绘图位点，并具有局部核-积闭性。这些结果包括所有有限唯一脊界及其并集。对于表示的导出模型，有限匹配和外棱镜过滤给出结构性的Brown演算。Nuiten的有限几何实现定理给出有限导出光滑结构及其局部化。在原始小Kan-富化位点上的富化实现基变换给出外无界超完备结构，具有正度几何脊条件和每一度的Reedy纤维性。对于Banach域，结构化谱、严格开粘合和分裂开超下降建立几何有限极限和超层可表示性。它们给出表示的Brown结构，在每个外边界使用相同的超完备弱等价类。一个平方零测试检测所选Banach图表的非普通自交，包括无限维图表。对于连通结合仿射，同伦分裂脊、Pridham的有限效应定理和逐点对角纤维化论证给出他的Artin和Deligne-Mumford类的纤维对象范畴。结合理论使用代数的拟同构和预层中的实现。
英文摘要
We study homotopy theories of higher groupoids in specified geometries, distinguishing geometric representability from realization and descent. Ordinary smooth and Banach-open sheaves admit full Brown categories of fibrant objects. For ordinary geometric groupoids, a crossing-axes obstruction motivates an empty-compatible incomplete Brown structure. Its split-Banach version requires a fixed chart-compatible plot site with local kernel-product closure. These results include all finite unique-horn bounds and their union. For represented derived models, finite matching and an outer-prism filtration give the structural Brown calculus. Nuiten's finite-geometric realization theorem yields the finite derived-smooth structure and its localization. Enriched realization base change on the original small Kan-enriched site gives the outer-unbounded hypercomplete structure, with positive-degree geometric horn conditions and Reedy fibrancy in every degree. For Banach domains, structured spectra, strict-open gluing and split-open hyperdescent establish geometric finite limits and hypersheaf representability. They give the represented Brown structure, using the same hypercompleted weak-equivalence class at every outer bound. A square-zero test detects nonordinary self-intersections of the chosen Banach charts, including infinite-dimensional charts. For connective associative affines, homotopy split horns, Pridham's finite effectivity theorem and a pointwise diagonal-fibration argument give categories of fibrant objects for his Artin and Deligne--Mumford classes. The associative theory uses quasi-isomorphisms of algebras and realization in presheaves.
