Derived Smooth and Banach Higher Groupoids: Representability and Descent
Abstract
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.
Related
- Damped stochastic mirror descent for non-smooth ill-posed problems in Banach space
- Empirical Bernstein in smooth Banach spaces
- Spectral theorem for compact self-adjoint operators on smooth Banach spaces
- A strictly convex and smooth Banach space with the Daugavet property
- Adaptive Schauder Stochastic Mirror Descent in Banach Spaces
BibTeX
@misc{zeng2026derived,
title = {{Derived Smooth and Banach Higher Groupoids: Representability and Descent}},
author = {Qingyun Zeng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28220},
eprint = {2609.28220},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28220},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Derived Smooth and Banach Higher Groupoids: Representability and Descent
AU - Qingyun Zeng
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28220
UR - https://arxiv.org/abs/2609.28220
AB - 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.
ER -