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Higher Orbifolds and Deligne-Mumford Stacks as Structured Infinity-Topoi
About this Title
David Joseph Carchedi
Publication: Memoirs of the American Mathematical Society
Publication Year:
2020; Volume 264, Number 1282
ISBNs: 978-1-4704-4144-9 (print); 978-1-4704-5810-2 (online)
DOI: https://doi.org/10.1090/memo/1282
Published electronically: March 11, 2020
Keywords: Infinity-topoi,
higher stacks,
Deligne-Mumford stacks,
orbifolds
MSC: Primary 18B25, 14D23; Secondary 58A03
Table of Contents
Chapters
- 1. Introduction
- 2. Preliminaries on higher topos theory
- 3. Local Homeomorphisms and Étale Maps of $\infty$-Topoi
- 4. Structured $\infty$-Topoi
- 5. Étendues: Gluing Local Models
- 6. Examples
Abstract
We develop a universal framework to study smooth higher orbifolds on the one hand and higher Deligne-Mumford stacks (as well as their derived and spectral variants) on the other, and use this framework to obtain a completely categorical description of which stacks arise as the functor of points of such objects. We choose to model higher orbifolds and Deligne-Mumford stacks as infinity-topoi equipped with a structure sheaf, thus naturally generalizing the work of Lurie (2004), but our approach applies not only to different settings of algebraic geometry such as classical algebraic geometry, derived algebraic geometry, and the algebraic geometry of commutative ring spectra as in Lurie (2004), but also to differential topology, complex geometry, the theory of supermanifolds, derived manifolds etc., where it produces a theory of higher generalized orbifolds appropriate for these settings. This universal framework yields new insights into the general theory of Deligne-Mumford stacks and orbifolds, including a representability criterion which gives a categorical characterization of such generalized Deligne-Mumford stacks. This specializes to a new categorical description of classical Deligne-Mumford stacks, a result sketched in Carchedi (2019), which extends to derived and spectral Deligne-Mumford stacks as well.- Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos, Lecture Notes in Mathematics, Vol. 269, Springer-Verlag, Berlin-New York, 1972 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4); Dirigé par M. Artin, A. Grothendieck, et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat. MR 0354652
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