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Higher Orbifolds and Deligne-Mumford Stacks as Structured Infinity-Topoi
 
David Carchedi George Mason University, Fairfax, VA
Higher Orbifolds and Deligne-Mumford Stacks as Structured Infinity-Topoi
Softcover ISBN:  978-1-4704-4144-9
Product Code:  MEMO/264/1282
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $51.00
eBook ISBN:  978-1-4704-5810-2
Product Code:  MEMO/264/1282.E
List Price: $85.00
AMS Member Price: $76.50
Softcover ISBN:  978-1-4704-4144-9
eBook: ISBN:  978-1-4704-5810-2
Product Code:  MEMO/264/1282.B
List Price: $170.00 $127.50
MAA Member Price: $114.75
AMS Member Price: $127.50 $76.50
Higher Orbifolds and Deligne-Mumford Stacks as Structured Infinity-Topoi
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Higher Orbifolds and Deligne-Mumford Stacks as Structured Infinity-Topoi
David Carchedi George Mason University, Fairfax, VA
Softcover ISBN:  978-1-4704-4144-9
Product Code:  MEMO/264/1282
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $51.00
eBook ISBN:  978-1-4704-5810-2
Product Code:  MEMO/264/1282.E
List Price: $85.00
AMS Member Price: $76.50
Softcover ISBN:  978-1-4704-4144-9
eBook ISBN:  978-1-4704-5810-2
Product Code:  MEMO/264/1282.B
List Price: $170.00 $127.50
MAA Member Price: $114.75
AMS Member Price: $127.50 $76.50
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2642020; 120 pp
    MSC: Primary 18; 14; Secondary 58;

    The author develops 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. He chooses to model higher orbifolds and Deligne-Mumford stacks as infinity-topoi equipped with a structure sheaf, thus naturally generalizing the work of Lurie, but his 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 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, which extends to derived and spectral Deligne-Mumford stacks as well.

  • 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
  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2642020; 120 pp
MSC: Primary 18; 14; Secondary 58;

The author develops 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. He chooses to model higher orbifolds and Deligne-Mumford stacks as infinity-topoi equipped with a structure sheaf, thus naturally generalizing the work of Lurie, but his 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 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, which extends to derived and spectral Deligne-Mumford stacks as well.

  • 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
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.