NDLI: Principal $$\infty $$ -bundles: presentations

Content Provider	Springer Nature Link
Author	Nikolaus, Thomas Stevenson, Danny Schreiber, Urs
Copyright Year	2014
Abstract	We discuss two aspects of the presentation of the theory of principal $$\infty $$ -bundles in an $$\infty $$ -topos, introduced in Nikolaus et al. (Principal $$\infty $$ -bundles: general theory, 2012), in terms of categories of simplicial (pre)sheaves. First we show that over a cohesive site $$C$$ and for $$G$$ a presheaf of simplicial groups which is $$C$$ -acyclic, $$G$$ -principal $$\infty $$ -bundles over any object in the $$\infty $$ -topos over $$C$$ are classified by hyper-Čech-cohomology with coefficients in $$G$$ . Then we show that over a site $$C$$ with enough points, principal $$\infty $$ -bundles in the $$\infty $$ -topos are presented by ordinary simplicial bundles in the sheaf topos that satisfy principality by stalkwise weak equivalences. Finally we discuss explicit details of these presentations for the discrete site (in discrete $$\infty $$ -groupoids) and the smooth site (in smooth $$\infty $$ -groupoids, generalizing Lie groupoids and differentiable stacks). In the companion article (Nikolaus et al. in Principal $$\infty $$ -bundles: examples and applications, 2012) we use these presentations for constructing classes of examples of (twisted) principal $$\infty $$ -bundles and for the discussion of various applications.
Starting Page	565
Ending Page	622
Page Count	58
File Format	PDF
ISSN	21938407
Journal	Journal of Homotopy and Related Structures
Volume Number	10
Issue Number	3
e-ISSN	15122891
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2014-03-12
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Homotopy theory Algebra Principal bundles Functional Analysis Differential geometry Algebraic Topology Topos theory Number Theory Algebraic topology
Content Type	Text
Resource Type	Article
Subject	Algebra and Number Theory Geometry and Topology

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