NDLI: Principal $$\infty $$ -bundles: general theory

Content Provider	Springer Nature Link
Author	Nikolaus, Thomas Stevenson, Danny Schreiber, Urs
Copyright Year	2014
Abstract	The theory of principal bundles makes sense in any $$\infty $$ -topos, such as the $$\infty $$ -topos of topological, of smooth, or of otherwise geometric $$\infty $$ -groupoids/ $$\infty $$ -stacks, and more generally in slices of these. It provides a natural geometric model for structured higher nonabelian cohomology and controls general fiber bundles in terms of associated bundles. For suitable choices of structure $$\infty $$ -group $$G$$ these $$G$$ -principal $$\infty $$ -bundles reproduce various higher structures that have been considered in the literature and further generalize these to a full geometric model for twisted higher nonabelian sheaf cohomology. We discuss here this general abstract theory of principal $$\infty $$ -bundles, observing that it is intimately related to the axioms that characterize $$\infty $$ -toposes. A central result is a natural equivalence between principal $$\infty $$ -bundles and intrinsic nonabelian cocycles, implying the classification of principal $$\infty $$ -bundles by nonabelian sheaf hyper-cohomology. We observe that the theory of geometric fiber $$\infty $$ -bundles associated to principal $$\infty $$ -bundles subsumes a theory of $$\infty $$ -gerbes and of twisted $$\infty $$ -bundles, with twists deriving from local coefficient $$\infty $$ -bundles, which we define, relate to extensions of principal $$\infty $$ -bundles and show to be classified by a corresponding notion of twisted cohomology, identified with the cohomology of a corresponding slice $$\infty $$ -topos.
Starting Page	749
Ending Page	801
Page Count	53
File Format	PDF
ISSN	21938407
Journal	Journal of Homotopy and Related Structures
Volume Number	10
Issue Number	4
e-ISSN	15122891
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2014-06-24
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Algebra Principal bundles Functional Analysis Algebraic Topology Nonabelian cohomology Number Theory Higher topos theory
Content Type	Text
Resource Type	Article
Subject	Algebra and Number Theory Geometry and Topology

Sl.	Authority	Responsibilities	Communication Details
1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
5	Website/Portal (Helpdesk)	Queries regarding NDLI and its services	support@ndl.gov.in
6	Contents and Copyright Issues	Queries related to content curation and copyright issues	content@ndl.gov.in
7	National Digital Library of India Club (NDLI Club)	Queries related to NDLI Club formation, support, user awareness program, seminar/symposium, collaboration, social media, promotion, and outreach	clubsupport@ndl.gov.in
8	Digital Preservation Centre (DPC)	Assistance with digitizing and archiving copyright-free printed books	dpc@ndl.gov.in
9	IDR Setup or Support	Queries related to establishment and support of Institutional Digital Repository (IDR) and IDR workshops	idr@ndl.gov.in

Principal $$\infty $$ -bundles: presentations

Involutive $$A_\infty $$ -algebras and dihedral cohomology

On the invariance and general cohomology comparison theorems

The BV formalism for $$L_\infty $$ -algebras

Maurer–Cartan spaces of filtered $$L_{\infty }$$ -algebras

Existence and uniqueness of $$E_{\infty }$$ structures on motivic $$K$$ -theory spectra

The category of equilogical spaces and the effective topos as homotopical quotients

Homomorphisms of $$L_{\infty }$$ modules

On the relation between continuous and combinatorial

Principal $$\infty $$ -bundles: general theory

Similar Documents

Principal $$\infty $$ -bundles: presentations

Involutive $$A_\infty $$ -algebras and dihedral cohomology

On the invariance and general cohomology comparison theorems

The BV formalism for $$L_\infty $$ -algebras

Maurer–Cartan spaces of filtered $$L_{\infty }$$ -algebras

Existence and uniqueness of $$E_{\infty }$$ structures on motivic $$K$$ -theory spectra

The category of equilogical spaces and the effective topos as homotopical quotients

Homomorphisms of $$L_{\infty }$$ modules

On the relation between continuous and combinatorial

Principal $$\infty $$ -bundles: general theory