Fiber bundles | Homotopy theory | Representable functors | Algebraic topology

Classifying space

In mathematics, specifically in homotopy theory, a classifying space BG of a topological group G is the quotient of a weakly contractible space EG (i.e. a topological space all of whose homotopy groups are trivial) by a proper free action of G. It has the property that any G principal bundle over a paracompact manifold is isomorphic to a pullback of the principal bundle EG → BG. As explained later, this means that classifying spaces represent a set-valued functor on the homotopy category of topological spaces. The term classifying space can also be used for spaces that represent a set-valued functor on the category of topological spaces, such as Sierpiński space. This notion is generalized by the notion of classifying topos. However, the rest of this article discusses the more commonly used notion of classifying space up to homotopy. For a discrete group G, BG is, roughly speaking, a path-connected topological space X such that the fundamental group of X is isomorphic to G and the higher homotopy groups of X are trivial, that is, BG is an Eilenberg–MacLane space, or a K(G,1). (Wikipedia).

Video thumbnail

What is space?

What exactly is space? Brian Greene explains what the "stuff" around us is. Subscribe to our YouTube Channel for all the latest from World Science U. Visit our Website: http://www.worldscienceu.com/ Like us on Facebook: https://www.facebook.com/worldscienceu Follow us on Twitter: https:

From playlist Science Unplugged: Physics

Video thumbnail

What is spacetime?

"Subscribe to our YouTube Channel for all the latest from World Science U. Visit our Website: http://www.worldscienceu.com/ Like us on Facebook: https://www.facebook.com/worldscienceu Follow us on Twitter: https://twitter.com/worldscienceu"

From playlist Science Unplugged: Special Relativity

Video thumbnail

Metric spaces -- Proofs

This lecture is on Introduction to Higher Mathematics (Proofs). For more see http://calculus123.com.

From playlist Proofs

Video thumbnail

A01 An introduction to a series on space medicine

A new series on space medicine.

From playlist Space Medicine

Video thumbnail

Introduction to Metric Spaces

Introduction to Metric Spaces - Definition of a Metric. - The metric on R - The Euclidean Metric on R^n - A metric on the set of all bounded functions - The discrete metric

From playlist Topology

Video thumbnail

Dimensions (1 of 3: The Traditional Definition - Directions)

More resources available at www.misterwootube.com

From playlist Exploring Mathematics: Fractals

Video thumbnail

What is a metric space ?

Metric space definition and examples. Welcome to the beautiful world of topology and analysis! In this video, I present the important concept of a metric space, and give 10 examples. The idea of a metric space is to generalize the concept of absolute values and distances to sets more gener

From playlist Topology

Video thumbnail

What is a metric space? An example

This is a basic introduction to the idea of a metric space. I introduce the idea of a metric and a metric space framed within the context of R^n. I show that a particular distance function satisfies the conditions of being a metric.

From playlist Mathematical analysis and applications

Video thumbnail

Foundations for Learning in the Age of Big Data II - Maria Florina Balcan

Topic: Foundations for Learning in the Age of Big Data Speaker: Maria Florina Balcan Affiliation: Carnegie Mellon University Date: May 24, 2022 Balcan-2022-05-24

From playlist Mathematics

Video thumbnail

Foundations for Learning in the Age of Big Data I - Maria Florina Balcan

2022 Program for Women and Mathematics: The Mathematics of Machine Learning Topic: Foundations for Learning in the Age of Big Data I Speaker: Maria Florina Balcan Affiliation: Carnegie Mellon University Date May 23, 2022 Balcan-2022-05-23

From playlist Mathematics

Video thumbnail

Fellow Short Talks: Professor Richard Samworth, Cambridge University

Bio Richard Samworth is Professor of Statistics in the Statistical Laboratory at the University of Cambridge and a Fellow of St John’s College. He received his PhD, also from the University of Cambridge, in 2004, and currently holds an EPSRC Early Career Fellowship. Research His main r

From playlist Short Talks

Video thumbnail

Porfirio Leandro Leon Alvarez: Virtually Abelian Dimension for 3-Manifold Groups

Porfirio Leandro Leon Alvarez, Instituto de Matematicas, UNAM Title: Virtually Abelian Dimension for 3-Manifold Groups Given a group $\Gamma$, we say a collection $\mc F$ of subgroups of $\Gamma$ is a family if it is non-empty, closed under conjugation and under taking subgroups. Fixing a

From playlist 39th Annual Geometric Topology Workshop (Online), June 6-8, 2022

Video thumbnail

Radioactive data: tracing through training (Paper Explained)

#ai #research #privacy Data is the modern gold. Neural classifiers can improve their performance by training on more data, but given a trained classifier, it's difficult to tell what data it was trained on. This is especially relevant if you have proprietary or personal data and you want

From playlist Papers Explained

Video thumbnail

Mark Grant (10/22/20): Bredon cohomology and LS-categorical invariants

Title: Bredon cohomology and LS-categorical invariants Abstract: Farber posed the problem of describing the topological complexity of aspherical spaces in terms of algebraic invariants of their fundamental groups. In Part One of this talk, I’ll discuss joint work with Farber, Lupton and O

From playlist Topological Complexity Seminar

Video thumbnail

What is a Vector Space?

This video explains the definition of a vector space and provides examples of vector spaces.

From playlist Vector Spaces

Video thumbnail

Felix Klein Lectures 2020: Quiver moduli and applications, Markus Reineke (Bochum), Lecture 1

Quiver moduli spaces are algebraic varieties encoding the continuous parameters of linear algebra type classification problems. In recent years their topological and geometric properties have been explored, and applications to, among others, Donaldson-Thomas and Gromov-Witten theory have

From playlist Felix Klein Lectures 2020: Quiver moduli and applications, Markus Reineke (Bochum)

Video thumbnail

Foundations for Learning in the Age of Big Data III - Maria Florina Balcan

2022 Program for Women and Mathematics: The Mathematics of Machine Learning Topic: Foundations for Learning in the Age of Big Data III Speaker: Maria Florina Balcan Affiliation: Carnegie Mellon University Date: May 26, 2022 In computer vision, generalization of neural representations is

From playlist Mathematics

Video thumbnail

Clara Löh (5/28/22): Lower multiplicity bounds via classifying spaces

We consider variations of the Lusternik--Schnirelmann category, based on open covers satisfying constraints on the level of the fundamental group. Such LS-category invariants can be analysed through equivariant methods. For example, classifying spaces for families of subgroups can be used

From playlist Topological Complexity Seminar

Video thumbnail

VOS: Learning What You Don't Know by Virtual Outlier Synthesis (Paper Explained)

#vos #outliers #deeplearning Sponsor: Assembly AI Check them out here: https://www.assemblyai.com/?utm_source=youtube&utm_medium=social&utm_campaign=yannic1 Outliers are data points that are highly unlikely to be seen in the training distribution, and therefore deep neural networks have t

From playlist Papers Explained

Video thumbnail

Ali Chamseddine - 2/4 Spectral Geometric Unification

Classification of finite spaces and basis for geometric unification.

From playlist Ali Chamseddine - Spectral Geometric Unification

Related pages

Category of sets | Topological space | Lie group | Borel's theorem | Principal bundle | Delta set | Fiber bundle | Homotopy group | Unitary group | Stiefel manifold | Classifying space for U(n) | Homological algebra | Brown's representability theorem | Free group | Topological group | Group cohomology | Algebraic topology | CW complex | Sierpiński space | Projective space | Quotient space (topology) | Discrete group | Trivial group | Grassmannian | Classifying space for O(n) | Homotopy category | Existence theorem | Pullback bundle | Homotopy theory | Genus (mathematics) | Eilenberg–MacLane space | Hyperbolic manifold | Configuration space (mathematics) | Cubical complex | Classifying topos | Closed manifold | Torus | Equivariant cohomology | Characteristic class | Real projective space | Braid group | Weak equivalence (homotopy theory) | Connected space | Free abelian group | Mathematics | Cyclic group | Helix | Representable functor | Weakly contractible | Functor | Compact space | Fundamental group | Hilbert space | Orthogonal group | Differential geometry | Intuitionistic logic | Universal bundle | Bott periodicity theorem | Simplicial complex | Universal property | Surface (topology) | Foliation | Circle