Homotopy theory | Topological spaces | Topology

Loop space

In topology, a branch of mathematics, the loop space ΩX of a pointed topological space X is the space of (based) loops in X, i.e. continuous pointed maps from the pointed circle S1 to X, equipped with the compact-open topology. Two loops can be multiplied by concatenation. With this operation, the loop space is an A∞-space. That is, the multiplication is homotopy-coherently associative. The set of path components of ΩX, i.e. the set of based-homotopy equivalence classes of based loops in X, is a group, the fundamental group π1(X). The iterated loop spaces of X are formed by applying Ω a number of times. There is an analogous construction for topological spaces without basepoint. The free loop space of a topological space X is the space of maps from the circle S1 to X with the compact-open topology. The free loop space of X is often denoted by . As a functor, the free loop space construction is right adjoint to cartesian product with the circle, while the loop space construction is right adjoint to the reduced suspension. This adjunction accounts for much of the importance of loop spaces in stable homotopy theory. (A related phenomenon in computer science is currying, where the cartesian product is adjoint to the hom functor.) Informally this is referred to as Eckmann–Hilton duality. (Wikipedia).

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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

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Vector spaces and subspaces

After our introduction to matrices and vectors and our first deeper dive into matrices, it is time for us to start the deeper dive into vectors. Vector spaces can be vectors, matrices, and even function. In this video I talk about vector spaces, subspaces, and the porperties of vector sp

From playlist Introducing linear algebra

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Null Space and Column Space of a Matrix

Given a matrix A(ie a linear transformation) there are several important related subspaces. In this video we investigate the Nullspace of A and the column space of A. The null space is the vectors that are "killed" by the transformation - ie sent to zero. The column space will be the image

From playlist Older Linear Algebra Videos

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Metric spaces -- Proofs

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

From playlist Proofs

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What is spacetime?

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From playlist Science Unplugged: Special Relativity

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What is a Vector Space?

What is a Vector Space? Definition of a Vector space.

From playlist Linear Algebra

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Column space of a matrix

We have already looked at the column view of a matrix. In this video lecture I want to expand on this topic to show you that each matrix has a column space. If a matrix is part of a linear system then a linear combination of the columns creates a column space. The vector created by the

From playlist Introducing linear algebra

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What is a Vector Space?

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

From playlist Vector Spaces

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Distiguished Lecture Series by Richard Melrose (Massachusetts Institute of Technology, USA)

From playlist Distinguished Visitors Lecture Series

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Higher Algebra 8: Spectra

In this video, we introduce and discuss spectra (in the sense of homotopy theory). We explain how they generalise abelian groups. Feel free to post comments and questions at our public forum at https://www.uni-muenster.de/TopologyQA/index.php?qa=tc-lecture Homepage with further informa

From playlist Higher Algebra

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From playlist 🔥Python | Python Tutorial For Beginners | Python Projects | Python Interview Questions And Answers | Updated Python Playlist 2023 | Simplilearn

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Loop products, closed geodesics and self-intersections - Nancy Hingston

Workshop on Geometric Functionals: Analysis and Applications Topic: Loop products, closed geodesics and self-intersections Speaker: Nancy Hingston Affiliation: The College of New Jersey Date: March 6, 2019 For more video please visit http://video.ias.edu

From playlist Mathematics

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Stable Homotopy Seminar, 9: Infinite Loop Spaces, and Homotopy Colimits

The fibrant spectra are the Ω-spectra, and we can give an elegant explicit description of the fibrant replacement. The "infinite loop space" functor, which is the derived right adjoint to the suspension spectrum, is then given by taking the 0th space of an equivalent Ω-spectrum. This allow

From playlist Stable Homotopy Seminar

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Pattern Programs In Java | Java Pattern Programs Tutorial | Java Tutorial For Beginners |Simplilearn

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From playlist 🔥Java Tutorial For Beginners | Java Full Course | Java Interview Questions And Answers | Java Programming | Updated Java Playlist 2023 | Simplilearn

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Floer theory by Mohammed Abouzaid

DATE & TIME: 25 December 2017 to 04 January 2018 VENUE: Madhava Lecture Hall, ICTS, Bangalore Holomorphic curves are a central object of study in complex algebraic geometry. Such curves are meaningful even when the target has an almost complex structure. The moduli space of these curves (

From playlist J-Holomorphic Curves and Gromov-Witten Invariants

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Homotopy Group - (1)Dan Licata, (2)Guillaume Brunerie, (3)Peter Lumsdaine

(1)Carnegie Mellon Univ.; Member, School of Math, (2)School of Math., IAS, (3)Dalhousie Univ.; Member, School of Math April 11, 2013 In this general survey talk, we will describe an approach to doing homotopy theory within Univalent Foundations. Whereas classical homotopy theory may be des

From playlist Mathematics

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Top 25 Pattern Programs In C Language | Pattern Program In C Explained | C For Beginners|Simplilearn

This video by Simplilearn will explain the Top 25 Pattern Programs In C Language. Pattern Program In C Explained For Beginners tutorial will include star pattern programs, Number pattern programs, Alphabet pattern programs, and special pattern programs in c. C pattern programs are critic

From playlist C++ Tutorial Videos

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Null space of a matrix using sympy

In this video I give a brief overview of the null space of a matrix and how to calculate (y hand) the column vectors that will make up the null space of the column. This becomes much easier in sympy, by simply using the nullspace function. If you want to learn more about the null space o

From playlist Modern linear algebra using Python instead of a textbook

Related pages

Topological space | Homotopy | Homeomorphism | Homotopy group | Quasigroup | Topology | Loop group | Currying | Group (mathematics) | Suspension (topology) | Path space (algebraic topology) | Pointed space | Eilenberg–MacLane space | Spectrum (topology) | Compact-open topology | Natural transformation | Path (topology) | Equivalence class | Associative property | Mathematics | Set (mathematics) | Gray's conjecture | Cartesian product | Eckmann–Hilton duality | Functor | Fundamental group | List of topologies | Hom functor | Free loop | Stable homotopy theory | Circle