Homotopy theory | Topological spaces | Topology
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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From playlist Science Unplugged: Physics
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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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
This lecture is on Introduction to Higher Mathematics (Proofs). For more see http://calculus123.com.
From playlist Proofs
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From playlist Linear Algebra
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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From playlist Center of Math Research: the Worldwide Lecture Seminar Series
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
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Loop products, closed geodesics and self-intersections - Nancy Hingston
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From playlist Mathematics
Stable Homotopy Seminar, 9: Infinite Loop Spaces, and Homotopy Colimits
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From playlist Stable Homotopy Seminar
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From playlist C++ Tutorial Videos
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From playlist Modern linear algebra using Python instead of a textbook