Homotopy category of chain complexes

Algebraic Topology - 11.3 - Homotopy Equivalence

We sketch why that the homotopy category is a category.

From playlist Algebraic Topology

Homotopy

Homotopy elements in the homotopy group π₂(S²) ≅ ℤ. Roman Gassmann and Tabea Méndez suggested some improvements to my original ideas.

From playlist Algebraic Topology

Introduction to Homotopy Theory- Part 5- Transition to Abstract Homotopy Theory

Credits: nLab: https://ncatlab.org/nlab/show/Introdu...​ Animation library: https://github.com/3b1b/manim​​​ Music: ► Artist Attribution • Music By: "KaizanBlu" • Track Name: "Remember (Extended Mix)" • YouTube Track Link: https://bit.ly/31Ma5s0​​​ • Spotify Track Link: https://spoti.fi/

From playlist Introduction to Homotopy Theory

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

Homomorphisms in abstract algebra

In this video we add some more definition to our toolbox before we go any further in our study into group theory and abstract algebra. The definition at hand is the homomorphism. A homomorphism is a function that maps the elements for one group to another whilst maintaining their structu

From playlist Abstract algebra

Homotopy type theory: working invariantly in homotopy theory -Guillaume Brunerie

Short talks by postdoctoral members Topic: Homotopy type theory: working invariantly in homotopy theory Speaker: Guillaume Brunerie Affiliation: Member, School of Mathematics Date: September 26, 2017 For more videos, please visit http://video.ias.edu

From playlist Mathematics

Introduction to Homotopy Theory- PART 1: UNIVERSAL CONSTRUCTIONS

The goal of this series is to develop homotopy theory from a categorical perspective, alongside the theory of model categories. We do this with the hope of eventually developing stable homotopy theory, a personal goal a passion of mine. I'm going to follow nLab's notes, but I hope to add t

From playlist Introduction to Homotopy Theory

Lie Groups and Lie Algebras: Lesson 34 -Introduction to Homotopy

Lie Groups and Lie Algebras: Introduction to Homotopy In order to proceed with Gilmore's study of Lie groups and Lie algebras we now need a concept from algebraic topology. That concept is the notion of homotopy and the Fundamental Group of a topological space. In this lecture we provide

From playlist Lie Groups and Lie Algebras

ITHT: Part 12- Model Structure on Topological Spaces

Credits: nLab: https://ncatlab.org/nlab/show/Introduction+to+Homotopy+Theory#TheClassicalModelStructureOfTopologicalSpaces Animation library: https://github.com/3b1b/manim​​​​​​​​​​ My own code/modified library: https://github.com/treemcgee42/youtub...

From playlist Introduction to Homotopy Theory

Model Categories by Rekha Santhanam

PROGRAM DUALITIES IN TOPOLOGY AND ALGEBRA (ONLINE) ORGANIZERS: Samik Basu (ISI Kolkata, India), Anita Naolekar (ISI Bangalore, India) and Rekha Santhanam (IIT Mumbai, India) DATE & TIME: 01 February 2021 to 13 February 2021 VENUE: Online Duality phenomena are ubiquitous in mathematics

From playlist Dualities in Topology and Algebra (Online)

Stable Homotopy Seminar, 11: Stable Model Categories and Triangulated Categories

(Note: I messed up the first recording and had to re-record the first 20 minutes of this.) I show that cofiber sequences agree with fiber sequences in Spectra, or indeed in any pointed model category where suspension is invertible. The homotopy category of such a model category is a highly

From playlist Stable Homotopy Seminar

Wojciech Chachólski (4/29/20): TDA invariants and model categories

Title: TDA invariants and model categories Abstract: Data analysis is a balancing act of simplification and ignoring most of the information available on the one hand, and retaining what might be meaningful for the particular task on the other. The same balancing act of extracting simplif

From playlist AATRN 2020

Duality In Higher Categories II by Pranav Pandit

PROGRAM DUALITIES IN TOPOLOGY AND ALGEBRA (ONLINE) ORGANIZERS: Samik Basu (ISI Kolkata, India), Anita Naolekar (ISI Bangalore, India) and Rekha Santhanam (IIT Mumbai, India) DATE & TIME: 01 February 2021 to 13 February 2021 VENUE: Online Duality phenomena are ubiquitous in mathematics

From playlist Dualities in Topology and Algebra (Online)

Higher algebra 4: Derived categories as ∞-categories

In this video, we construct the ∞-categorical refinement of the derived category of an abelian category. This is the fourth video in our introduction to ∞-categories and Higher Algebra. Feel free to post comments and questions at our public forum at https://www.uni-muenster.de/TopologyQA

From playlist Higher Algebra

Graham ELLIS - Computational group theory, cohomology of groups and topological methods 4

The lecture series will give an introduction to the computer algebra system GAP, focussing on calculations involving cohomology. We will describe the mathematics underlying the algorithms, and how to use them within GAP. Alexander Hulpke's lectures will being with some general computation

From playlist École d'Été 2022 - Cohomology Geometry and Explicit Number Theory

Lie Algebras and Homotopy Theory - Jacob Lurie

Members' Seminar Topic: Lie Algebras and Homotopy Theory Speaker: Jacob Lurie Affiliation: Professor, School of Mathematics Date: November 11, 2019 For more video please visit http://video.ias.edu

From playlist Mathematics

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

Spatial refinements and Khovanov homology – Robert Lipshitz & Sucharit Sarkar – ICM2018

Topology Invited Lecture 6.11 Spatial refinements and Khovanov homology Robert Lipshitz & Sucharit Sarkar Abstract: We review the construction and context of a stable homotopy refinement of Khovanov homology. © International Congress of Mathematicians – ICM www.icm2018.org     Os direi

From playlist Topology

Stable Homotopy Seminar, 4: Model categories (Ivo Vekemans)

This talk by Ivo Vekemans is a thorough introduction to model categories, presenting: weak factorization systems; the definition of model category and major examples (simplicial sets, topological spaces, and chain complexes); notions of homotopy in a model category, and the homotopy catego

From playlist Stable Homotopy Seminar

Homomorphisms in abstract algebra examples

Yesterday we took a look at the definition of a homomorphism. In today's lecture I want to show you a couple of example of homomorphisms. One example gives us a group, but I take the time to prove that it is a group just to remind ourselves of the properties of a group. In this video th

From playlist Abstract algebra