Lemmas in category theory | Articles containing proofs | Representable functors
In mathematics, the Yoneda lemma is arguably the most important result in category theory. It is an abstract result on functors of the type morphisms into a fixed object. It is a vast generalisation of Cayley's theorem from group theory (viewing a group as a miniature category with just one object and only isomorphisms). It allows the embedding of any locally small category into a category of functors (contravariant set-valued functors) defined on that category. It also clarifies how the embedded category, of representable functors and their natural transformations, relates to the other objects in the larger functor category. It is an important tool that underlies several modern developments in algebraic geometry and representation theory. It is named after Nobuo Yoneda. (Wikipedia).
This lecture is part of an online course on categories. Any object of a category can be thought of as a representable functor in the category of presheaves. We give several examples of representable functors. Then we state Yoneda's lemma, which roughly that morphisms of objects are he sa
From playlist Categories for the idle mathematician
Higher Algebra 7: Non-abelian derived functors
In this video, we discuss the notion of non-abelian derived functors and Animation. Along the way, we discuss the Yoneda lemma. Warning: The Yoneda exercises stated at 35:00 is a bit hard given the technology we have, so I recommend simply proving the analogous statement for ordinary cat
From playlist Higher Algebra
FEMINISM PT 6 - "Backlash" by ChristyOMisty - Good stuff!
EXCELLENT Analysis by the smart, knowledgeable and beautiful,user ChristyOMisty. This is a real woman! What is feminism, really? You may be surprised to find it's only the tool of a much larger, more dangerous global agenda. FUN FACT: The term "feminism" was coined in the 1880s in France
From playlist Christy0Misty - Feminism
Noncommutative Geometric Invariant Theory (Lecture 3) by Arvid Siqveland
PROGRAM :SCHOOL ON CLUSTER ALGEBRAS ORGANIZERS :Ashish Gupta and Ashish K Srivastava DATE :08 December 2018 to 22 December 2018 VENUE :Madhava Lecture Hall, ICTS Bangalore In 2000, S. Fomin and A. Zelevinsky introduced Cluster Algebras as abstractions of a combinatoro-algebra
From playlist School on Cluster Algebras 2018
Sona Jobarteh hails from a long West African tradition of Griots and kora players; her grandfather was the master Griot Amadu Bansang Jobarteh. Creating her own history, she has broken from the male-dominated kora tradition to become the family's first female virtuoso of the instrument.
From playlist World
Serge Bouc: Correspondence functors
Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, b
From playlist Algebraic and Complex Geometry
This course is on Lemma: http://lem.ma Lemma looking for developers: http://lem.ma/jobs Other than http://lem.ma, I recommend Strang http://bit.ly/StrangYT, Gelfand http://bit.ly/GelfandYT, and my short book of essays http://bit.ly/HALAYT Questions and comments below will be prompt
From playlist Problems, Paradoxes, and Sophisms
Higher Algebra 13: The Tate diagonal
In this video we discuss the Tate diagonal, which is a surprising feature of the world of spectra. For further details on this construction, see https://arxiv.org/pdf/1707.01799.pdf, section III.1. Feel free to post comments and questions at our public forum at https://www.uni-muenster
From playlist Higher Algebra
Lecture 6: Sheaves of sets (Part 1)
The most important examples of topoi are categories of sheaves of sets on a small category. Patrick Eilliott introduced this class of examples over two talks, of which is the first. In this talk he defines presheaves and sheaves on a topological space, and explains using the Yoneda lemma h
From playlist Topos theory seminar
Shadows of Computation - Lecture 6 - The line is part of a circle
Welcome to Shadows of Computation, an online course taught by Will Troiani and Billy Snikkers, covering the foundations of category theory and how it is used by computer scientists to abstract computing systems to reveal their intrinsic mathematical properties. In the sixth lecture Will sp
From playlist Shadows of Computation
This course is on Lemma: http://lem.ma Lemma looking for developers: http://lem.ma/jobs Other than http://lem.ma, I recommend Strang http://bit.ly/StrangYT, Gelfand http://bit.ly/GelfandYT, and my short book of essays http://bit.ly/HALAYT Questions and comments below will be prompt
From playlist Problems, Paradoxes, and Sophisms
Linear Algebra Vignette 2d: RREF And The Inverse Matrix
This course is on Lemma: http://lem.ma Lemma looking for developers: http://lem.ma/jobs Other than http://lem.ma, I recommend Strang http://bit.ly/StrangYT, Gelfand http://bit.ly/GelfandYT, and my short book of essays http://bit.ly/HALAYT Questions and comments below will be prompt
From playlist Linear Algebra Vignettes
Persistence and Triangulation in Lagrangian Topology - Paul Biran
IAS/PU-Montreal-Paris-Tel-Aviv Symplectic Geometry Topic: Persistence and Triangulation in Lagrangian Topology Speaker: Paul Biran Affiliation: Eidgenössische Technische Hochschule (ETH) Zürich Date: November 20, 2020 For more video please visit http://video.ias.edu
From playlist Mathematics
早稲田大学 生医&建築(1) 数学B1(微分積分) 第1回・第2回 学習の道案内(担当教員:早水 桃子)【日本語字幕有り】
2020年度5月から早稲田大学の生命医科学科と建築学科の一年生向けに開講される早水桃子の数学B1(微分積分)の履修者向け動画です(※授業の主たるコンテンツはWaseda Moodleにアップロードしているので,YouTubeの動画はあくまでも「学習を促進するオマケ」である点にご注意ください).今回はリクエストにお応えして第1回・第2回の「学習の道案内」をお送りします.動画で軽く内容を把握した後に教科書やPDF資料を読むと,学習が少しスムーズになるかもしれません.授業内容の復習にも活用してみてください.なお,この動画は字幕表示にも完全対応しています.音声の聴き取りが難しい方や
From playlist 2020年度 早稲田大学 生医&建築(1) 数学B1(微分積分)(担当教員:早水桃子)
Linear Algebra: Continuing with function properties of linear transformations, we recall the definition of an onto function and give a rule for onto linear transformations.
From playlist MathDoctorBob: Linear Algebra I: From Linear Equations to Eigenspaces | CosmoLearning.org Mathematics
Math 060 101317C Linear Transformations: Isomorphisms
Lemma: Linear transformations that agree on a basis are identical. Definition: one-to-one (injective). Examples and non-examples. Lemma: T is one-to-one iff its kernel is {0}. Definition: onto (surjective). Examples and non-examples. Definition: isomorphism; isomorphic. Theorem: T
From playlist Course 4: Linear Algebra (Fall 2017)
Reducible Second Order Differential Equations, Missing Y (Differential Equations 26)
https://www.patreon.com/ProfessorLeonard How so solve Reducible Second Order Differential Equations by making a substitution when missing the "y" variable.
From playlist Differential Equations
I relate the languages of "wondrous wisdom" and "advanced mathematics". As an example, I express the mental framework of One, All, Many with the minimization operator μ from computability theory. (This example starts at 8:45, ends at 43:00, and is the part that is relevant for the SOME2
From playlist Summer of Math Exposition 2 videos