In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems (i.e., constructions of objects having a certain universal property), such as the construction of a free group on a set in algebra, or the construction of the Stone–Čech compactification of a topological space in topology. By definition, an adjunction between categories and is a pair of functors (assumed to be covariant) and and, for all objects in and in a bijection between the respective morphism sets such that this family of bijections is natural in and . Naturality here means that there are natural isomorphisms between the pair of functors and for a fixed in , and also the pair of functors and for a fixed in . The functor is called a left adjoint functor or left adjoint to , while is called a right adjoint functor or right adjoint to . An adjunction between categories and is somewhat akin to a "weak form" of an equivalence between and , and indeed every equivalence is an adjunction. In many situations, an adjunction can be "upgraded" to an equivalence, by a suitable natural modification of the involved categories and functors. (Wikipedia).

We show the connection between the method of adjoints in optimal control to the implicit function theorem ansatz. We relate the costate or adjoint state variable to Lagrange multipliers.

From playlist There and Back Again: A Tale of Slopes and Expectations (NeurIPS-2020 Tutorial)

In this video, I define the notion of adjugate matrix and use it to calculate A-1 using determinants. This is again beautiful in theory, but inefficient in examples. Adjugate matrix example: https://youtu.be/OFykHi0idnQ Check out my Determinants Playlist: https://www.youtube.com/playlist

From playlist Determinants

Adjoint / Daggered Operators in Quantum Mechanics

In this video, we will explain adjoint operators in quantum mechanics. First of all, for any operator A, we can define its adjoint, A-dagger, via this equation. The idea behind this is, that while operators in quantum mechanics usually act towards the right, adjoint operators act to the le

From playlist Quantum Mechanics, Quantum Field Theory

The Inverse of a 2 by 2 Matrix Using the Adjoint Method

This video explains how to find the inverse matrix of a 2 by 2 matrix using the adjoint method.

From playlist Inverse Matrices

Algebraic properties of the adjoint. Null space and range of the adjoint. The matrix of T* is the conjugate transpose of the matrix of T.

From playlist Linear Algebra Done Right

In this veideo we continue our look in to the dihedral groups, specifically, the dihedral group with six elements. We note that two of the permutation in the group are special in that they commute with all the other elements in the group. In the next video I'll show you that these two el

From playlist Abstract algebra

Lecture with Mads Jakobsen. Kapitler: 00:00 - Introduction; 00:30 - Homework; 04:30 - Normed Vector Spaces; 08:30 - The Adjoint Operator; 18:30 - Theorem 4.5.1; 19:30 - Proof; 24:00 - Lema 4.4.2; 32:30 - Example Week 2;

From playlist DTU: Mathematics 4 Real Analysis | CosmoLearning.org Math

Self-adjoint operators. All eigenvalues of a self-adjoint operator are real. On a complex vector space, if the inner product of Tv and v is real for every vector v, then T is self-adjoint.

From playlist Linear Algebra Done Right

Higher Algebra 6: Derived Functors

In this video, we define and discuss derived functors between derived categories of abelian categories. Additionally we discuss the notion of adjoint functors and Kan extensions. Feel free to post comments and questions at our public forum at https://www.uni-muenster.de/TopologyQA/index.

From playlist Higher Algebra

Algebraic Topology - 8.1 - Right Adjoints Preserve Limits (RAPL)

Right adjoints preserve limits. Errata: Arkamouli Debnath points out @ 10:04 I mean to write Sets and not Top on the RHS.

From playlist Category Theory Crash Course

Mr LIMA de CARVALHO e SILVA - From Essential Inclusions to Local Geometric Morphisms

It is well known that, given a site of denition, a subtopos of Grothendieck topos can be obtained by strengthening the Grothendieck topology, thus obtaining an inclusion of toposes. An essential inclusion is one where the inverse image functor of this inclusion has a left adjoint. Kelly an

From playlist Topos à l'IHES

Duality In Higher Categories IV 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)

Find the Cofactor Matrix and Adjoint Matrix (2 by 2)

This video explains how to find the cofactor matrix and adjoint matrix for a 2 by 2 matrix.

From playlist The Determinant of a Matrix

Charles Rezk - 4/4 Higher Topos Theory

Course at the school and conference “Toposes online” (24-30 June 2021): https://aroundtoposes.com/toposesonline/ Slides: https://aroundtoposes.com/wp-content/uploads/2021/07/RezkNotesToposesOnlinePart4.pdf In this series of lectures I will give an introduction to the concept of "infinity

From playlist Toposes online

ITHT: Part 11- Quillen Adjunctions

Credits: nLab: https://ncatlab.org/nlab/show/Introduction+to+Homotopy+Theory#QuillenAdjunctions Animation library: https://github.com/3b1b/manim My own code/modified library: https://github.com/treemcgee42/youtub... Music: ► Artist Attribution • Music By: "KaizanBlu" • Track Na

From playlist Introduction to Homotopy Theory

Charles Rezk - 2/4 Higher Topos Theory

Course at the school and conference “Toposes online” (24-30 June 2021): https://aroundtoposes.com/toposesonline/ Slides: https://aroundtoposes.com/wp-content/uploads/2021/07/RezkNotesToposesOnlinePart2.pdf In this series of lectures I will give an introduction to the concept of "infinity

From playlist Toposes online

Model Theory - part 06 - Quantifiers as Adjoints

In this video we start to talk about how one can view quantifiers as adjoints of certain functors.

From playlist Model Theory

The synthetic theory of ∞-categories vs the synthetic theory of ∞-categories - Emily Riehl

Vladimir Voevodsky Memorial Conference Topic: The synthetic theory of ∞-categories vs the synthetic theory of ∞-categories Speaker: Emily Riehl Affiliation: Johns Hopkins University Date: September 12, 2018 For more video please visit http://video.ias.edu

From playlist Mathematics

Multiply Two Binomials Represent the Area of a Rectangle - Math Tutorial

👉 Learn how to multiply polynomials. To multiply polynomials, we use the distributive property. The distributive property is essential for multiplying polynomials. The distributive property is the use of each term of one of the polynomials to multiply all the terms of the other polynomial.

From playlist How to Multiply Polynomials