Duality theories | Category theory

Dual (category theory)

In category theory, a branch of mathematics, duality is a correspondence between the properties of a category C and the dual properties of the opposite category Cop. Given a statement regarding the category C, by interchanging the source and target of each morphism as well as interchanging the order of composing two morphisms, a corresponding dual statement is obtained regarding the opposite category Cop. Duality, as such, is the assertion that truth is invariant under this operation on statements. In other words, if a statement is true about C, then its dual statement is true about Cop. Also, if a statement is false about C, then its dual has to be false about Cop. Given a concrete category C, it is often the case that the opposite category Cop per se is abstract. Cop need not be a category that arises from mathematical practice. In this case, another category D is also termed to be in duality with C if D and Cop are equivalent as categories. In the case when C and its opposite Cop are equivalent, such a category is self-dual. (Wikipedia).

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Category Theory 1.2: What is a category?

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From playlist Category Theory

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Category Theory: The Beginner’s Introduction (Lesson 1 Video 4)

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From playlist Category Theory: The Beginner’s Introduction

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Category Theory 2.1: Functions, epimorphisms

Functions, epimorphisms

From playlist Category Theory

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Category Theory 3.1: Examples of categories, orders, monoids

Examples of categories, orders, monoids.

From playlist Category Theory

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Category Theory 9.1: Natural transformations

Natural transformations

From playlist Category Theory

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Category Theory: The Beginner’s Introduction (Lesson 1 Video 5)

Lesson 1 is concerned with defining the category of Abstract Sets and Arbitrary Mappings. We also define our first Limit and Co-Limit: The Terminal Object, and the Initial Object. Other topics discussed include Duality and the Opposite (or Mirror) Category. These videos will be discussed

From playlist Category Theory: The Beginner’s Introduction

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Duality In Higher Categories IV by Pranav Pandit

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Gauge Theory and the Analytic Approach to Geometric Langlands - Edward Witten

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From playlist Mathematics

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David Ben-Zvi: Boundary conditions and hamiltonian actions in geometric Langlands

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Category Theory: The Beginner’s Introduction (Lesson 1 Video 2)

Lesson 1 is concerned with defining the category of Abstract Sets and Arbitrary Mappings. We also define our first Limit and Co-Limit: The Terminal Object, and the Initial Object. Other topics discussed include Duality and the Opposite (or Mirror) Category. Follow me on Twitter: @mjmcodr

From playlist Category Theory: The Beginner’s Introduction

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Spanier Whitehead Duality by Samik Basu

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

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From playlist Quantum Fields, Geometry and Representation Theory 2021 (ONLINE)

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3D Gauge Theories: Vortices and Vertex Algebras (Lecture 2) by Tudor Dimofte

PROGRAM: VORTEX MODULI ORGANIZERS: Nuno Romão (University of Augsburg, Germany) and Sushmita Venugopalan (IMSc, India) DATE & TIME: 06 February 2023 to 17 February 2023 VENUE: Ramanujan Lecture Hall, ICTS Bengaluru For a long time, the vortex equations and their associated self-dual fie

From playlist Vortex Moduli - 2023

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Duality in Algebraic Geometry by Suresh Nayak

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From playlist Dualities in Topology and Algebra (Online)

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David Ben-Zvi: Geometric Langlands correspondence and topological field theory - Part 1

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Codomain | Dual object | Algebraic topology | Duality (mathematics) | Domain of a function | Duality (order theory) | Homotopy theory | Mathematics | Set (mathematics) | De Morgan's laws | Equivalence of categories | Cofibration | Category theory | Morphism | Concrete category | Eckmann–Hilton duality | Limit (category theory) | Monomorphism | Fibration | Epimorphism | Function composition | Opposite category