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  Dual (category Theory)

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Dual (category theory)

In category theory, a branch of mathematics, duality is a correspondence between properties of a category C and so-called dual properties of the opposite category C^op. 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 C^op. 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 C^op. Also, if a statement is false about C, then its dual has to be false about C^op.

Given a concrete category C, it is often the case that the opposite category C^op per se is abstract. C^op 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 C^op are equivalent as categories.

In the case when C and its opposite C^op are equivalent, such a category is self-dual.

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This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Dual_(category_theory)

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  What is…duality in categories?

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15:12

What is…duality in categories?

• Order: Reorder
• Duration: 15:12
• Uploaded Date: 08 Mar 2022
• views: 695

Goal. Explaining basic concepts of category theory in an intuitive way. This time. What is…duality in categories? Or: Rigid and pivotal categories. Disclaim...

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https://wn.com/What_Is…Duality_In_Categories

Goal. Explaining basic concepts of category theory in an intuitive way. This time. What is…duality in categories? Or: Rigid and pivotal categories. Disclaimer. Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references. Disclaimer. The distinction between “large classes” and “small classes (sets)” turns out is crucial for many categorical considerations, but somehow makes the language more cumbersome without too much gain imho. So I will strategically ignore all set-theoretical issues. Slides. http://www.dtubbenhauer.com/youtube.html Website with exercises. http://www.dtubbenhauer.com/lecture-ct-2022.html Rigid and pivotal categories. https://en.wikipedia.org/wiki/Rigid_category https://ncatlab.org/nlab/show/rigid+monoidal+category https://ncatlab.org/nlab/show/pivotal+category https://math.stackexchange.com/questions/4010759/pivotal-category-why-that-name Monoidal categories. https://en.wikipedia.org/wiki/Monoidal_category https://nlab-pages.s3.us-east-2.amazonaws.com/nlab/show/monoidal+category https://ocw.mit.edu/courses/mathematics/18-769-topics-in-lie-theory-tensor-categories-spring-2009/lecture-notes/MIT18_769S09_lec01.pdf https://www.dtubbenhauer.com/qinvariants.pdf String diagrams. https://en.wikipedia.org/wiki/String_diagram https://ncatlab.org/nlab/show/string+diagram https://core.ac.uk/download/pdf/21747055.pdf Pictures used. Pictures from From http://www.dtubbenhauer.com/qinvariants.pdf Some books I am using (I sometimes steal some pictures from there). https://en.wikipedia.org/wiki/Categories_for_the_Working_Mathematician https://www.cambridge.org/core/books/an-introduction-to-category-theory/38C6B02892C2FE7408F52975756AC88D http://www.tac.mta.ca/tac/reprints/articles/17/tr17.pdf https://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf https://math.jhu.edu/~eriehl/context.pdf https://github.com/hmemcpy/milewski-ctfp-pdf Nlab. https://ncatlab.org/nlab/show/HomePage TheCatsters. https://www.youtube.com/channel/UC5Y9H2KDRHZZTWZJtlH4VbA Mathematica. https://wildcatsformma.wordpress.com/ #categorytheory #categoricalalgebra #mathematics

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1:07:25

Category Theory For Beginners: Duality And Functors

• Order: Reorder
• Duration: 1:07:25
• Uploaded Date: 03 Jun 2019
• views: 11945

In this video we introduce the idea of duality, and the notion of the opposite category. We define the initial object of a category, and show that it can be vie...

In this video we introduce the idea of duality, and the notion of the opposite category. We define the initial object of a category, and show that it can be viewed as the empty set, within the category of sets. We also introduce the idea of the coproduct and show that it corresponds to the discriminated union, within the category of sets. We also show how the Cartesian product can be viewed as a functor in the category of sets. More generally, we show that the categorical product can be viewed as a functor, when the categorical product of each pair of objects is defined. This also gives us a notion of the categorical product of a pair of arrows.

https://wn.com/Category_Theory_For_Beginners_Duality_And_Functors

In this video we introduce the idea of duality, and the notion of the opposite category. We define the initial object of a category, and show that it can be viewed as the empty set, within the category of sets. We also introduce the idea of the coproduct and show that it corresponds to the discriminated union, within the category of sets. We also show how the Cartesian product can be viewed as a functor in the category of sets. More generally, we show that the categorical product can be viewed as a functor, when the categorical product of each pair of objects is defined. This also gives us a notion of the categorical product of a pair of arrows.

• published: 03 Jun 2019
• views: 11945

44:04

Takt Category Theory Session 13: Duality and Coproducts

• Order: Reorder
• Duration: 44:04
• Uploaded Date: 02 Feb 2017
• views: 264

We go over the first two sections of chapter 3 in Awodey's "Category Theory".

We go over the first two sections of chapter 3 in Awodey's "Category Theory".

https://wn.com/Takt_Category_Theory_Session_13_Duality_And_Coproducts

We go over the first two sections of chapter 3 in Awodey's "Category Theory".

• published: 02 Feb 2017
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26:20

A Sensible Introduction to Category Theory

• Order: Reorder
• Duration: 26:20
• Uploaded Date: 22 Jun 2022
• views: 449007

Remember when I used a video with a coconut in the thumbnail to drive a stake through the heart of mathematical structure? Today, in this introduction to the ba...

Remember when I used a video with a coconut in the thumbnail to drive a stake through the heart of mathematical structure? Today, in this introduction to the basics of category theory, I attempt to remove it. 27 Unhelpful Facts About Category Theory: https://www.youtube.com/watch?v=H0Ek86IH-3Y MetaMaths on category theory: https://www.youtube.com/watch?v=ZG6t0-JMrw0 My dissertation on the equivalence between the category of monoidal categories and the category of representable multicategories: https://drive.google.com/file/d/1hAkV1qSnUutzQMMQi48yo_fXsgb1YnbL/view?usp=sharing FURTHER READING Basic Category Theory (Tom Leinster): https://arxiv.org/pdf/1612.09375.pdf Categories for the Working Mathematician (Saunders Mac Lane): http://www.mtm.ufsc.br/~ebatista/2016-2/maclanecat.pdf Category Theory for Computing Science (Michael Barr and Charles Wells): https://www.math.mcgill.ca/triples/Barr-Wells-ctcs.pdf Category Theory for the Sciences (David Spivak): https://math.mit.edu/~dspivak/CT4S.pdf Bartosz Milewski on category theory: https://www.youtube.com/watch?v=I8LbkfSSR58&list=PLbgaMIhjbmEnaH_LTkxLI7FMa2HsnawM_ Emily Riehl on category theory: https://www.youtube.com/watch?v=WLkMBMUk48E MUSIC Meditation Aquatic 369 (Epidemic Sound) Nights Full of Overthinking Lionel Quick (Epidemic Sound) Oregano Vendla (Epidemic Sound) Wash Timothy Infinite (Epidemic Sound) Wind Osoku (Epidemic Sound)

https://wn.com/A_Sensible_Introduction_To_Category_Theory

Remember when I used a video with a coconut in the thumbnail to drive a stake through the heart of mathematical structure? Today, in this introduction to the basics of category theory, I attempt to remove it. 27 Unhelpful Facts About Category Theory: https://www.youtube.com/watch?v=H0Ek86IH-3Y MetaMaths on category theory: https://www.youtube.com/watch?v=ZG6t0-JMrw0 My dissertation on the equivalence between the category of monoidal categories and the category of representable multicategories: https://drive.google.com/file/d/1hAkV1qSnUutzQMMQi48yo_fXsgb1YnbL/view?usp=sharing FURTHER READING Basic Category Theory (Tom Leinster): https://arxiv.org/pdf/1612.09375.pdf Categories for the Working Mathematician (Saunders Mac Lane): http://www.mtm.ufsc.br/~ebatista/2016-2/maclanecat.pdf Category Theory for Computing Science (Michael Barr and Charles Wells): https://www.math.mcgill.ca/triples/Barr-Wells-ctcs.pdf Category Theory for the Sciences (David Spivak): https://math.mit.edu/~dspivak/CT4S.pdf Bartosz Milewski on category theory: https://www.youtube.com/watch?v=I8LbkfSSR58&list=PLbgaMIhjbmEnaH_LTkxLI7FMa2HsnawM_ Emily Riehl on category theory: https://www.youtube.com/watch?v=WLkMBMUk48E MUSIC Meditation Aquatic 369 (Epidemic Sound) Nights Full of Overthinking Lionel Quick (Epidemic Sound) Oregano Vendla (Epidemic Sound) Wash Timothy Infinite (Epidemic Sound) Wind Osoku (Epidemic Sound)

• published: 22 Jun 2022
• views: 449007

6:14

Dual Category/Opposite Category | Category theory | Msc maths | @HigherMathemagic

• Order: Reorder
• Duration: 6:14
• Uploaded Date: 11 Jun 2024
• views: 118

Dual Category/Opposite Category | Category theory | Msc maths | ‎@HigherMathemagic  @HigherMathemagic Category theory-1 Full playlist link- https://youtube...

Dual Category/Opposite Category | Category theory | Msc maths | ‎@HigherMathemagic  @HigherMathemagic Category theory-1 Full playlist link- https://youtube.com/playlist?list=PLWwrtchTZqjkYZxH1SA2Xew1co0_v_QL4&feature=shared #categorytheory #categories #cat #category #functor #math #mathematical #mathematics #maths #subcategory #functor #covariantfunctor #contravariantfunctor #homfunctororrepresentablfunctor #functors #representable #covariant #covariance #homfunctors #forgetfulfunctor #contravariant #contravariance #categories #functor #functors #freefunctor #functorsvsfunctions #forgetfulfunctor #whatarefunctors #adjointfunctors #functorsvsmonads #functorsbyexample #applicativefunctors #functorswithexamples #contravariantfunctor #covariantfunctor #categorytheory #categories #cat #category #functor #math #mathematical #mathematics #maths #subcategory #oppositecategory #dualcategory #category #dual #opposite #categorytheory #categories #cat #category #functor #math #mathematical #mathematics #maths #arrow #pup Category theory-1 Full playlist link- https://youtube.com/playlist?list=PLWwrtchTZqjkYZxH1SA2Xew1co0_v_QL4&feature=shared @HigherMathemagic

https://wn.com/Dual_Category_Opposite_Category_|_Category_Theory_|_Msc_Maths_|_Highermathemagic

Dual Category/Opposite Category | Category theory | Msc maths | ‎@HigherMathemagic  @HigherMathemagic Category theory-1 Full playlist link- https://youtube.com/playlist?list=PLWwrtchTZqjkYZxH1SA2Xew1co0_v_QL4&feature=shared #categorytheory #categories #cat #category #functor #math #mathematical #mathematics #maths #subcategory #functor #covariantfunctor #contravariantfunctor #homfunctororrepresentablfunctor #functors #representable #covariant #covariance #homfunctors #forgetfulfunctor #contravariant #contravariance #categories #functor #functors #freefunctor #functorsvsfunctions #forgetfulfunctor #whatarefunctors #adjointfunctors #functorsvsmonads #functorsbyexample #applicativefunctors #functorswithexamples #contravariantfunctor #covariantfunctor #categorytheory #categories #cat #category #functor #math #mathematical #mathematics #maths #subcategory #oppositecategory #dualcategory #category #dual #opposite #categorytheory #categories #cat #category #functor #math #mathematical #mathematics #maths #arrow #pup Category theory-1 Full playlist link- https://youtube.com/playlist?list=PLWwrtchTZqjkYZxH1SA2Xew1co0_v_QL4&feature=shared @HigherMathemagic

• published: 11 Jun 2024
• views: 118

3:31:42

Limits and Adjoints in Category theory ; Stone duality in propositional logic

• Order: Reorder
• Duration: 3:31:42
• Uploaded Date: 18 May 2022
• views: 255

Limits and Adjoints in Category theory by Dipankar Maity and Animesh Renanse; Stone duality in propositional logic by Prof. Mohua Banerjee

Limits and Adjoints in Category theory by Dipankar Maity and Animesh Renanse; Stone duality in propositional logic by Prof. Mohua Banerjee

https://wn.com/Limits_And_Adjoints_In_Category_Theory_Stone_Duality_In_Propositional_Logic

Limits and Adjoints in Category theory by Dipankar Maity and Animesh Renanse; Stone duality in propositional logic by Prof. Mohua Banerjee

• published: 18 May 2022
• views: 255

15:07

What is...the duality principle?

• Order: Reorder
• Duration: 15:07
• Uploaded Date: 23 Nov 2021
• views: 2684

Goal. Explaining basic concepts of category theory in an intuitive way. This time. What is...the duality principle? Or: Flipping arrows. Disclaimer. Nobody...

Goal. Explaining basic concepts of category theory in an intuitive way. This time. What is...the duality principle? Or: Flipping arrows. Disclaimer. Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references. Disclaimer. The distinction between “large classes” and “small classes (sets)” turns out is crucial for many categorical considerations, but somehow makes the language more cumbersome without too much gain imho. So I will strategically ignore all set-theoretical issues. Nonsense. On the “Transposing matrices” slide: Transposing matrices does not change entries from 3→2. No idea how that typo happened... Slides. http://www.dtubbenhauer.com/youtube.html Website with exercises. http://www.dtubbenhauer.com/lecture-ct-2022.html Dual and opposite. https://en.wikipedia.org/wiki/Dual_(category_theory) https://ncatlab.org/nlab/show/duality https://en.wikipedia.org/wiki/Opposite_category https://ncatlab.org/nlab/show/opposite+category Duality. https://en.wikipedia.org/wiki/Duality_(mathematics) Pictures used. http://www.euclideanspace.com/maths/topology/algtop/cohomology/innerProduct.png Picture from Chapter 1 of http://www.tac.mta.ca/tac/reprints/articles/17/tr17.pdf https://en.wikipedia.org/wiki/Injective_function#/media/File:Injection.svg https://en.wikipedia.org/wiki/Injective_function#/media/File:Surjection.svg Some books I am using (I sometimes steal some pictures from there). https://en.wikipedia.org/wiki/Categories_for_the_Working_Mathematician https://www.cambridge.org/core/books/an-introduction-to-category-theory/38C6B02892C2FE7408F52975756AC88D http://www.tac.mta.ca/tac/reprints/articles/17/tr17.pdf https://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf https://math.jhu.edu/~eriehl/context.pdf https://github.com/hmemcpy/milewski-ctfp-pdf Nlab. https://ncatlab.org/nlab/show/HomePage TheCatsters. https://www.youtube.com/channel/UC5Y9H2KDRHZZTWZJtlH4VbA Mathematica. https://wildcatsformma.wordpress.com/ #categorytheory #categoricalalgebra #mathematics

https://wn.com/What_Is...The_Duality_Principle

Goal. Explaining basic concepts of category theory in an intuitive way. This time. What is...the duality principle? Or: Flipping arrows. Disclaimer. Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references. Disclaimer. The distinction between “large classes” and “small classes (sets)” turns out is crucial for many categorical considerations, but somehow makes the language more cumbersome without too much gain imho. So I will strategically ignore all set-theoretical issues. Nonsense. On the “Transposing matrices” slide: Transposing matrices does not change entries from 3→2. No idea how that typo happened... Slides. http://www.dtubbenhauer.com/youtube.html Website with exercises. http://www.dtubbenhauer.com/lecture-ct-2022.html Dual and opposite. https://en.wikipedia.org/wiki/Dual_(category_theory) https://ncatlab.org/nlab/show/duality https://en.wikipedia.org/wiki/Opposite_category https://ncatlab.org/nlab/show/opposite+category Duality. https://en.wikipedia.org/wiki/Duality_(mathematics) Pictures used. http://www.euclideanspace.com/maths/topology/algtop/cohomology/innerProduct.png Picture from Chapter 1 of http://www.tac.mta.ca/tac/reprints/articles/17/tr17.pdf https://en.wikipedia.org/wiki/Injective_function#/media/File:Injection.svg https://en.wikipedia.org/wiki/Injective_function#/media/File:Surjection.svg Some books I am using (I sometimes steal some pictures from there). https://en.wikipedia.org/wiki/Categories_for_the_Working_Mathematician https://www.cambridge.org/core/books/an-introduction-to-category-theory/38C6B02892C2FE7408F52975756AC88D http://www.tac.mta.ca/tac/reprints/articles/17/tr17.pdf https://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf https://math.jhu.edu/~eriehl/context.pdf https://github.com/hmemcpy/milewski-ctfp-pdf Nlab. https://ncatlab.org/nlab/show/HomePage TheCatsters. https://www.youtube.com/channel/UC5Y9H2KDRHZZTWZJtlH4VbA Mathematica. https://wildcatsformma.wordpress.com/ #categorytheory #categoricalalgebra #mathematics

• published: 23 Nov 2021
• views: 2684

13:23

Category Theory Part 1 of 3: Categories

• Order: Reorder
• Duration: 13:23
• Uploaded Date: 25 May 2020
• views: 11560

An introduction to categories, functors, universal properties, natural transformations, and monads with applications to the lambda calculus and functional progr...

An introduction to categories, functors, universal properties, natural transformations, and monads with applications to the lambda calculus and functional programming. This video is part 1 of a series: https://youtube.com/playlist?list=PL6kPvEdcJ4jTXsLMBy-1E8CIalh5DCc6B Read more here: https://github.com/blargoner/math-categories/blob/main/categories.pdf

https://wn.com/Category_Theory_Part_1_Of_3_Categories

An introduction to categories, functors, universal properties, natural transformations, and monads with applications to the lambda calculus and functional programming. This video is part 1 of a series: https://youtube.com/playlist?list=PL6kPvEdcJ4jTXsLMBy-1E8CIalh5DCc6B Read more here: https://github.com/blargoner/math-categories/blob/main/categories.pdf

• published: 25 May 2020
• views: 11560

24:56

Categories 1 Introduction

• Order: Reorder
• Duration: 24:56
• Uploaded Date: 20 Sep 2021
• views: 56099

This lecture is part of an online course on Category theory This is the introductory lecture, where we give a few examples of categories and define them. The ...

This lecture is part of an online course on Category theory This is the introductory lecture, where we give a few examples of categories and define them. The lectures were originally part of a graduate algebra course, and give a quick overview of the basic category theory that is useful in algebra. Recommended textbook: "Categories for the working mathematician", by S. Mac Lane. For the other lectures in the course see https://www.youtube.com/playlist?list=PL8yHsr3EFj51F9XZ_Ka4bLnQoxTdMx0AL

https://wn.com/Categories_1_Introduction

This lecture is part of an online course on Category theory This is the introductory lecture, where we give a few examples of categories and define them. The lectures were originally part of a graduate algebra course, and give a quick overview of the basic category theory that is useful in algebra. Recommended textbook: "Categories for the working mathematician", by S. Mac Lane. For the other lectures in the course see https://www.youtube.com/playlist?list=PL8yHsr3EFj51F9XZ_Ka4bLnQoxTdMx0AL

• published: 20 Sep 2021
• views: 56099

0:41

What does categorical dual mean?

• Order: Reorder
• Duration: 0:41
• Uploaded Date: 21 Jun 2017
• views: 5

What does categorical dual mean? A spoken definition of categorical dual. Intro Sound: Typewriter - Tamskp Licensed under CC:BA 3.0 Outro Music: Groov...

What does categorical dual mean? A spoken definition of categorical dual. Intro Sound: Typewriter - Tamskp Licensed under CC:BA 3.0 Outro Music: Groove Groove - Kevin MacLeod (incompetech.com) Licensed under CC:BA 3.0 Intro/Outro Photo: The best days are not planned - Marcus Hansson Licensed under CC-BY-2.0 Book Image: Open Book template PSD - DougitDesign Licensed under CC:BA 3.0 Text derived from: http://en.wiktionary.org/wiki/categorical_dual

https://wn.com/What_Does_Categorical_Dual_Mean

What does categorical dual mean? A spoken definition of categorical dual. Intro Sound: Typewriter - Tamskp Licensed under CC:BA 3.0 Outro Music: Groove Groove - Kevin MacLeod (incompetech.com) Licensed under CC:BA 3.0 Intro/Outro Photo: The best days are not planned - Marcus Hansson Licensed under CC-BY-2.0 Book Image: Open Book template PSD - DougitDesign Licensed under CC:BA 3.0 Text derived from: http://en.wiktionary.org/wiki/categorical_dual

• published: 21 Jun 2017
• views: 5

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15:12

What is…duality in categories?

Goal. Explaining basic concepts of category theory in an intuitive way. This time. What...

published: 08 Mar 2022

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What is…duality in categories?

What is…duality in categories?

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• published: 08 Mar 2022
• views: 695

Goal. Explaining basic concepts of category theory in an intuitive way. This time. What is…duality in categories? Or: Rigid and pivotal categories. Disclaimer. Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references. Disclaimer. The distinction between “large classes” and “small classes (sets)” turns out is crucial for many categorical considerations, but somehow makes the language more cumbersome without too much gain imho. So I will strategically ignore all set-theoretical issues. Slides. http://www.dtubbenhauer.com/youtube.html Website with exercises. http://www.dtubbenhauer.com/lecture-ct-2022.html Rigid and pivotal categories. https://en.wikipedia.org/wiki/Rigid_category https://ncatlab.org/nlab/show/rigid+monoidal+category https://ncatlab.org/nlab/show/pivotal+category https://math.stackexchange.com/questions/4010759/pivotal-category-why-that-name Monoidal categories. https://en.wikipedia.org/wiki/Monoidal_category https://nlab-pages.s3.us-east-2.amazonaws.com/nlab/show/monoidal+category https://ocw.mit.edu/courses/mathematics/18-769-topics-in-lie-theory-tensor-categories-spring-2009/lecture-notes/MIT18_769S09_lec01.pdf https://www.dtubbenhauer.com/qinvariants.pdf String diagrams. https://en.wikipedia.org/wiki/String_diagram https://ncatlab.org/nlab/show/string+diagram https://core.ac.uk/download/pdf/21747055.pdf Pictures used. Pictures from From http://www.dtubbenhauer.com/qinvariants.pdf Some books I am using (I sometimes steal some pictures from there). https://en.wikipedia.org/wiki/Categories_for_the_Working_Mathematician https://www.cambridge.org/core/books/an-introduction-to-category-theory/38C6B02892C2FE7408F52975756AC88D http://www.tac.mta.ca/tac/reprints/articles/17/tr17.pdf https://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf https://math.jhu.edu/~eriehl/context.pdf https://github.com/hmemcpy/milewski-ctfp-pdf Nlab. https://ncatlab.org/nlab/show/HomePage TheCatsters. https://www.youtube.com/channel/UC5Y9H2KDRHZZTWZJtlH4VbA Mathematica. https://wildcatsformma.wordpress.com/ #categorytheory #categoricalalgebra #mathematics

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1:07:25

Category Theory For Beginners: Duality And Functors

In this video we introduce the idea of duality, and the notion of the opposite category. W...

published: 03 Jun 2019

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Category Theory For Beginners: Duality And Functors

Category Theory For Beginners: Duality And Functors

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• published: 03 Jun 2019
• views: 11945

In this video we introduce the idea of duality, and the notion of the opposite category. We define the initial object of a category, and show that it can be viewed as the empty set, within the category of sets. We also introduce the idea of the coproduct and show that it corresponds to the discriminated union, within the category of sets. We also show how the Cartesian product can be viewed as a functor in the category of sets. More generally, we show that the categorical product can be viewed as a functor, when the categorical product of each pair of objects is defined. This also gives us a notion of the categorical product of a pair of arrows.

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44:04

Takt Category Theory Session 13: Duality and Coproducts

We go over the first two sections of chapter 3 in Awodey's "Category Theory".

published: 02 Feb 2017

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Takt Category Theory Session 13: Duality and Coproducts

Takt Category Theory Session 13: Duality and Coproducts

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• published: 02 Feb 2017
• views: 264

We go over the first two sections of chapter 3 in Awodey's "Category Theory".

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26:20

A Sensible Introduction to Category Theory

Remember when I used a video with a coconut in the thumbnail to drive a stake through the ...

published: 22 Jun 2022

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A Sensible Introduction to Category Theory

A Sensible Introduction to Category Theory

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• published: 22 Jun 2022
• views: 449007

Remember when I used a video with a coconut in the thumbnail to drive a stake through the heart of mathematical structure? Today, in this introduction to the basics of category theory, I attempt to remove it. 27 Unhelpful Facts About Category Theory: https://www.youtube.com/watch?v=H0Ek86IH-3Y MetaMaths on category theory: https://www.youtube.com/watch?v=ZG6t0-JMrw0 My dissertation on the equivalence between the category of monoidal categories and the category of representable multicategories: https://drive.google.com/file/d/1hAkV1qSnUutzQMMQi48yo_fXsgb1YnbL/view?usp=sharing FURTHER READING Basic Category Theory (Tom Leinster): https://arxiv.org/pdf/1612.09375.pdf Categories for the Working Mathematician (Saunders Mac Lane): http://www.mtm.ufsc.br/~ebatista/2016-2/maclanecat.pdf Category Theory for Computing Science (Michael Barr and Charles Wells): https://www.math.mcgill.ca/triples/Barr-Wells-ctcs.pdf Category Theory for the Sciences (David Spivak): https://math.mit.edu/~dspivak/CT4S.pdf Bartosz Milewski on category theory: https://www.youtube.com/watch?v=I8LbkfSSR58&list=PLbgaMIhjbmEnaH_LTkxLI7FMa2HsnawM_ Emily Riehl on category theory: https://www.youtube.com/watch?v=WLkMBMUk48E MUSIC Meditation Aquatic 369 (Epidemic Sound) Nights Full of Overthinking Lionel Quick (Epidemic Sound) Oregano Vendla (Epidemic Sound) Wash Timothy Infinite (Epidemic Sound) Wind Osoku (Epidemic Sound)

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6:14

Dual Category/Opposite Category | Category theory | Msc maths | @HigherMathemagic

Dual Category/Opposite Category | Category theory | Msc maths | ‎@HigherMathemagic  @High...

published: 11 Jun 2024

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Dual Category/Opposite Category | Category theory | Msc maths | @HigherMathemagic

Dual Category/Opposite Category | Category theory | Msc maths | @HigherMathemagic

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• published: 11 Jun 2024
• views: 118

Dual Category/Opposite Category | Category theory | Msc maths | ‎@HigherMathemagic  @HigherMathemagic Category theory-1 Full playlist link- https://youtube.com/playlist?list=PLWwrtchTZqjkYZxH1SA2Xew1co0_v_QL4&feature=shared #categorytheory #categories #cat #category #functor #math #mathematical #mathematics #maths #subcategory #functor #covariantfunctor #contravariantfunctor #homfunctororrepresentablfunctor #functors #representable #covariant #covariance #homfunctors #forgetfulfunctor #contravariant #contravariance #categories #functor #functors #freefunctor #functorsvsfunctions #forgetfulfunctor #whatarefunctors #adjointfunctors #functorsvsmonads #functorsbyexample #applicativefunctors #functorswithexamples #contravariantfunctor #covariantfunctor #categorytheory #categories #cat #category #functor #math #mathematical #mathematics #maths #subcategory #oppositecategory #dualcategory #category #dual #opposite #categorytheory #categories #cat #category #functor #math #mathematical #mathematics #maths #arrow #pup Category theory-1 Full playlist link- https://youtube.com/playlist?list=PLWwrtchTZqjkYZxH1SA2Xew1co0_v_QL4&feature=shared @HigherMathemagic

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3:31:42

Limits and Adjoints in Category theory ; Stone duality in propositional logic

Limits and Adjoints in Category theory by Dipankar Maity and Animesh Renanse; Stone dualit...

published: 18 May 2022

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Limits and Adjoints in Category theory ; Stone duality in propositional logic

Limits and Adjoints in Category theory ; Stone duality in propositional logic

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• published: 18 May 2022
• views: 255

Limits and Adjoints in Category theory by Dipankar Maity and Animesh Renanse; Stone duality in propositional logic by Prof. Mohua Banerjee

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15:07

What is...the duality principle?

Goal. Explaining basic concepts of category theory in an intuitive way. This time. What...

published: 23 Nov 2021

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What is...the duality principle?

What is...the duality principle?

• Report rights infringement
• published: 23 Nov 2021
• views: 2684

Goal. Explaining basic concepts of category theory in an intuitive way. This time. What is...the duality principle? Or: Flipping arrows. Disclaimer. Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references. Disclaimer. The distinction between “large classes” and “small classes (sets)” turns out is crucial for many categorical considerations, but somehow makes the language more cumbersome without too much gain imho. So I will strategically ignore all set-theoretical issues. Nonsense. On the “Transposing matrices” slide: Transposing matrices does not change entries from 3→2. No idea how that typo happened... Slides. http://www.dtubbenhauer.com/youtube.html Website with exercises. http://www.dtubbenhauer.com/lecture-ct-2022.html Dual and opposite. https://en.wikipedia.org/wiki/Dual_(category_theory) https://ncatlab.org/nlab/show/duality https://en.wikipedia.org/wiki/Opposite_category https://ncatlab.org/nlab/show/opposite+category Duality. https://en.wikipedia.org/wiki/Duality_(mathematics) Pictures used. http://www.euclideanspace.com/maths/topology/algtop/cohomology/innerProduct.png Picture from Chapter 1 of http://www.tac.mta.ca/tac/reprints/articles/17/tr17.pdf https://en.wikipedia.org/wiki/Injective_function#/media/File:Injection.svg https://en.wikipedia.org/wiki/Injective_function#/media/File:Surjection.svg Some books I am using (I sometimes steal some pictures from there). https://en.wikipedia.org/wiki/Categories_for_the_Working_Mathematician https://www.cambridge.org/core/books/an-introduction-to-category-theory/38C6B02892C2FE7408F52975756AC88D http://www.tac.mta.ca/tac/reprints/articles/17/tr17.pdf https://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf https://math.jhu.edu/~eriehl/context.pdf https://github.com/hmemcpy/milewski-ctfp-pdf Nlab. https://ncatlab.org/nlab/show/HomePage TheCatsters. https://www.youtube.com/channel/UC5Y9H2KDRHZZTWZJtlH4VbA Mathematica. https://wildcatsformma.wordpress.com/ #categorytheory #categoricalalgebra #mathematics

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13:23

Category Theory Part 1 of 3: Categories

An introduction to categories, functors, universal properties, natural transformations, an...

published: 25 May 2020

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Category Theory Part 1 of 3: Categories

Category Theory Part 1 of 3: Categories

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• published: 25 May 2020
• views: 11560

An introduction to categories, functors, universal properties, natural transformations, and monads with applications to the lambda calculus and functional programming. This video is part 1 of a series: https://youtube.com/playlist?list=PL6kPvEdcJ4jTXsLMBy-1E8CIalh5DCc6B Read more here: https://github.com/blargoner/math-categories/blob/main/categories.pdf

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24:56

Categories 1 Introduction

This lecture is part of an online course on Category theory This is the introductory lect...

published: 20 Sep 2021

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Categories 1 Introduction

Categories 1 Introduction

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• published: 20 Sep 2021
• views: 56099

This lecture is part of an online course on Category theory This is the introductory lecture, where we give a few examples of categories and define them. The lectures were originally part of a graduate algebra course, and give a quick overview of the basic category theory that is useful in algebra. Recommended textbook: "Categories for the working mathematician", by S. Mac Lane. For the other lectures in the course see https://www.youtube.com/playlist?list=PL8yHsr3EFj51F9XZ_Ka4bLnQoxTdMx0AL

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0:41

What does categorical dual mean?

What does categorical dual mean? A spoken definition of categorical dual. Intro Sound:...

published: 21 Jun 2017

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What does categorical dual mean?

What does categorical dual mean?

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• published: 21 Jun 2017
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What does categorical dual mean? A spoken definition of categorical dual. Intro Sound: Typewriter - Tamskp Licensed under CC:BA 3.0 Outro Music: Groove Groove - Kevin MacLeod (incompetech.com) Licensed under CC:BA 3.0 Intro/Outro Photo: The best days are not planned - Marcus Hansson Licensed under CC-BY-2.0 Book Image: Open Book template PSD - DougitDesign Licensed under CC:BA 3.0 Text derived from: http://en.wiktionary.org/wiki/categorical_dual

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Dual (category theory)

In category theory, a branch of mathematics, duality is a correspondence between properties of a category C and so-called dual properties of the opposite category C^op. 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 C^op. 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 C^op. Also, if a statement is false about C, then its dual has to be false about C^op.

Given a concrete category C, it is often the case that the opposite category C^op per se is abstract. C^op 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 C^op are equivalent as categories.

In the case when C and its opposite C^op are equivalent, such a category is self-dual.

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This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Dual_(category_theory)

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