In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions. On the other hand, any monoid can be understood as a special sort of category, and so can any preorder. In general, the objects and arrows may be abstract entities of any kind, and the notion of category provides a fundamental and abstract way to describe mathematical entities and their relationships. This is the central idea of category theory, a branch of mathematics which seeks to generalize all of mathematics in terms of objects and arrows, independent of what the objects and arrows represent. Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. For more extensive motivational background and historical notes, see category theory and the list of category theory topics.
In mathematics, the Lyusternik–Schnirelmann category (or, Lusternik–Schnirelmann category, LS-category) of a topological space is the homotopical invariant defined to be the smallest integer number such that there is an open covering of with the property that each inclusion map is nullhomotopic. For example, if is the circle, this takes the value two.
Sometimes a different normalization of the invariant is adopted, which is one less than the definition above Such a normalization has been adopted in the definitive monograph by Cornea, Lupton, Oprea, and Tanré (see below).
In general it is not easy to compute this invariant, which was initially introduced by Lazar Lyusternik and Lev Schnirelmann in connection with variational problems. It has a close connection with algebraic topology, in particular cup-length. In the modern normalization, the cup-length is a lower bound for LS category.
It was, as originally defined for the case of X a manifold, the lower bound for the number of critical points that a real-valued function on X could possess (this should be compared with the result in Morse theory that shows that the sum of the Betti numbers is a lower bound for the number of critical points of a Morse function).
In mathematics, higher category theory is the part of category theory at a higher order, which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities.
Strict higher categories
An ordinary category has objects and morphisms. A 2-category generalizes this by also including 2-morphisms between the 1-morphisms. Continuing this up to n-morphisms between (n-1)-morphisms gives an n-category.
Just as the category known as Cat, which is the category of small categories and functors is actually a 2-category with natural transformations as its 2-morphisms, the category n-Cat of (small) n-categories is actually an n+1-category.
An (n+1)-category is a category enriched over the category n-Cat.
So a 1-category is just a (locally small) category.
The monoidal structure of Set is the one given by the cartesian product as tensor and a singleton as unit. In fact any category with finite products can be given a monoidal structure. The recursive construction of n-Cat works fine because if a category C has finite products, the category of C-enriched categories has finite products too.
A goal is a desired result that a person or a system envisions, plans and commits to achieve: a personal or organizational desired end-point in some sort of assumed development. Many people endeavor to reach goals within a finite time by setting deadlines.
Goal setting may involve establishing specific, measurable, achievable, relevant, and time-bounded (SMART) objectives, but not all researchers agree that these SMART criteria are necessary.
Research on goal setting by Edwin A. Locke and his colleagues suggests that goal setting can serve as an effective tool for making progress when it ensures that group members have a clear awareness of what each person must do to achieve a shared objective. On a personal level, the process of setting goals allows individuals to specify and then work toward their own objectives (such as financial or career-based goals). Goal-setting comprises a major component of personal development and management.
Category Theory: An Introduction to Abstract Nonsense
Correction: Universal Property of Quotients requires ker(f) to contain ker(pi)
0:00 Motivation
1:33 Basics in Category Theory
4:14 Group Objects
5:08 Functors
8:17 Universal Properties
11:57 Proof using Category Theory
13:27 Shortcomings of Category Theory
published: 02 Jan 2023
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
Catego...
published: 22 Jun 2022
27 Unhelpful Facts About Category Theory
Category theory is the heart of mathematical structure. In this video, I will drive a stake through that heart. I don't know why I made this.
Grothendieck Googling: https://mobile.twitter.com/grothendieckg
Join my Discord server to discuss this video and more: https://discord.gg/AVcU9w5gVW
MUSIC
Oregano
Vendla (Epidemic Sound)
Penumbra
Kevin MacLeod (incompetech.com)
published: 31 Dec 2021
Intro to Category Theory
Please watch with subtitles. Errata noted in transcript and at bottom of description.
Some content may require a little background in abstract algebra, but there are no topology heavy examples included.
This was originally written for the oral presentation component of my essay module, but the script ended up being way too long. I'd already made the animations, so I've decided to turn it into a crappy video due to the sunk cost fallacy. The audio was recorded in 1-2 takes at 3am, so the quality isn't great (most of the audio is taken from a recording I took purely to time out how long it would take for me to present it). I might update and remake the video in higher quality and in more detail if I have the motivation, but I have too much work right now.
Despite the first subtitle, we on...
published: 01 Feb 2023
Category Theory Part 3 of 3: Universal Properties
An introduction to categories, functors, universal properties, natural transformations, and monads with applications to the lambda calculus and functional programming.
This video is part 3 of a series:
https://youtube.com/playlist?list=PL6kPvEdcJ4jTXsLMBy-1E8CIalh5DCc6B
Read more here: https://github.com/blargoner/math-categories/blob/main/categories.pdf
published: 14 Jun 2020
Category Theory 4.1: Terminal and initial objects
Terminal and initial objects
published: 15 Sep 2016
What is Category Theory?
published: 05 Jan 2018
Category Theory 8.1: Function objects, exponentials
published: 13 Oct 2016
Category Theory for Neuroscience (pure math to combat scientific stagnation)
sources and references, in temporal order:
Nature paper on the decline in disruptive science:
https://pubmed.ncbi.nlm.nih.gov/36599999/
Gordon Shepherd's book on the revolutionary 1950s "Creating Modern Neuroscience":
https://www.amazon.com/Creating-Modern-Neuroscience-Revolutionary-1950s/dp/0195391500
Group theory, SU(3), hadrons, quarks and particle physics:
https://tinyurl.com/quarksymmetry
Alexander Unzicker's video on how science moves from numbers (measurements) to equations (laws):
https://www.youtube.com/watch?v=yfmSujRhqCk&ab_channel=Unzicker%27sRealPhysics
Andrei Rodin on pure vs. applied math:
https://www.youtube.com/watch?v=FD472NfobQ0&ab_channel=ToposInstitute
JC Gorman on "What is a topology in why is it in my neuroscience?":
https://neuwritesd.org/2021/06/10/what-is-a-...
published: 26 Jun 2023
Category Theory 3.1: Examples of categories, orders, monoids
Correction: Universal Property of Quotients requires ker(f) to contain ker(pi)
0:00 Motivation
1:33 Basics in Category Theory
4:14 Group Objects
5:08 Functors
...
Correction: Universal Property of Quotients requires ker(f) to contain ker(pi)
0:00 Motivation
1:33 Basics in Category Theory
4:14 Group Objects
5:08 Functors
8:17 Universal Properties
11:57 Proof using Category Theory
13:27 Shortcomings of Category Theory
Correction: Universal Property of Quotients requires ker(f) to contain ker(pi)
0:00 Motivation
1:33 Basics in Category Theory
4:14 Group Objects
5:08 Functors
8:17 Universal Properties
11:57 Proof using Category Theory
13:27 Shortcomings of 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 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)
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)
Category theory is the heart of mathematical structure. In this video, I will drive a stake through that heart. I don't know why I made this.
Grothendieck Goog...
Category theory is the heart of mathematical structure. In this video, I will drive a stake through that heart. I don't know why I made this.
Grothendieck Googling: https://mobile.twitter.com/grothendieckg
Join my Discord server to discuss this video and more: https://discord.gg/AVcU9w5gVW
MUSIC
Oregano
Vendla (Epidemic Sound)
Penumbra
Kevin MacLeod (incompetech.com)
Category theory is the heart of mathematical structure. In this video, I will drive a stake through that heart. I don't know why I made this.
Grothendieck Googling: https://mobile.twitter.com/grothendieckg
Join my Discord server to discuss this video and more: https://discord.gg/AVcU9w5gVW
MUSIC
Oregano
Vendla (Epidemic Sound)
Penumbra
Kevin MacLeod (incompetech.com)
Please watch with subtitles. Errata noted in transcript and at bottom of description.
Some content may require a little background in abstract algebra, but the...
Please watch with subtitles. Errata noted in transcript and at bottom of description.
Some content may require a little background in abstract algebra, but there are no topology heavy examples included.
This was originally written for the oral presentation component of my essay module, but the script ended up being way too long. I'd already made the animations, so I've decided to turn it into a crappy video due to the sunk cost fallacy. The audio was recorded in 1-2 takes at 3am, so the quality isn't great (most of the audio is taken from a recording I took purely to time out how long it would take for me to present it). I might update and remake the video in higher quality and in more detail if I have the motivation, but I have too much work right now.
Despite the first subtitle, we only briefly cover the Yoneda lemma in this presentation. Actually, we only briefly cover most of the content in here - I was intending this to be a 15 minute talk, so a lot of material is glossed over.
I am aware I speak quickly - I kept it in mind when recording, but I'll try harder next time. The pacing of transitions is also a bit quick in certain places upon rewatching - I'll make sure to pause more. In the meantime, I have included subtitles which might be helpful if you prefer reading.
Graphics inspired by Oliver Lugg's 27 Unhelpful Facts about Category Theory: https://www.youtube.com/watch?v=H0Ek86IH-3Y.
Main reference during video creation was Basic Category Theory by Leinster and Category Theory in Context by Riehl. Examples of representable functors sourced from notes by Dr. Emanuele Dotto.
Timeline:
00:00 - Introduction
01:08 - Objects
01:40 - Morphisms
02:44 - Compositions
03:01 - Identity
03:22 - Associativity
03:30 - Examples of Categories
06:18 - Product and Dual Categories
07:12 - Duality
07:44 - Commutative Diagrams
08:17 - Isomorphism
09:02 - Functors
10:40 - Covariance and Contravariance
11:15 - Examples of Functors
13:25 - Natural Transformations
15:31 - Vertical Composition
16:53 - Functor Categories
17:18 - Natural Isomorphism
18:22 - Hom Functors
22:19 - Representables
22:40 - Examples of Representables
25:30 - Classifying Spaces
28:19 - The Yoneda Lemma
Errata:
11;13 - "...that a functor is [contravariant], than to...", not "covariant".
18;12 - "...corresponding [objects] are isomorphic...", not "morphisms".
21;57 - the upper string of mappings should be g mapsto hom(h,X)(g) = g o h mapsto hom(B,f)(g o h) = f o (g o h). That is, B and X are the wrong way around in the hom morphisms.
26;59 - "...between the [functions] 1 to R and...", not "functors". (Though, if we treat 1 as the trivial category, and R as a category under ordering, then this does hold for functors in the category of categories. But I really do just mean functions here.)
28;38 - accidentally cut audio, see transcript.
ree
Please watch with subtitles. Errata noted in transcript and at bottom of description.
Some content may require a little background in abstract algebra, but there are no topology heavy examples included.
This was originally written for the oral presentation component of my essay module, but the script ended up being way too long. I'd already made the animations, so I've decided to turn it into a crappy video due to the sunk cost fallacy. The audio was recorded in 1-2 takes at 3am, so the quality isn't great (most of the audio is taken from a recording I took purely to time out how long it would take for me to present it). I might update and remake the video in higher quality and in more detail if I have the motivation, but I have too much work right now.
Despite the first subtitle, we only briefly cover the Yoneda lemma in this presentation. Actually, we only briefly cover most of the content in here - I was intending this to be a 15 minute talk, so a lot of material is glossed over.
I am aware I speak quickly - I kept it in mind when recording, but I'll try harder next time. The pacing of transitions is also a bit quick in certain places upon rewatching - I'll make sure to pause more. In the meantime, I have included subtitles which might be helpful if you prefer reading.
Graphics inspired by Oliver Lugg's 27 Unhelpful Facts about Category Theory: https://www.youtube.com/watch?v=H0Ek86IH-3Y.
Main reference during video creation was Basic Category Theory by Leinster and Category Theory in Context by Riehl. Examples of representable functors sourced from notes by Dr. Emanuele Dotto.
Timeline:
00:00 - Introduction
01:08 - Objects
01:40 - Morphisms
02:44 - Compositions
03:01 - Identity
03:22 - Associativity
03:30 - Examples of Categories
06:18 - Product and Dual Categories
07:12 - Duality
07:44 - Commutative Diagrams
08:17 - Isomorphism
09:02 - Functors
10:40 - Covariance and Contravariance
11:15 - Examples of Functors
13:25 - Natural Transformations
15:31 - Vertical Composition
16:53 - Functor Categories
17:18 - Natural Isomorphism
18:22 - Hom Functors
22:19 - Representables
22:40 - Examples of Representables
25:30 - Classifying Spaces
28:19 - The Yoneda Lemma
Errata:
11;13 - "...that a functor is [contravariant], than to...", not "covariant".
18;12 - "...corresponding [objects] are isomorphic...", not "morphisms".
21;57 - the upper string of mappings should be g mapsto hom(h,X)(g) = g o h mapsto hom(B,f)(g o h) = f o (g o h). That is, B and X are the wrong way around in the hom morphisms.
26;59 - "...between the [functions] 1 to R and...", not "functors". (Though, if we treat 1 as the trivial category, and R as a category under ordering, then this does hold for functors in the category of categories. But I really do just mean functions here.)
28;38 - accidentally cut audio, see transcript.
ree
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 3 of a series:
https://youtube.com/playlist?list=PL6kPvEdcJ4jTXsLMBy-1E8CIalh5DCc6B
Read more here: https://github.com/blargoner/math-categories/blob/main/categories.pdf
An introduction to categories, functors, universal properties, natural transformations, and monads with applications to the lambda calculus and functional programming.
This video is part 3 of a series:
https://youtube.com/playlist?list=PL6kPvEdcJ4jTXsLMBy-1E8CIalh5DCc6B
Read more here: https://github.com/blargoner/math-categories/blob/main/categories.pdf
sources and references, in temporal order:
Nature paper on the decline in disruptive science:
https://pubmed.ncbi.nlm.nih.gov/36599999/
Gordon Shepherd's book...
sources and references, in temporal order:
Nature paper on the decline in disruptive science:
https://pubmed.ncbi.nlm.nih.gov/36599999/
Gordon Shepherd's book on the revolutionary 1950s "Creating Modern Neuroscience":
https://www.amazon.com/Creating-Modern-Neuroscience-Revolutionary-1950s/dp/0195391500
Group theory, SU(3), hadrons, quarks and particle physics:
https://tinyurl.com/quarksymmetry
Alexander Unzicker's video on how science moves from numbers (measurements) to equations (laws):
https://www.youtube.com/watch?v=yfmSujRhqCk&ab_channel=Unzicker%27sRealPhysics
Andrei Rodin on pure vs. applied math:
https://www.youtube.com/watch?v=FD472NfobQ0&ab_channel=ToposInstitute
JC Gorman on "What is a topology in why is it in my neuroscience?":
https://neuwritesd.org/2021/06/10/what-is-a-topology-and-why-is-it-in-my-neuroscience/
Tai-Danae Bradley's excellent blog explaining category theory and the Yoneda Lemma
https://www.math3ma.com/categories/category-theory
Bartosz Milewski's explantion of hom-sets and the hom-functor:
https://bartoszmilewski.com/2015/07/29/representable-functors/
John Baez' overview of 'applied category theory':
https://www.youtube.com/watch?v=tfiour4xJ7o&ab_channel=JohnBaez
The inverted spectrum problem:
https://en.wikipedia.org/wiki/Inverted_spectrum
Tsuchiya & Saigo (2021) on the Yoneda Lemma and consciousness:
https://academic.oup.com/nc/article/2021/2/niab034/6397521
https://osf.io/68nhy/download
Nao Tsuchiya's excellent YouTube channel:
https://www.youtube.com/@neuralbasisofconsciousness
Other YouTube channels covering pure math and consciousness:
https://www.youtube.com/@MCS_lectures
https://www.youtube.com/@models-of-consciousness
Math-themed Thank You:
https://www.etsy.com/listing/400131963/thank-you-math-themed-thank-you-card
sources and references, in temporal order:
Nature paper on the decline in disruptive science:
https://pubmed.ncbi.nlm.nih.gov/36599999/
Gordon Shepherd's book on the revolutionary 1950s "Creating Modern Neuroscience":
https://www.amazon.com/Creating-Modern-Neuroscience-Revolutionary-1950s/dp/0195391500
Group theory, SU(3), hadrons, quarks and particle physics:
https://tinyurl.com/quarksymmetry
Alexander Unzicker's video on how science moves from numbers (measurements) to equations (laws):
https://www.youtube.com/watch?v=yfmSujRhqCk&ab_channel=Unzicker%27sRealPhysics
Andrei Rodin on pure vs. applied math:
https://www.youtube.com/watch?v=FD472NfobQ0&ab_channel=ToposInstitute
JC Gorman on "What is a topology in why is it in my neuroscience?":
https://neuwritesd.org/2021/06/10/what-is-a-topology-and-why-is-it-in-my-neuroscience/
Tai-Danae Bradley's excellent blog explaining category theory and the Yoneda Lemma
https://www.math3ma.com/categories/category-theory
Bartosz Milewski's explantion of hom-sets and the hom-functor:
https://bartoszmilewski.com/2015/07/29/representable-functors/
John Baez' overview of 'applied category theory':
https://www.youtube.com/watch?v=tfiour4xJ7o&ab_channel=JohnBaez
The inverted spectrum problem:
https://en.wikipedia.org/wiki/Inverted_spectrum
Tsuchiya & Saigo (2021) on the Yoneda Lemma and consciousness:
https://academic.oup.com/nc/article/2021/2/niab034/6397521
https://osf.io/68nhy/download
Nao Tsuchiya's excellent YouTube channel:
https://www.youtube.com/@neuralbasisofconsciousness
Other YouTube channels covering pure math and consciousness:
https://www.youtube.com/@MCS_lectures
https://www.youtube.com/@models-of-consciousness
Math-themed Thank You:
https://www.etsy.com/listing/400131963/thank-you-math-themed-thank-you-card
Correction: Universal Property of Quotients requires ker(f) to contain ker(pi)
0:00 Motivation
1:33 Basics in Category Theory
4:14 Group Objects
5:08 Functors
8:17 Universal Properties
11:57 Proof using Category Theory
13:27 Shortcomings of 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)
Category theory is the heart of mathematical structure. In this video, I will drive a stake through that heart. I don't know why I made this.
Grothendieck Googling: https://mobile.twitter.com/grothendieckg
Join my Discord server to discuss this video and more: https://discord.gg/AVcU9w5gVW
MUSIC
Oregano
Vendla (Epidemic Sound)
Penumbra
Kevin MacLeod (incompetech.com)
Please watch with subtitles. Errata noted in transcript and at bottom of description.
Some content may require a little background in abstract algebra, but there are no topology heavy examples included.
This was originally written for the oral presentation component of my essay module, but the script ended up being way too long. I'd already made the animations, so I've decided to turn it into a crappy video due to the sunk cost fallacy. The audio was recorded in 1-2 takes at 3am, so the quality isn't great (most of the audio is taken from a recording I took purely to time out how long it would take for me to present it). I might update and remake the video in higher quality and in more detail if I have the motivation, but I have too much work right now.
Despite the first subtitle, we only briefly cover the Yoneda lemma in this presentation. Actually, we only briefly cover most of the content in here - I was intending this to be a 15 minute talk, so a lot of material is glossed over.
I am aware I speak quickly - I kept it in mind when recording, but I'll try harder next time. The pacing of transitions is also a bit quick in certain places upon rewatching - I'll make sure to pause more. In the meantime, I have included subtitles which might be helpful if you prefer reading.
Graphics inspired by Oliver Lugg's 27 Unhelpful Facts about Category Theory: https://www.youtube.com/watch?v=H0Ek86IH-3Y.
Main reference during video creation was Basic Category Theory by Leinster and Category Theory in Context by Riehl. Examples of representable functors sourced from notes by Dr. Emanuele Dotto.
Timeline:
00:00 - Introduction
01:08 - Objects
01:40 - Morphisms
02:44 - Compositions
03:01 - Identity
03:22 - Associativity
03:30 - Examples of Categories
06:18 - Product and Dual Categories
07:12 - Duality
07:44 - Commutative Diagrams
08:17 - Isomorphism
09:02 - Functors
10:40 - Covariance and Contravariance
11:15 - Examples of Functors
13:25 - Natural Transformations
15:31 - Vertical Composition
16:53 - Functor Categories
17:18 - Natural Isomorphism
18:22 - Hom Functors
22:19 - Representables
22:40 - Examples of Representables
25:30 - Classifying Spaces
28:19 - The Yoneda Lemma
Errata:
11;13 - "...that a functor is [contravariant], than to...", not "covariant".
18;12 - "...corresponding [objects] are isomorphic...", not "morphisms".
21;57 - the upper string of mappings should be g mapsto hom(h,X)(g) = g o h mapsto hom(B,f)(g o h) = f o (g o h). That is, B and X are the wrong way around in the hom morphisms.
26;59 - "...between the [functions] 1 to R and...", not "functors". (Though, if we treat 1 as the trivial category, and R as a category under ordering, then this does hold for functors in the category of categories. But I really do just mean functions here.)
28;38 - accidentally cut audio, see transcript.
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An introduction to categories, functors, universal properties, natural transformations, and monads with applications to the lambda calculus and functional programming.
This video is part 3 of a series:
https://youtube.com/playlist?list=PL6kPvEdcJ4jTXsLMBy-1E8CIalh5DCc6B
Read more here: https://github.com/blargoner/math-categories/blob/main/categories.pdf
sources and references, in temporal order:
Nature paper on the decline in disruptive science:
https://pubmed.ncbi.nlm.nih.gov/36599999/
Gordon Shepherd's book on the revolutionary 1950s "Creating Modern Neuroscience":
https://www.amazon.com/Creating-Modern-Neuroscience-Revolutionary-1950s/dp/0195391500
Group theory, SU(3), hadrons, quarks and particle physics:
https://tinyurl.com/quarksymmetry
Alexander Unzicker's video on how science moves from numbers (measurements) to equations (laws):
https://www.youtube.com/watch?v=yfmSujRhqCk&ab_channel=Unzicker%27sRealPhysics
Andrei Rodin on pure vs. applied math:
https://www.youtube.com/watch?v=FD472NfobQ0&ab_channel=ToposInstitute
JC Gorman on "What is a topology in why is it in my neuroscience?":
https://neuwritesd.org/2021/06/10/what-is-a-topology-and-why-is-it-in-my-neuroscience/
Tai-Danae Bradley's excellent blog explaining category theory and the Yoneda Lemma
https://www.math3ma.com/categories/category-theory
Bartosz Milewski's explantion of hom-sets and the hom-functor:
https://bartoszmilewski.com/2015/07/29/representable-functors/
John Baez' overview of 'applied category theory':
https://www.youtube.com/watch?v=tfiour4xJ7o&ab_channel=JohnBaez
The inverted spectrum problem:
https://en.wikipedia.org/wiki/Inverted_spectrum
Tsuchiya & Saigo (2021) on the Yoneda Lemma and consciousness:
https://academic.oup.com/nc/article/2021/2/niab034/6397521
https://osf.io/68nhy/download
Nao Tsuchiya's excellent YouTube channel:
https://www.youtube.com/@neuralbasisofconsciousness
Other YouTube channels covering pure math and consciousness:
https://www.youtube.com/@MCS_lectures
https://www.youtube.com/@models-of-consciousness
Math-themed Thank You:
https://www.etsy.com/listing/400131963/thank-you-math-themed-thank-you-card
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions. On the other hand, any monoid can be understood as a special sort of category, and so can any preorder. In general, the objects and arrows may be abstract entities of any kind, and the notion of category provides a fundamental and abstract way to describe mathematical entities and their relationships. This is the central idea of category theory, a branch of mathematics which seeks to generalize all of mathematics in terms of objects and arrows, independent of what the objects and arrows represent. Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. For more extensive motivational background and historical notes, see category theory and the list of category theory topics.