27 Unhelpful Facts About Category Theory

27 unhelpful facts about category theory fact one this video is a joke please don't take any of it seriously i am in fact studying category theory properly and i do enjoy it but this video idea came to me and couldn't be stopped so please enjoy this context-free exploration of category theory's weirdest quirks most of which are actually true i may make a series video on the topic at some point but for now i apologize for the picture i'm about to paint of what's generally a beautiful mathematical subject anyway fact two a category consists of a collection of objects for any two objects a hom set of morphisms from one to the other and a composition operation such that composition is associative and has identities for each object easy right fact three not easy homssets morphisms the hell is an ob you don't need any of those to understand category theory no you need that meme with two pictures that pam from the office says are the same picture that's all mathematics is right you take two things and find a theory that abstracts away their particulars to deal with them together you know that whole thing about coffee mugs and donuts and how topologists can't tell the difference between them well category theory does that but meta to a category theorist all of maths is the exact same thing in all constructions you have some kind of stuff and stuff that relates that stuff and you can chain those relations together to get further relations with the same properties and sure you can call them objects and morphisms if you really want but that's just going to confuse people and we don't want to confuse people right to that end here are some illustrative examples the category of sets and set functions the category of groups and group homomorphisms and the fundamental groupoid of a topological space where objects are points and morphosims are path homotopic classes and if you still aren't clear on what a category is after that good luck fact four category theory was invented by american mathematicians samuel islandberg and saunders mac lane and officially appeared in mathematical literature for the first time in september 1945 coinciding exactly with the formal end of world war ii category theory has been keeping the world safe from global conflicts ever since fact five you can take some objects and morphisms from a category and arrange them to make a diagram if all roots through the diagram between the same two objects evaluates to the same morphism what you have there is a commutative diagram we say the diagram commutes commutative diagrams are equations for the category theorist you thought a picture wasn't a proof well think again commutative diagrams are everywhere too many places probably all right uh that's enough stop stop stop fact six you might have noticed the word class in the definition of a category you probably didn't and that's okay we all make mistakes why though use classes rather than sets sets are nice and simple classes are weird and technical and annoying in practice almost everyone uses sets and ignores the distinction we just put class in there to appease the set theorists who would otherwise throw their paradoxes at us it's a good idea to stay on the correct side of the set theorists they can construct a set of all things that cause you pain fact seven you can turn categories into other categories with functors yes not function functer one who functs functors take objects to objects and morphs them to morphism subject to sensible composition laws but what that means for you is that they change your perspective on the world you're looking at one thing you apply a functor and suddenly you're looking at something else to use the meme analogy again it's uh this one sure some of these functions are called forgetful functors they do nothing except forget some of the properties you cared about in the last category a functor from the category of edible things to the category of all things would for instance make you star to death because despite doing nothing to the food it would make you forget how to eat fact eight there are five ways to say things are the same in category theory the same basically the same basically basically the same basically the same but spicy and i don't even know how to explain this one fact nine if you know any maths you'll likely find lots of familiar words in category theory but because we've abstracted beyond abstraction they don't all look right this is apparently a product yeah sure category theorists whatever you say and this i'm told is a cone hey uh mac lane i think you need to go back to primary school mate this sort of thing does however mean you can study tensors without ever needing to know what a tensor actually is you just need to understand this commutative diagram fact 10 the opposite of a functor is not called a defunctor this makes me sad fact 11 you can do anything backwards by adding the prefix co a backwards product is a co-product a backwards cone is a cocoon a backwards bra is a cobra and i can't think of anything worse for support than a cobra so the theory checks out qed fact 12 one of the most famous category theorists was a guy named alexander grotendik some people believe he's responsible for making category theory its own field of study rather than a tool occasionally used in other places groton deak was a staunch pacifist and even gave a lecture on category theory in the forests surrounding hanoi as it was being bombed during the vietnam war once again category theory is truly the ultimate tool for peace later in life groton deak gave up on society and went to live as a spiritual recluse in the french pyrenees he died in 2014 but it's okay because his spirit lives on through the twitter account groton deak googling fact 13 remember functors they're maps between categories but what about maps between functors mind-blowing i know right these are called natural transformations natural transformations require that all diagrams of this form commute simple fact 14. there's a thing called the yaneda lemma which says that if you know what a thing looks like from all perspectives you know what it is students can get stuck flip-flopping between believing the una dilemma is entirely trivial and believing it's the most complex thing they've ever seen fact 15. remember natural transformations they're maps between functors but what about maps between natural transformations you see the category of categories together with functors and natural transformations is in fact an example of a two category these have objects morphisms and morphisms between morphisms ah but we've started counting now and once mathematicians start counting they can't stop literally you've got your three categories with morphisms between morphisms between morphisms and your four categories with morphisms between morphisms between morphisms between morphisms and your five categories between morphemes between morphological simply reports of between these morphisms and then somehow you go all the way up to infinity categories of what the hell even are those fact 16 have a break fact 17 a coconut is just a nut fact 18 due to its extended isolation from the rest of the world australian category theory has developed into a strange beast that even other category theorists fear its separate evolutionary lineage allowed it to reach levels of abstraction never dreamed of in the old world so basically it's the mathematical equivalent of marsupials also all their commutative diagrams are upside down factor 19 there's a thing called the pentagonator i don't really understand what it is but knowing a word like that is out there somewhere makes me happy fact 20 a monad is a monoid in the category of endo functors nobody actually knows what that means so they'll just parrot a monad is a monod in the category of endofunctors at you repeatedly to make themselves look smart fact 21 initial objects have morphsums to everything and terminal objects have morphisms from everything fact 22 category theory has lots of exciting real-world applications such as uh well um there's there's i'm sure there was something fact 23 what'd you call someone who reads a paper on category theory a co-author fact 24 oh yeah uh functional programming right functional programming is an increasingly popular programming paradigm that owes its development to category theory huzzah we've found a real world application in functional programming you pass functions as arguments to other functions and nothing keeps any state and you can have type into inter and there's this thing called maybe oh how did that get in there fact 25 some people are tentatively applying category theory to the arts art music literature with a framework called carter curry theory fact 26 in certain categorical constructions of mathematical logic things can be true false or bottom and finally fact 27 the terminal object in the category of people and sexual relations is your mum

Channel: Oliver Lugg

Views: 416,688

Keywords: Category theory, Functor, Grothendieck, Natural transformation, Mathematics

Id: H0Ek86IH-3Y

Length: 9min 26sec (566 seconds)

Published: Fri Dec 31 2021