Category Therory: A Sensible Introduction to Category Theory

Transcript

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[Music]

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about half a year ago i published a

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video titled 27 unhelpful facts about

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category theory a silly spur of the

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moment kind of thing i neglected to

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realize that making a thumbnail which

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simply featured a coconut next to the

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word nut would make me the darling of

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the youtube algorithm for the next week

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and so it was that much of the

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internet's first introduction to my

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channel was a 9 minute long your mum

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joke including several people i knew in

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real life who weren't aware i even had a

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youtube channel

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still not sure how to process that in an

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effort to recover some respectability

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i've decided to remain faithful and full

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and essentially subjective and objects

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by fulfilling my obligation to a serious

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follow-up

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today i'll be teaching you the very

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basics of category theory

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properly

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ideally i'd like anyone at any level of

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mathematical education to come away from

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this video feeling like they learned

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something so i'll be keeping jargon and

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technicalities to a minimum pretty much

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everything i say will thus have a giant

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implicit asterisk attached proclaiming

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that yes things are more complicated

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than i've made them seem but if i stop

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to clarify everything will be here

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forever

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i'll also be focusing a lot on intuition

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and in particular my own intuition

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i'm nowhere near an expert even if the

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almighty algorithm disagrees so if you

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find my explanations lacking i've linked

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some more authoritative perspectives

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below

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i'd also like to give a shout out to

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metamaths a channel currently working

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through a parallel introduction to the

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field so go check that out if you want

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more

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right let's begin then with the obvious

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question

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what is category theory

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[Music]

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mathematicians work with a large variety

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of mathematical constructions

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each has properties specific to a

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particular area of study properties that

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make each area distinct

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topology studies topological spaces

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group theory studies groups linear

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algebra studies vector spaces and so on

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these disciplines all care about

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different structures and they've all

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built different machinery to deal with

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each of them

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even so more often than not such

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structures have an awful lot in common

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they may look and feel different but the

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foundations are more alike than you

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might think

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there appears to be a standard way of

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building contexts in which we do

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mathematics

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category theory is all about formalizing

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this observation

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if we make precise the notion of a

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mathematical context we have the tools

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in place to examine contexts as a whole

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and in particular translations between

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contexts we get a bird's-eye view of the

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whole subject

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to understand what category theory is it

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might help to understand what it is not

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firstly it's not really about

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categorizing things

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i know a lot of people saw the coconut

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in my previous videos thumbnail and

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assumed category theory was a way to

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classify real world objects or something

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but no it's a mathematical theory about

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mathematical structure coconuts involved

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only for comedic purposes

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second it's not some new age third eye

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opening secret of the universe type of

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thing you laugh but i have actually seen

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people claim this

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third and related category theory is not

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a theory of everything

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granted it definitely looks like it

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sometimes

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if category theory subsumes every

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mathematical framework is it not the

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ultimate language of the universe

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i mean i know i was drawn to category

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theory on a quest to know everything and

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i somehow managed to make brief mentions

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of general relativity quantum mechanics

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russell's paradox and linguistics in my

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master's dissertation about moving

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brackets around

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more on that later but category theory

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on its own is in fact rather limited in

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scope

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some have said it's merely scaffolding

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surrounding buildings made of actual

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maths but my preferred analogy is that

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if maths is a story category theory is

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the setting essential but far from the

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full picture

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some stories are all about the setting

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while others don't give it a second

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thought preferring to focus on

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characters or plot instead the details a

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bird's eye view might ignore to tell

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certain stories you can get away with

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only one approach to tell every story

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though you need both the high level

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abstraction and the low level details

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plus as we'll see shortly not everything

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is a category

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while the idea covers most frameworks we

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consider useful or interesting there are

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plenty of exceptions

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speaking of which if we're about to talk

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about category theory we need to know

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what a category is

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let's meet the definition that's

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responsible for all of this

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a category which we might name curly c

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consists of four things

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objects morphisms composition and

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identities subject to two requirements

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the identity and associativity laws

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let's look at each of these in turn

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every category has a collection of

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objects

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in principle objects can be anything

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though we generally take them to be

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mathematical things like numbers or

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collections of numbers or collections of

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numbers with extra structure

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we generally specify the objects of a

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category by writing them in a set

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a set for those unaware is a collection

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of things called elements written

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between curly brackets where we don't

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care about the order and we don't care

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about duplicates

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so for instance these sets are the same

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and these sets are the same

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there you go you now know set theory

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categories also have morphisms

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for any two objects a and b the category

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has a set of morphisms from a to b

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morphisms are such a general idea that

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it can be hard to intuitively pin down

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what we mean by this

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i like to think of morphisms as

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relationships with a direction there is

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a morphism from a to b if a is related

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to b

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but b doesn't have to be related to a in

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return

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it could be a one-way connection

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it could be the case that the set of

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morphisms from a to b is empty so a is

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not related to b

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or we could have more than one morphism

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from a to b so a is related to b in

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multiple different ways

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morphisms are commonly some kind of

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function and we'll see an example

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shortly but they don't have to be

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they can be much more abstract

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once you have morphisms you can compose

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them

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if f is a morphism from a to b and g is

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a morphism from b to c the definition of

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a category says there must be some way

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of combining them to produce amorphism

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from a to c

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in other words we can stick compatible

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morphisms end to end to make new ones

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and the composition operation in the

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category gives a recipe for making them

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we usually write composition with a

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small circle like this

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so g composed with f here is a morphism

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from a to c

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a category also requires certain special

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morphisms to exist called identities

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identities are morphisms from one object

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to itself

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in a category we must have one identity

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for each and every object which in

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particular means that the set of

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morphisms from any object to itself is

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not empty

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these identities are then subject to the

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identity law

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composing any morphism with an identity

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on either side does nothing to that

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morphism

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it's a bit like how one times any number

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equals that number unless you're terence

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howard finally we require the

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composition operation to satisfy the

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associativity law this one's a bit more

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complicated if you haven't met

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associativity before but i'll do my best

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most operations we use in day-to-day

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maths are binary this means they take

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two inputs and produce one output

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but we often use them as if they can

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take any number of inputs

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strictly speaking we should choose an

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order in which we evaluate such

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expressions

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luckily for us it often doesn't matter

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which order we choose no matter our

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choice we get the same answer so we

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ignore order entirely and write the

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expression without brackets this is

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shorthand for choose any order you like

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and successively evaluates the binary

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operation

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if a binary operation has this property

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we say it's associative

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addition and multiplication are the most

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familiar examples but some operations

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are not associative

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the most obvious is subtraction

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bracketing in different ways produces

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different outputs

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but my favorite non-associative

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operation is rock paper scissors

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ponder this if you wish

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in a category we require composition

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which is a binary operation to be

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associative we can therefore write any

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number of compositions in a row without

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brackets

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hurrah

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putting these all together you have

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yourself a category

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tell me what its objects are what its

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morphisms are how you compose morphisms

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and what its identities are and if the

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stuff you've given me satisfies the

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identity and associativity laws the

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category is well defined

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but all this is so far extremely

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abstract

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categories are meant to be general so we

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can apply them to lots of different

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scenarios but that approach has its

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drawbacks when you come to understand

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them for the first time

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let's make it a bit more concrete before

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i lose you

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let's build one of the most fundamental

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categories

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set the category of sets and set

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functions

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objects in set are done surprisingly

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sets

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morphisms in set are set functions

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again for those unaware a set function

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is a way of assigning to each element in

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one set one element in the second

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we can define a function by writing down

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its effect on every element or if

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there's an easier shorthand we could use

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that

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composition of functions is pretty

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straightforward

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for each element the composite function

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applies the first function then applies

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the second

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and for any set we can create a function

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that does nothing to each element these

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will be our identities

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we can then check that composition is

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associative and the proposed identities

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do act as identities to verify that set

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is indeed a category

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many of the most common and most useful

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categories follow the model of set

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that is their objects are sets with

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extra structure and their morphisms are

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special functions which respect that

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structure

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take the category group for instance

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a group is a set equipped with a binary

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operation which follows certain rules

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this operation adds a certain structure

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to the set

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we can then consider specialized set

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functions which respect the binary

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operation combining two elements then

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applying such a function is the same as

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applying the function to each element

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then combining

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functions with this property are called

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group homomorphisms in the category

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group objects are groups and morphisms

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are group homomorphisms composition and

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identities are exactly the same as in

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set since everything in group is a

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specialized version of something in set

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but the power of categories becomes

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apparent when you move outside this

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model of structured sets and

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structure-preserving functions let's

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look at an example that's somehow both

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more down to earth and more confusing

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construct a category as follows

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define its objects to be all the numbers

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you can place on a number line which

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mathematicians call real numbers for

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numbers x and y define a unique morphism

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from x to y if and only if x is less

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than or equal to y

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the difficulty here compared to set is

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that we can't really say what a morphism

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is

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it's not a thing you can write out like

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a function and it doesn't seem to do

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anything other than representing the

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fact that one number is less than or

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equal to another how then do we compose

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such porphisms well let's work through

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it

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if we have morphisms x to y and y to z

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this means x is less than or equal to y

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and y is less than or equal to z thus x

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is less than or equal to z

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but we know what that means it means

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there is a unique morphism from x to z

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we're looking for a morphism from x to z

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to be the composition of our two

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original morphisms so we can just define

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it to be this one

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since such morphisms are unique this

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definition uniquely determines the

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composition

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what about identities though we need

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amorphism from x to itself

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since x equals x x is in particular less

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than or equal to x so there is a unique

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morphism from xx and ta-da we have found

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our identities

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as before one can then follow through

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the definitions to see this proposed

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category abides by the identity and

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associativity laws so we have ourselves

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another category

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at this point it might help to consider

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a few constructions which aren't

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categories what if we modify the last

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example to be real numbers with less

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than instead of less than or equal to

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well then we can't have identities we

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would need every number to be strictly

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less than itself which of course isn't

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possible so this isn't a category

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what if we try to build a category where

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objects are numbers and there is a

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unique morphism from x to y if and only

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if x equals y plus one

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again identities don't work here no

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number equals itself plus one all right

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you might say let's adjust that to say

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there's a morphism from x to y if and

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only if x equals y or x equals y plus

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one

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this fixes the identity issue so maybe

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it works as a category

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but consider composition

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if we have morphisms x to y and y to z

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which aren't identities then x equals y

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plus one and y equals z plus one

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so x equals z plus two hence there

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cannot be a morphism from x to z

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according to our definition so we can't

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define the composition of these two

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morphisms

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alas this isn't a category either

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even though the definition of a category

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is highly general it's still pretty

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restrictive

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the fact that it nonetheless applies to

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so much of our mathematical landscape

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highlights just how much we're drawn to

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this kind of structure and how useful it

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can be

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categories are even more useful when we

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depict them visually

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i've already been using a load of arrows

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to help communicate these ideas but we

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can make things a bit more formal from

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here

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we can take some objects from a category

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and arrange them in a diagram then draw

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morphisms as arrows between them

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we can then compose morphisms by joining

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their corresponding arrows nose to tail

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thanks to the associativity law we can

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do this multiple times without brackets

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tracing out a root through the diagram

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this alone is helpful but certain

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diagrams have an extra property which

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means we don't have to write out reams

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of equations to prove things

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a diagram is commutative if for any two

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objects all routes between them are

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equal

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for instance this diagram commutes if

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and only if h is equal to g composed

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with f

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now commutative diagrams are infamous

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for being

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overly elaborate but you have to

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remember that under the hood a

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commutative diagram is simply a list of

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equations each pair of roots between two

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objects corresponds to one equation no

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matter how convoluted the roots may be

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so in fact despite their complexity

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commutative diagrams are stunningly

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efficient imagine writing down every

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single pair of paths in a diagram like

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this

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not fun

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instead we can translate the problem of

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solving multiple equations into the

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problem of gluing shapes together along

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shared morphisms much more satisfying

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one important commutative diagram to

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consider is this one

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suppose we have objects a and b in a

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category and morphism's f from a to b

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and g from b to a such that this diagram

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commutes

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in other words f composed with g and g

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composed with f are equal to identities

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in this case f is known as an

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isomorphism

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isomorphisms are morphisms with inverses

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which act as a kind of undo button for

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the morphism

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doing both does the same thing as doing

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neither if there exists an isomorphism

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between two objects we say those objects

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are isomorphic written like this

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if you've done any university level

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maths you'll be well acquainted with

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isomorphism in some guys

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isomorphisms of groups isomorphisms of

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vector spaces whatever these are all the

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result of applying the definition of

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isomorphism in the category setup for

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each of these contexts

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in our set example for instance an

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isomorphism is a set function that can

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be reversed using another set function

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equivalently two sets are isomorphic if

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they contain the same elements labeled

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in different ways

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the actual names of the elements and any

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extra structure aren't important in set

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only the set theoretic perspective is

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isomorphism captures the idea that

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objects are functionally the same where

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the meaning of functionally depends on

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the working category

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to the category itself the objects look

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the same because they share the only

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properties the category actually cares

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about

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so we've defined categories we've

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represented them visually and we've

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looked at an important concept you can

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find inside them but there's a lot more

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to category theory than categories

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first let's take a quick look at

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functors

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a functor is in some sense a translation

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between categories we can turn one

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category into another look at what's

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changed and draw conclusions about how

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the categories themselves are related

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i like to think of functors as rotations

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in perspective we value different

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properties of objects and morphisms in

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different contexts and functors provide

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a way of changing these priorities

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rotating your worldview

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let c and d be categories

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a functor f from c to d consists of two

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things a function on objects and a

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function on morphisms subject to two

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requirements functors respect identities

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and functors respect composition

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again let's look at these one by one

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a functor takes objects from c and turns

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them into objects in d usually written

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like this

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similarly a functor takes morphisms from

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c and turns them into morphisms in d

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with the endpoints determined by the

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function on objects

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by respecting identities we mean the

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functor takes identities to identities

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as for respecting composition this means

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that f applied to a composition in c is

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the composition in d of f applied to

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each morphism hey that kinda looks like

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a group homomorphism

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i wonder if there's a reason for that

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and that's all you need to make a

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functor two functions which do sensible

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things to identities and composition

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as before however all rather abstract

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let's see some examples to bring it all

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back down to earth

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one notorious type of functor is the

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forgetful functor these do nothing to

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objects and morphisms except forget some

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of the properties we cared about in the

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original category

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take the forgetful functor from group to

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set

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as we saw earlier any group is a set

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with extra structure the forgetful

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functor

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forgets that structure as the name might

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imply a group homomorphism is similarly

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a set function with extra structure

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ignoring that structure still leaves a

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normal set function i.e a morphism in

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set

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since we're doing nothing to morphisms

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identity and composition preservation

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follow immediately demonstrating that u

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is a well-defined functor

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it's also possible to construct a

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functor going the other way this is

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called the free group functor which

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builds a group out of any set you give

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it

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i'll spare you the details here though

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we can also build a functor out of our

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less than or equal to example consider a

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category whose objects are real numbers

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with a unique morphism from x to y if

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and only if x is greater than or equal

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to y

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i'll leave you to dot the eyes and cross

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the t's to verify that this is a

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category

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how can we translate between these two

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contexts in a way that preserves

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composition of morphisms

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how can we

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rotate our perspective

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oh well that was easy

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because we've flipped the direction of

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the order to preserve the right stuff we

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merely need to flip the sign of the

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objects that is map everything to its

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negative and the rest falls out

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at their core functors are maps between

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categories which respect categorical

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structure hopefully there are some bells

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ringing in your head right now because

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we've met that sort of thing before

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remember i said many categories are

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structured objects and functions

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respecting their structure

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well what if we just

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move this around a bit and add a few

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more and

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whoops

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i i appear to have created the category

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of categories

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yep category theory has gone full meta

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everything we've looked at so far

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applies not only to things inside

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categories but to categories themselves

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here we begin to glimpse the true power

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of category theory

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now that our morphisms are functors if

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we have two functors which compose to

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produce identity functors we found

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ourselves some isomorphic categories

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categories which differ in decoration

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alone

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unfortunately isomorphism of categories

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tends to be too strong a requirement for

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anything useful it's a bit like

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comparing strings of text and saying

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they're only equal if both are written

21:47

in the same font when really we care

21:49

about something more fundamental but

21:51

there is a way to fix this

21:53

the category of categories comes

21:55

prepared with a weakened form of

21:57

isomorphism called equivalence

22:00

for equivalence we don't require the

22:02

composite functors to equal identity

22:05

functors they need only be naturally

22:08

isomorphic

22:10

now i'm not going to rigorously define

22:12

natural isomorphisms here because lord

22:14

knows i've been going too long already

22:15

and i can tell some of you are drifting

22:17

off put simply just as functors are

22:20

translations between categories natural

22:23

transformations are translations between

22:25

functors

22:26

and natural isomorphisms are natural

22:29

transformations with inverses

22:31

natural isomorphisms are then

22:33

isomorphisms in the category where

22:35

objects are functions between two fixed

22:37

categories and morphisms are natural

22:38

transformations between such functors is

22:40

your head hurting yet miners point is

22:43

you already know the most important

22:45

takeaway if g composed with f and the

22:48

identity functor are naturally

22:49

isomorphic then they may not be equal

22:52

but they're close enough to equal for

22:54

most things that matter

22:55

equivalence of categories turns out to

22:57

be much more common and much more

22:59

informative than isomorphism if two

23:02

categories are equivalent then they

23:04

encode essentially the same information

23:06

in different ways for example i wrote my

23:08

dissertation on the equivalence between

23:10

the category of minority categories and

23:12

the category of representable

23:14

multicategories try saying that five

23:15

times fast monodual categories and multi

23:18

categories are category-like structures

23:20

which emerged somewhat independently

23:23

until one day someone noticed all

23:25

minority categories were

23:26

multi-categories in disguise and a

23:28

certain subset of multi-categories

23:30

representable ones were monogal

23:32

categories in disguise what's more the

23:34

categories containing each had matching

23:37

structure in their morphisms or in more

23:40

precise terms there is an equivalence of

23:42

categories

23:43

people were looking at two perspectives

23:45

on the same thing all along

23:47

and if equivalence isn't enough for you

23:49

it's possible to generalize further to

23:52

adjunction two categories connected by

23:54

adjoint functors might not be equal they

23:57

might not be almost equal and they might

23:59

not be almost almost equal but they are

24:02

almost almost almost equal kind of like

24:06

evil twins i guess

24:08

i don't know i i've honestly never read

24:10

a description of a junction that makes

24:12

real intuitive sense to me

24:14

and that i think is the signal i've

24:16

reached the limits of what's explainable

24:18

in a casual quippy youtube video believe

24:21

me there are many many more advanced

24:23

topics i could talk about from limits

24:26

and co-limits to the yena dilemma but i

24:28

think i've made good on my promise to

24:30

fill in the most prominent gaps in my

24:31

previous overview category theory is a

24:34

deep and diverse subject and if you've

24:36

made it this far i'm sure you'd enjoy

24:38

reading into it further as i said there

24:41

are several sources in the description

24:43

and there's plenty of other interesting

24:44

content to find out there i just hope

24:46

i've managed to add something worthwhile

24:48

to that list

24:49

all this does however leave one question

24:52

i'm sure some of you have been itching

24:54

to write a snarky comment about the

24:55

whole time

24:56

what's all this actually for

24:59

even compared to other subjects in pure

25:01

maths category theory has a reputation

25:04

for being unadulterated abstract

25:06

nonsense

25:07

despite the memes however there are some

25:09

concrete applications

25:11

functional programming is the best known

25:13

any time you use a lambda expression in

25:15

your programming language of choice you

25:17

are in fact paying homage to among

25:20

others category theory

25:22

the subject has also had minor impacts

25:24

in fields as disparate as quantum

25:26

mechanics mathematical modelling and

25:28

linguistics as the subject has grown

25:31

ever trendier in the past few years

25:33

there have even been conferences

25:35

dedicated to applied category theory

25:38

the essence of category theory is an

25:40

organizing principle

25:41

it's a formalized method to recognize

25:43

patterns in the way we structure our

25:45

problem-solving machinery wherever that

25:47

turns up

25:48

while we do have to avoid the traps of

25:50

romanticizing it or asking too much of

25:52

it there's no denying a categorical

25:54

approach is a powerful one

25:57

providing you're willing to melt your

25:58

brain a bit in the process

26:01

for those curious i've put a link to my

26:03

dissertation which is a journey from the

26:05

basics of category theory to a long and

26:07

painful proof in the description i

26:10

haven't yet received final feedback so i

26:12

can't guarantee correctness but it might

26:14

be an interesting read if you'd like to

26:16

see category theory in action