Intuitive Introduction to Category Theory

Category Theory offers a different style of thinking about mathematics. I describe how using functions and sets as examples.

(00:00) - elements and operations
(01:04) - functions are sets ?
(02:20) - injection and surjection
(04:00) - alternative definitions
(06:05) - crucial conclusion !

#categorytheory
#algebra
#manim

[thestemgamer]

This is a very tricky concept that is critically important to functional programming.
You explained it beautifully.

[Champs]

omg! very elegant and simple explanation, amazing!

[angelmendez-rivera]

Also, it should be noted that the statement that f is surjective if and only if it has a right-inverse requires the axiom of choice.

[PowerhouseCell]

Woah, this is such an underrated channel! As a fellow educational YouTuber, I understand how much work must have gone into this— amazing job!! Liked and subscribed :)

[mathematicalmachinery]

Thank you for reading the notation out aloud at 4:32, it was really helpful :)

[acortis]

Wonderful explanation! Please keep them coming!

[luffis]

Great Video. Would've liked if you also covered surjection's three definitions aswell.

[joshuatilley]

Perhaps worth stating that the equivalence of left(right) invertible, in(sur)jective, and mono(epi)morphism is false in most categories other than Set.

[rog]

Im so glad I found this video, could you maybe also take a look at type theory? :)

[tombouie]

Hmmmm .... most excellent;
However I thought a function was a group of indep/dep pairs such that any pairs with =indep values all have =dep values
Of course you can use a math expression to produce the above group of indep/dep pairs
Oh I must apolozies but I missed what exactly category theory was, sorry.

[tradingishard]

What is the analogue of a monomorphism for surjections ?

[KevinFlowersJr]

Who needs university when you have Youtube & Discord

[blinded]

Why do mathematicians love to multiply from right to left, when we read everything from left to right, most programming languages are executed from left to right, and when division works from left to right ((a*b)/b = a)?

[skit]

This vid will only grow in value and views through time. This is great, thanks!

[skit]

Also, I'm seeing that math only treats lossless compressions. Using more natural domains, that perfectly bijective identity is the rare exception, not the norm.

[angelmendez-rivera]

5:50 - 5:56 That is a misleading statement, if not an inaccurate one. You are still dealing with sets here. The set of all functions from A to B is denoted B^A, and the set of all functions from Z to A is denoted A^Z. The functions f°α and f°α' are functions from Z to B, and so they are elements of the set B^Z. In this context, composition ° is then a function from the Cartesian product of B^A with A^Z to B^Z. This is a necessary caveat, as otherwise, composition of functions is not even well-defined.

There is some extended sense here in which one uses abstract algebraic terminology, in which we say f is cancellable if and only if, all α, α' in A^Z, f°α = f°α' implies α = α'. Thus, f is a monomorphism if and only if it is cancellable.

[lhpl]

Hmm. Is this merely an advertisement for a book? First you talk about injection, surjection and bijection based in sets, then you rephrase them in left and right inverses, so far so good, then finally you go full abstract nonsense, but suddenly it's just about injections. Where did surjection go? I have watched a few "introductions to category theory", and it is my impression that most of them appear to only be understandable if one knows everything in advance - but then one would not need to watch an intro...

I am still looking for an intro video on category theory that is as good and thorough as sudgylacmoe's _A Swift Introduction to Geometric Algebra_ is for geometric algebra...

[deyanirasaez]

Perhaps his explanation can be extrapolated to the study of the human sciences, between sociology (focused on the whole, ignoring the individual, such as freedom, passions... -->objects, elements, identities) and psychology, focused on the individual (elements, objects). And perhaps to look at the tree or the forest. In this way, every theory looks at either the tree or the forest as a method to draw conclusions where both "looks" are needed to reach more general "truths." The social "kills" the individual, and looking only at the individual does not explain certain social facts (the important thing is what the person does, his social function, not his motivations or whys). The family is the measure of all things, because it is the middle state between the individual and the social. Maybe we have to look for the comparable concept "family" in mathematics. Sorry for the possible nonsense, I come from the human sciences

[angelmendez-rivera]

1:28 - 1:40 This is technically inaccurate. What this is describing is a functional graph, not a function. A function F is a set ((X, Y), G) such that G is a subset of the Cartesian product of X with Y, and such that for all elements x in X, there exists exactly one y in Y, such that (x, y) is in G. Here, I would like to clarify that the ordered pairs (x, y), (X, Y), and ((X, Y), G), are Kuratowski pairs, in set theory. The Kuratowski pair is defined as (x, y) := {{x}, {x, y}}, and it can be proven that this is always a well-defined object, by the axiom of extensionality and the axiom of pairing. The axiom of power set and the axiom of union guarantee the existence of the Cartesian product of X with Y, and the restricted axiom of comprehension guarantees the existence of the uniqueness restriction above. Therefore, this is a well-definition of a function.

[dembeto]

I do not understand why french-speaking scholars dont explain things like that. It’s always a mess!!