Category Theory is Impossible Without These 6 Things

😎 Become a member to have exclusive access:

🔔 Subscribe:

🥹 Consider supporting us on Patreon:

---

Thanks for watching, and we’ll see you in the next video! 🌟👋

—-

#CategoryTheory #Mathematics #MathExplained #MathTheory #MathematicalStructures #AbstractMath #MathEducation #MathLovers #STEM #Science #Technology #Engineering #Math #PureMath #AppliedMath #CategoryTheoryBasics #MathConcepts #MathFramework #MathConnections #MathInnovations #MathCommunity #MathResearch #MathDiscovery #MathFundamentals #MathApplications #ComputerScience #Physics #Linguistics #QuantumMechanics #Algorithms #ToposTheory #AbstractAlgebra #GroupTheory #Topology #Functor #Morphism #HigherCategoryTheory #MathVideo #EducationalVideo #LearningMath #MathStudents #MathTeachers #MathEnthusiasts #STEMEducation #STEMCommunity #MathPassion #MathGeek #MathNerd #MathChannel #MathTutorial #ExploringMath

—-

Image credits:

Keys

Eilenberg and Mac Lane

William Lawvere

Columbia University

Alexander Grothendieck

[kindreddarkness]

One of the many triumphs of this channel is how rigorously it treats "the roadmap."

Obviously, any one aspect must necessarily lose a lot of resolution... but anyone new to these regions of math were going to forget all of that anyways.

Rather, your depth is the width of the entire territory itself... meaning that I will waste SIGNIFICANTLY less time once I begin the journey of rigor through any one specific aspect.

I really wish more channels would not only take this approach, but build on it. Thank you!

[sourisdiyorigamiandcrafts]

Please make a video on functional analysis

[dailymemigzugxoyditsi]

You guys don't counter question each. Instead asks questions which will lead you guys to the next segment of the video. Which fails the method of Socrates.

[shaneri]

Category theory at last! Amazing!

Thank you dear Di Beo's

[PUP-vc]

This just seems like a concrete math definition of a program.

[vnshngpnt]

Amazing video for such complex topic, really a good middleground for those who is not a professional mathematician but still has basic understanding. Actually love how you managed to explain functor in 1 minute

[scottychen]

I would say it’s the heart of geometry, rather than saying topology is distinct from geometry.

It loses its meaning when it’s distinction from geometry is emphasized:

Important though this first order concept may be for the logician/computer scientist.

The question of what the essence of geometry is, is not answered by geometry itself!

There is true maturity that develops when one uses the ‘system’ to transcend itself:

Into the field of group actions on topological structures.


To emphasize this vocabulary’s distinction from geometry: removes its life.

Though one has the opportunity to indeed

Emphasize the distinction.


@1:30




As someone who does construct characteristic classes in the name of this deeper form of geometry:

Topology: and in particular the study of topological invariants,

It is backwards … or, to me, strange to say it’s done ‘for category theory’:

Unless maybe youre a computer scientist with a different sense of the value this concept has -

The beginning of categroy theory, more naturally, start’s with grothendieck’s

Concept of there being classes of objects: which reinterprets the euler characteristic in many different ways:

e.g. (1) the edges of a 2-manifold’s subdivision are thought of as a vector space….


Although, my appreciation is probably shallow in comparison to a computer scientist:

So maybe eventually in my understanding it’s not truly backwards,


But as a geometer, yes it actually is backwards.


I apologize for accusing you of emphasizing a distinction: I see that’s what the rest of the video is for.


@3:33

One is bringing up noether: one is interested in the property of an abstract ring -

Of every ideal : for Z, nZ, is finitely generated.

This is simply an observation for this particular ring, but is a coveted property for an abstract ring.

There is an implication of there being a readily available construction.


I actually work for a literal construction worker in exploration of this concept in mathematics:

‘Construction’



Obviously, the conclusion here is that you have a computer scientists understanding of topology.

These chains in homology,

Or ‘diagrams’ - which is a degrading term and not representative of the deeper meaning that the diagrams encode - is an excellent technology in understanding the primary structures, … but is only one high-way system on a fertile earfh which is richer a priori:

This is the sheaf-theoretical topology, perhaps one could say in the sense of chebyshev: of there truly being Data of epsilons;

Instead of using the noncommutativr algebra to hide from analysis.

The fertile earth is richer a priori:

And the diagrams is a culture built on a very fine fabric of one facet of wealth held in the kernel a prior.

[neildutoit]

My professor was never a fan of the closure requirement and I agree with his reasoning. An operation is a function. A function is a triple (G, X, Y) where X is the domain and Y is the target. If you say that a group has a set and an operation "on that set", you've already stated the requirement that the domain and target of the function of the operation must be that set (or products of it). There is nothing left to check or require as far as closure is concerned.

[Mahle-fb]

Thank you very much for sharing this information with.

[capitainek]

Wow, your work in explaining such an abstract topic as category theory is incredible! I love the video!

[enpeacemusic]

Right, I'm convinced that this video is made with AI.

To quote: "The group is the set of keys, and the operation is trying to fit the key into the keyhole"

First of all, wrong terminology, second of all, this is obviously a language model misinterpreting what "operation" means in this context, as it is absurd. From that it is also clear that you guys do not know what you are talking about in favour of pumping out as many videos as you can. Also, the description reeks of generative AI, it's genuinely disgusting to read.

Please, do better, fact check the stuff that is in these videos, for fuck's sake.

[catgbo]

1.Calculator
2.Pencil case
3.Pen
4.Math notebook (preferably with grids)
5.Formula sheet
6.Textbook

[logosecho]

Very informative accessible video. Thank you :)

[SobTim-euxu]

My lovely show(I mean an rubric) come back
As always :❤

[error.]

That's the worst explanation of what a group is I've ever heard