Foundations 2: Category Theory
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- เผยแพร่เมื่อ 25 ก.ค. 2024
- In this series we develop an understanding of the modern foundations of pure mathematics, starting from first principles. We start with intuitive ideas about set theory, and introduce notions from category theory, logic and type theory, until we are in a position to understand dependent type theory, and in particular, homotopy type theory, which promises to replace set theory as the foundation of modern mathematics. We also take an interest in computer science, and how to write computer programming languages to formalize mathematics.
In this video, we introduce category theory, the category Set, and the idea of terminal objects.
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great video
thank you for this!
Hey there Richard, I have been watching some of your videos for fun and I must say your way of presenting is superb: clear, rigorous but also exciting. On that point, have you been thinking of turning this into a book? I would definitely like to have all this knowledge in a book format! In any case , very nice job, keep it up :)
You're a great teacher, keep it up :)
Richard, I really admire your work.
Your comparison to wave-particle-duality was wonderful. Every video I have seen so far is excellent and I very much thank you for your superb work!
thank you. this series is awesome, including the thumbnails. i wish my uni took this approach to their math foundations course
Thank you for the great explenation! Everything just clicked in the last part
The usefulness of category theory can even have reverberations in a completely different field, which is metaphysics and philosophy of physics. Because more things can be expressed in structures, pure relationships, without needing concrete atoms of units, than is other way around, this hints that the true nature of reality is fundamentally structuralist. Neither particles nor space nor quantum fields would be the true denominators of reality at the bedrock of it, but simply bare structures (at the absolute border between the material and immaterial, conceptual, but just one insect step over in the material) that over sufficiently many transformations appear to us as these semi-concrete things.
IMHO reality is what you experienced instead of something absolutely universal and objective. On that note, the "persistent" experiences cannot be structureless, otherwise, we wouldn't be able to express them in a logical form. In another word, structuralism is the language of logic. Just some ideas, I don't study philosophy :)
We can describe reality with a useful abstraction and bare structures, that doesn't make reality itself an abstraction
This is quite elegant. But how come Set Theory is not a suitable foundation of mathematics? You only listed Category Theory, Type Theory, and Logic. Is there something these 3 can express that Set Theory cannot?
Thank you RIchard
Amazing channel
Regarding the example for a category without a terminal object (at 32:00), isn't the object on the right a terminal object?
Oops, yes you are right. I meant to draw two arrows from the far left object to the far right one.
It stands to reason that category theory should be able to be subsumed under a wider philosophical category, maybe one that isn't mathematical per see, but distinguishing between what is mathematical vs. philosophical becomes increasingly difficult the more layers one builds around mathematical objects for ultimate perspective (outside looking in) on those objects. In other words, category theory should be a point on the board as well, with a wider reach around it. What is that wider reach? Foundationally, it's philosophical, but after that it's probably perceptual and a function of how our brains perceive the world and the limitations of our cognitive structures to see the bigger picture.
Thanks for these videos. There’s something I don’t understand. I thought a key definition of a function was that for any given element or input in a set it maps to only one element or output in another set. However, you have at the end of the video one set, set 1 with a single element mapping onto multiple other elements, how is it still a function?
You are correct about the definition of a function. But consider the case where X={1,2}. There are two functions from the terminal object (the singleton set) to X. One called u which sends the element of the singleton set to 1. The other function, which I shall call v sends the element of the singleton set to 2. So there are two functions /arrows from the terminal object to X, but each function just involves mapping one element. I meant similarly at the end of the video.
@@RichardSouthwell Thank you so much for responding. I think I understand.
So, in an attempt to wrap my head around this, if you go to a grocery store and have a set of food items and a set of prices, for any object of the set of food items, say banana, for two people shopping that same day and time under the same circumstances that banana should have a function which maps it to a single price. However, if one of them happens to have a coupon, then that represent a different function/rule/operation, mapping the same item from the set of food to a different price from the set of prices. Am I headed in the right direction?
I didn't quite get why there has to be only one arrow from each object into a terminal object. What happens if we have two? Does encoding set elements as arrows somehow break down? Isn't the existence of an object / set with one element enough for such purpose, why this additional constraint?
Good question. The definition of a terminal object as having exactly one arrow into it, from any object is useful in many categories. For example, it implies any two terminal objects are isomorphic. In Set, let T be a set with two elements. T has the feature that there is at least one function into it, from any set. Only singleton sets have the feature that there is exactly one function into them from any set. Hence we need this detailed definition of a terminal object to pick out the singleton set categorically.
Are functions like f∘f and f∘g∘f included in these diagrams?
Is it tempting to use a more convenient "before" operator rather than "after"?
I want to know how to apply category theory in database. That is the main problem in bigdata
anyway, i want to say thankyou to you because bring the basic knowledge for me about category theory
Hope you can create more awesome video like this
Spivak writes about databases quite a lot in his great book 'Category Theory for Scientists'. There is lots of room for you to develop powerful new ideas in this direction. Good luck.
Have I missed something or is do you guys agree that the name 'terminal' is unintuitive? In the example with (A,B,C), (A->B, B->C, A->C), C seems like a 'terminal' to me (in the common sense of the word). Also, does a terminal need to have just one element? I'm quite confused:)
C is a terminal object of that category. I made a mistake. There is also another comment here discussing that.
@@RichardSouthwell Haha, I didn't notice that it didn't violate the definition. Anyway, I basically wanted to ask the same thing as
@Luka Hadžiegrić (i.e. why we define it that way) and you've already answered that. Thank you so much for doing this! :)
Hold on, the "function" from the terminal object to the set Q is a relation not a function. What am I missing?
Hello Richard, really exciting so far, very well articulated. A question: is it necessary for the selected terminal object "1" to launch arrows corresponding to all the elements of all the corresponding sets or is that only an imposed condition on the category if it is to be informationally complete(lossless)?
I'm not sure I understand the question. There are many singleton sets. We can pick any, and use it to pick out other elements. This is a feature of the category Set, and of Set theory. There are plenty of similar categories that don't have terminal objects, or where things work similarly, but not quite the same. But they don't correspond to set theory. I'd say having a terminal object like this is a natural feature of Set. There are other categories where similar things happen.
Is * {} ?
Seems to have the right properties.
Are categories then similar to groups but without the requirement of the inclusion of an inverse?
One object categories (monoids) are like groups without inverses. Categories with many objects have the added complication that one cannot always compose one arrow with another.
On the board of 38:43 ..
Why isn't Z the terminal object in that case ? It seems like it obeys the rules of the terminal object. I understand that Z itself is a set with more than one object which doesn't follow the definition .. but you also said that in category theory we don't care about the content of the individual objects and that they are just dots ...which seems contradictory to me ... Should I treat them as just dots with no internal structure and thus declare Z as a terminal object which you should have done ... Or should I care about their content and thus deciding that Z is not a terminal object ?
Am I missing something ?
Z is terminal. That was a mistake I made. Apologies. This issue is also discussed in other comments.
Same question
Nepali transporont langugangnges .
↔︎ must have an inverse, otherwise it'd be ⇄.
↺
Curry-Howard correspondence
Called Curry-Howard-Lambek when inciuding Category theory, as originally Curry-Howard just related logic and computation (type theory)