These lectures are so great! Usually I put it at 1.5 speed playback; not on this one, but if universality were easy then category theory would not be worth much. Without universality category theory would just be a bunch of jargon. So, thank you Professor Southwell, these lectures are the best I've seen.
Very nice introduction to the concept; I'm actually trying myself to make a course on category theory with the gimmick being that it is "from the ground up". That is, every new concept I introduce is a special case of what came before. However, I'm running into trouble with applying this approach to the concepts of comma categories, functor categories, terminal/initial objects, and universal properties/morphisms. The problem is essentially circular reasoning involved with needing to define universal morphisms themselves in terms of constructs which themselves satisfy universal properties. For example, let's say I introduce universal morphisms as terminal/initial objects in a comma category. However, comma categories are comma objects in the category of categories, which are determined by a kind of limit, which is a universal property defined by a universal morphism, which is a terminal/initial object in a comma category, creating a circular reasoning logic. There doesn't seem to be any way to introduce these concepts in a non-arbitrary way such that this kind of thing does not happen. If you have any suggestions I would love to hear them! Thanks and keep up the great work :)
This is a hard one for me, but I think I understood the main concepts (after watching it 2 times) :-) Your lectures are priceless, thank you very much!
Yea universal morphisms are amongst the deepest ideas in category theory in my opinion. Well worth close study. I've just finished a video on limits, which hopefully sheds a bit more light on them. The Wikipedia article on universal properties is very useful. I think I should have explained them better, by going from familiar examples, rather than diving straight in with the definitions. Anyhow, I think studying how products may be considered to be universal morphisms is a good way to get a handle on them. I'm currently making a video on how universal morphisms relate to adjunctions. There is so much going on with the ideas that it can be bewildering. Especially in cases where there is a terminal morphism to each object in the target category.
@@RichardSouthwell I have finally fully understood the example with product and diagonal functor. Very beautiful construct, thank you for the explanation! I am starting to see the power of universality :-)
@@gucker Yes universality is a really really nice idea. Good job on understanding about the product and diagonal functor. An analogous, but generalized construction is in the up and coming video on limits. You will be able to refer to your understanding of the product to make sense of it. There are so many useful universal morphisms. I'm getting convinced that they may be the most important idea in category theory.
@@RichardSouthwell I for one think it would be great if you could add a 10 min example lecture of examples of initial/terminal morphisms/universal properties now you're back. :) Thanks for the series btw, every one I've seen is great.
Thank you for the video. When I was watching it, I was wondering, is it possible to formulate the homotopy lifting property via the notions of the initial morphism and initial property? If yes, what will be the objects and functors in this case?
Those videos are amazing! thanks a lot. BTW coming from a programming background I better understand the Natural number object as the coproduct of unit and itself (in programming terms a natural can be defined inductively as 'type Nat = 1 + Nat')
Wow thanks for that. Is there a way to define the successor function/arrow in those terms ? Another thing I keep wondering about is how to define a real number object
@@RichardSouthwell In programming terms Successor is just a function: 'Nat -> Nat', In a functional language like Haskell could be defined as 'data Nat = Zero | Succ Nat' (the vertical bar acts as '+', 'Zero' and 'Succ' serve both for tagging and also as functions also called data constructors). in category theory this could mean the second arrow of the coproduct is self referencing and have the required universal property.
@@RichardSouthwell for constructing real numbers I'm not knowledgeable but maybe this is relevant pdfs.semanticscholar.org/8f77/bdeca524acc7aaa824bc5331402aa629490a.pdf (Chapter 2)
@@yassineelouafi5162 Thank you so much for the information. How nice it is to live in times where it's possible to understand such issues properly. Regarding natural numbers, they also have a nice connection with the category of dynamical systems (I will make a video on the Yoneda Lemma that goes over this soon, but many aspects are discussed in Lawvere's 'Conceptual Mathematics'). I also saw there are things called infinesimal objects. At some point I will read about them in parrellel with the interesting document you gave a link to
The initial value theorem (IVT) is dual to the final (terminal) theorem (FVT) Infinity is dual to zero Poles or eigenvalues are dual to zeroes in control theory Duality is being conserved. the fifth law of thermodynamics Same is dual to different Energy is dual to mass -- Einstein Dark energy is dual to dark matter Genes are dual to memes
These lectures are so great! Usually I put it at 1.5 speed playback; not on this one, but if universality were easy then category theory would not be worth much. Without universality category theory would just be a bunch of jargon. So, thank you Professor Southwell, these lectures are the best I've seen.
Such a great lecture! I'm a maths graduate and applying for a PhD and these videos are invaluable to me thank you!
Very nice introduction to the concept; I'm actually trying myself to make a course on category theory with the gimmick being that it is "from the ground up". That is, every new concept I introduce is a special case of what came before. However, I'm running into trouble with applying this approach to the concepts of comma categories, functor categories, terminal/initial objects, and universal properties/morphisms.
The problem is essentially circular reasoning involved with needing to define universal morphisms themselves in terms of constructs which themselves satisfy universal properties. For example, let's say I introduce universal morphisms as terminal/initial objects in a comma category. However, comma categories are comma objects in the category of categories, which are determined by a kind of limit, which is a universal property defined by a universal morphism, which is a terminal/initial object in a comma category, creating a circular reasoning logic.
There doesn't seem to be any way to introduce these concepts in a non-arbitrary way such that this kind of thing does not happen. If you have any suggestions I would love to hear them! Thanks and keep up the great work :)
I am very curious of the answers
This is a hard one for me, but I think I understood the main concepts (after watching it 2 times) :-) Your lectures are priceless, thank you very much!
Yea universal morphisms are amongst the deepest ideas in category theory in my opinion. Well worth close study. I've just finished a video on limits, which hopefully sheds a bit more light on them. The Wikipedia article on universal properties is very useful. I think I should have explained them better, by going from familiar examples, rather than diving straight in with the definitions. Anyhow, I think studying how products may be considered to be universal morphisms is a good way to get a handle on them. I'm currently making a video on how universal morphisms relate to adjunctions. There is so much going on with the ideas that it can be bewildering. Especially in cases where there is a terminal morphism to each object in the target category.
@@RichardSouthwell I have finally fully understood the example with product and diagonal functor. Very beautiful construct, thank you for the explanation!
I am starting to see the power of universality :-)
@@gucker Yes universality is a really really nice idea. Good job on understanding about the product and diagonal functor. An analogous, but generalized construction is in the up and coming video on limits. You will be able to refer to your understanding of the product to make sense of it. There are so many useful universal morphisms. I'm getting convinced that they may be the most important idea in category theory.
@@RichardSouthwell I for one think it would be great if you could add a 10 min example lecture of examples of initial/terminal morphisms/universal properties now you're back. :) Thanks for the series btw, every one I've seen is great.
@@simond9076 Agreed. Thank you for your amazingly instructive videos, Richard. People like you are the best of mankind.
Exponential , diagonal and terminal initial morphisms are defined in this... Thanks
Wow my favorite so far, brilliant. I'm very intrigued and will keep watching.
"This is a little bit abstract" ≃ Understatement of the year.
Ikr, this is a whole new level of abstraction for me XD
Thank you for the video. When I was watching it, I was wondering, is it possible to formulate the homotopy lifting property via the notions of the initial morphism and initial property? If yes, what will be the objects and functors in this case?
Those videos are amazing! thanks a lot. BTW coming from a programming background I better understand the Natural number object as the coproduct of unit and itself (in programming terms a natural can be defined inductively as 'type Nat = 1 + Nat')
Wow thanks for that. Is there a way to define the successor function/arrow in those terms ? Another thing I keep wondering about is how to define a real number object
@@RichardSouthwell In programming terms Successor is just a function: 'Nat -> Nat', In a functional language like Haskell could be defined as 'data Nat = Zero | Succ Nat' (the vertical bar acts as '+', 'Zero' and 'Succ' serve both for tagging and also as functions also called data constructors). in category theory this could mean the second arrow of the coproduct is self referencing and have the required universal property.
@@RichardSouthwell for constructing real numbers I'm not knowledgeable but maybe this is relevant pdfs.semanticscholar.org/8f77/bdeca524acc7aaa824bc5331402aa629490a.pdf (Chapter 2)
@@yassineelouafi5162 Thank you so much for the information. How nice it is to live in times where it's possible to understand such issues properly. Regarding natural numbers, they also have a nice connection with the category of dynamical systems (I will make a video on the Yoneda Lemma that goes over this soon, but many aspects are discussed in Lawvere's 'Conceptual Mathematics'). I also saw there are things called infinesimal
objects. At some point I will read about them in parrellel with the interesting document you gave a link to
i have a question around the 46:00 mark, why does f(b) and f(c) points to a?
Church numerals!
The initial value theorem (IVT) is dual to the final (terminal) theorem (FVT)
Infinity is dual to zero
Poles or eigenvalues are dual to zeroes in control theory
Duality is being conserved. the fifth law of thermodynamics
Same is dual to different
Energy is dual to mass -- Einstein
Dark energy is dual to dark matter
Genes are dual to memes
FVT = Final value theorem.
Initial is dual to final
Start is dual to finish
Beginning is dual to end
Vectors are dual to co-vectors (forms)