What makes this series great is that you, 1) make it simple, 2) show extremes (i.e., category with a single object, many arrows versus many objects, few arrows), 3) Use both positive and negative examples.
Your video is the most straightforward explanation of covariance and contravariance that I have seen. I ran into these words as a brief aside in a programming book, and every tutorial I've looked up dives into implementation details in C# and brushes over the intuition. Thinking of it as a "functor that reverses the direction of the arrows" makes the concept a lot more digestible. Thank you for these videos!
Having studied algebra from a set-theoretic point of view, category theory gives such an elegant new perspective! Thanks for these lessons Prof. Southwell!
Thank you for these lectures! I just started watching along, and I'm very glad to have found a thorough and accessible introduction to category theory. Generally I really appreciate your style, a lot of your notation is the notation that I prefer. One small difference I prefer is to talk about functors, full stop--i.e. "covariant" functors--and then to define the opposite category by reversing the arrows of a category, so that "contravariant functors" are just ordinary functors from the opposite category.
I find it so interesting to take "objects" as atomic objects and think of mathematics in terms of a vast landscape of arrows or maps betwen objects, where the identity map on an object can always be (trivially I guess is the word) defined... But it begs the question, what actually is an "object", a variable in a predicate language? Or a location in memory/in a whiteboard? You could define an object as "a collection of isomorphic representations", but another feature of an "object" is that it always must exist in some context, which is kind of what set theory captures.. Hm its so weird
Yes it is very interesting. I think its good to be flexible about what objects mean, just like one is for elements in a set. There is also an "arrows only" way of setting up category theory, so you don't have objects. You just have identity arrow representing them. You can also extend that idea, to get arrows between arrows between arrows etc., and it gives an approach to higher dimensional category theory (although I have not done any videos on that yet).
I'm glad you find the series helpful. I hope you enjoy it. Also, if you are starting out learning category theory, I recommend Lawvere's book 'Conceptual Mathematics'
Thank you so much for doing this. Yours is the best introduction to category theory on youtube and better than any book I've come across. I'm finally understanding category theory, and liking it. What makes it is your very diagrammatic approach. Once I saw you draw all those arrows on a singleton I finally got what a moniod was. I feel it's how all math should be conceptualized, via diagrammatics. Albeit delimited and computed via relations and equations leading to numbers. I'm looking forward to viewing all your videos. But I'm a bit dismayed at seeing that the last four or so videos from this playlist on category theory have been removed as 'private videos'. What happened there, is that you or youtube who did that? I hope you will upload them again.
Thank you very much for your kind comment. I think so much of maths is actually about pictures (at least the way I think of it), and I don't understand why so many people think it should only be taught using symbols (although symbols obviously have a big part to play). I like to schedule videos for release many months in the future, so those videos will eventually become public.
Thank you for another great video, Prof. Southwell! At about 10:00 , when you're talking about the singleton set terminal object, how is it that there is one function (2 arrows from the set view and one arrow from the category view) from the 2-element set to the singleton set, but 2 functions from the singleton set to the 2-element set (2 arrows from the set view and 2 arrows from the category view)? Like, why/how is there not just one arrow from the singleton to the 2 element set, or why/how is there not 2 arrows from the 2 element set to the singleton in the category view?
@16:50 Why doesn't the number of isomorphisms scale combinatorially with the sizes of sets A and B? It seems we should also have a function mapping T to 1 and F to 2, and inverses of those.
I think they do. I think the number of isomorphisms from one two finite set to another (both having n elements) will be n factorial. You are correct about that other function being an isomorphism too.
What are the terminal objects for the category of Mon, Pre, or Graphs? Or whatever the terminal object from the category of Sets equivalence is. I'd like to draw those examples from the category view. I suppose the part of this that trips me up is how to represent the functor of morphisms.
For some mathematical structures, such as groups, vector spaces and so on, bijective homomorphism immediately means isomorphism because its inverse is also a homomorphism, always. But for other structures, such as topological spaces and differentiable manifolds, a bijective continuous/differentiable map isn't necessarily an isomorphism, because its inverse might not be continuous/differentiable
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What makes this series great is that you, 1) make it simple, 2) show extremes (i.e., category with a single object, many arrows versus many objects, few arrows), 3) Use both positive and negative examples.
This series is astoundingly good. I can't wait to watch more.
Very nice of you to say
Your explanations are very clear and understandable, thank you!
You are very welcome. I enjoyed making these videos, and I am glad if people are enjoying watching them.
Your video is the most straightforward explanation of covariance and contravariance that I have seen. I ran into these words as a brief aside in a programming book, and every tutorial I've looked up dives into implementation details in C# and brushes over the intuition. Thinking of it as a "functor that reverses the direction of the arrows" makes the concept a lot more digestible. Thank you for these videos!
Having studied algebra from a set-theoretic point of view, category theory gives such an elegant new perspective! Thanks for these lessons Prof. Southwell!
I've trying to learn category theory for algebraic topology and your videos have been a "safe haven" for clearer and fun explanations
Thank you for your effort in making these category theory videos.
You are very welcome
Thank you for these lectures! I just started watching along, and I'm very glad to have found a thorough and accessible introduction to category theory. Generally I really appreciate your style, a lot of your notation is the notation that I prefer. One small difference I prefer is to talk about functors, full stop--i.e. "covariant" functors--and then to define the opposite category by reversing the arrows of a category, so that "contravariant functors" are just ordinary functors from the opposite category.
Amazing. You’re an incredible teacher who obviously is passionate about category theory
Your videos give me pure joy. I love your energy. :)
I find it so interesting to take "objects" as atomic objects and think of mathematics in terms of a vast landscape of arrows or maps betwen objects, where the identity map on an object can always be (trivially I guess is the word) defined... But it begs the question, what actually is an "object", a variable in a predicate language? Or a location in memory/in a whiteboard?
You could define an object as "a collection of isomorphic representations", but another feature of an "object" is that it always must exist in some context, which is kind of what set theory captures.. Hm its so weird
Yes it is very interesting. I think its good to be flexible about what objects mean, just like one is for elements in a set. There is also an "arrows only" way of setting up category theory, so you don't have objects. You just have identity arrow representing them. You can also extend that idea, to get arrows between arrows between arrows etc., and it gives an approach to higher dimensional category theory (although I have not done any videos on that yet).
wow, this series is a great find!
Amazing teacher.
I wish I found this series earlier - your explanations are very clear and helpful!
I'm glad you find the series helpful. I hope you enjoy it. Also, if you are starting out learning category theory, I recommend Lawvere's book 'Conceptual Mathematics'
@@RichardSouthwell thanks for the tip!
Thank you for taking your time to share your knowledge, I am really enjoying these videos and I love the way you teach, very calm, clear and precise!
Thank you for your very kind comment. It is very motivating.
Thank you so much for doing this. Yours is the best introduction to category theory on youtube and better than any book I've come across. I'm finally understanding category theory, and liking it. What makes it is your very diagrammatic approach. Once I saw you draw all those arrows on a singleton I finally got what a moniod was. I feel it's how all math should be conceptualized, via diagrammatics. Albeit delimited and computed via relations and equations leading to numbers. I'm looking forward to viewing all your videos. But I'm a bit dismayed at seeing that the last four or so videos from this playlist on category theory have been removed as 'private videos'. What happened there, is that you or youtube who did that? I hope you will upload them again.
Thank you very much for your kind comment. I think so much of maths is actually about pictures (at least the way I think of it), and I don't understand why so many people think it should only be taught using symbols (although symbols obviously have a big part to play). I like to schedule videos for release many months in the future, so those videos will eventually become public.
Great vid
21:18 Definition of Category and Functor
Thank you for another great video, Prof. Southwell! At about 10:00 , when you're talking about the singleton set terminal object, how is it that there is one function (2 arrows from the set view and one arrow from the category view) from the 2-element set to the singleton set, but 2 functions from the singleton set to the 2-element set (2 arrows from the set view and 2 arrows from the category view)? Like, why/how is there not just one arrow from the singleton to the 2 element set, or why/how is there not 2 arrows from the 2 element set to the singleton in the category view?
@16:50 Why doesn't the number of isomorphisms scale combinatorially with the sizes of sets A and B? It seems we should also have a function mapping T to 1 and F to 2, and inverses of those.
I think they do. I think the number of isomorphisms from one two finite set to another (both having n elements) will be n factorial. You are correct about that other function being an isomorphism too.
Finally I understand what "structure preserving mean", I 'm laughing...
thanks!! insightful and useful;
Perfect !
Very glad you like it
Is the contravariant functor related to cohomolgy or a sheaf(sheafification or presheaf)? The reason I ask is they also have reversed arrows.
Yes, they are used in presheafs and in many places.
Wish he gave us practice questions to try!!
What are the terminal objects for the category of Mon, Pre, or Graphs? Or whatever the terminal object from the category of Sets equivalence is. I'd like to draw those examples from the category view. I suppose the part of this that trips me up is how to represent the functor of morphisms.
If CAT is the category of all categories, then is Cat an object in it, and/or is CAT an object in itself?
Thank you sir .
You are most welcome sir.
22:22 functors are defined.
Endomorphisms is efficient in astrophysic algebra.This is efficient
So an isomorphism in the category of sets will be a bijection, but an isomorphism of the category of groups may or may not be a group isomorphism?
For some mathematical structures, such as groups, vector spaces and so on, bijective homomorphism immediately means isomorphism because its inverse is also a homomorphism, always. But for other structures, such as topological spaces and differentiable manifolds, a bijective continuous/differentiable map isn't necessarily an isomorphism, because its inverse might not be continuous/differentiable
the video image is too poor, you need to fix it more