Your content is more amazing the more abstract and higher level it goes. There are lots of people that could give a course on elementary complex analysis, only a few can give a course on etale cohomology. If you make a survey, or a poll, you will find that the results always ask for the most elementary topic of the ones listed. This is just because of the pyramid nature of learning mathematics, so there are always more people confused at a lower level than at a higher one, so following popular opinion will not always be the best, it will mean you will not cover the most advanced topics. Despite opinion polls, the highest level topics are the ones that are most valuable to cover, because these cannot be picked up by other means. Thank you again for your amazing lectures.
Yes, there will be more people confused about the lower-level topics but enough people may be more confused about the higher-level topics, so I agree it is valuable to cover the latter.
Dr. Borcherds, I was excited to see you starting a series on category theory. I'm glad to see a new video today and I greatly look forward to the rest of the series. Thanks so much for this.
I am really glad that you came back! It’s the best math channel on youtube! At the same time I am kind of upset that you stopped your topology course. Look forward to see more of the topology lectures. Thank you for your work!
13:16 Forgetful functor 5:32 Fix a group G. Each element of G defines a morphism g: G --> G (which sends each element of x to gx ..it should be ignored?). If G acts on a set, say F(G), then each element of G defines a morphism F(g): F(G) --> F(G). This correspondence is a functor. 19:05 A pre-sheave can be regarded as a functor from category of posets to category of abelian groups.
Wonderful explanations, as usual. I come from computer science (or actually software engineering with an interest in information modeling) and have been circling around category theory for some time while I try to get enough math to get through the usual introductory examples. This series is giving me the courage try reading Mac Lane, Categories for the Working Mathematician. I may not be able to do the exercises, but it may give me more a perspective as I review the other introductions I have been looking at (Leinster, Riehl, Spivak, Ed Morehouse's videos at OPLSS, Awodey, ...)
For a computer-science point of view, take a look at Category Theory for Programmers by Bartosz Milewski. You can find free pdf online. It's more mathematical than you might assume, but takes a Haskell centered point of view. It's a good companion to the pure mathematical texts like MacLane or Wells & Barr (Toposes Triples and Theories).
Functors preserve or map structure from one category to another. In some cases the relationship of something in the first category has to be mapped to something that moves in the opposite direction in the second category. E.g., say there is a morphism representing "up" in the first category but in "opposite world" where "up" is "down" then one would have a contravariant functor. E.g., think of a sociopath that thinks good is bad and bad is good. In their world everything is "backwards" to yours(assuming you are not a sociopath). If you try and map the structure of your world(which has the same underlying objects) to theirs covariant then it won't make sense. You have to reverse the logic and their for have a contravariant functor. Alternatively you can think of having a covariant functor to their op-category. (a contravariant functor is a functor to the op-category) E.g., we see how politicians have a logic where they say one thing but always do the other. They say things like "We will do X to get Y" such as "we will give universal healthcare to all citizens so we can have a healthy society"(this is what they say) but then it actually turns out to do the opposite "we will do not want a healthy society and so will not give universal healthcare"(this is what they do). So to set up a functor to understand what they are saying you map universal healthcare to no universal healthcare and healthy to unhealthy. The morphism is from universal healthcare to healthy and it is mapped to the politicians actions of unhealthy->no universal healthcare. For one morphism one can't see any structure but the idea is that everything is reversed so if you had other morphisms attached it would be traced in the opposite order. For politicians they know how to lie and if you think of "truth" as your first category then mapping the entire structure backwards gives you "false". They speak the "language" of the common person but in their minds they revers the arrows(e.g., they say "I will help you" but in reality they are thinking "I will hurt you"). The common person, generally not being a sociopath, does not understand there is this "opposite world" that exists and so generally is easily manipulated(less so today, I guess but still relatively common). One can see this logic, say U = universal healthcare and H = healthy. Then the logic they say: U->H while what they do is ~H->~U where ~ = not. The idea is that functors map structure of A to structure of B and for some things the structure of B is sort of mirrored in some way of that of B and contravariant specifies that mirroring, although it is somewhat meaningless(it's just a short hand semantic to specify there is a sort of mirrored structure going on). One can always treat contravariant functors as functors(since they are functors) and as covariant functors(since they are covariant functors) but then they would have to specify the codomain as being the opposite category of the codomain of some functor. So in some sense you need a reference frame for covariant and contravariant functors. It only makes sense to talk about them when you have defined the underlying categories. E.g., if you are talking to a sane person you might say you are using a covariant functor to map their logical structure to yours which will be close to the identity functor. But if you are talking with an insane person you might say you have to use a contravariant functor to map their logical structure to yours so you can make sense of what they really mean. The contravariant functor is in reference to a "normal" way something is being done somewhere else. It has no innate meaning in and of itself.
One has to have either of the two. Some part of the audience, the one equipped with great mathematical erudition, must be finding it satisfactory and comforting, while those of us, who lack broad mathematical horizons, see a day or two of hard work to go through the examples thoroughly. Would that be the "working mathematicians"? Complementarity with "idle" is not assumed.
Thank you, Richard, pointing out the contravariant nature of cohomology. Covariant times contravariant = contravariant. +1 times -1 = -1. Richard's lectures should be read before Mac Lane and Moedjick. Lovely comment on Alexander Grothendieck! I don't think he will mention Verdier's thesis. I think Richard would be remembered if he gave an account of mathematics starting from this!
@@carlgauss1702 like I have studied math for like last 20 years ... I still don't know what category theory does ... still don't know what it tells us.
A comathematician is a machine for turning cotheorems into ffee
😂😂😂😂 took me a moment there
This is hilarious. I’ll have to remember that when I take a category theory class.
I am a mathematician. Comathematicians are different from non mathematicians Wiles and especially Langlands.
I know you said that the terminology for homology/cohomology and covariant/contravariant is contradictory.... But are we sure it's not codictory?
Your content is more amazing the more abstract and higher level it goes. There are lots of people that could give a course on elementary complex analysis, only a few can give a course on etale cohomology. If you make a survey, or a poll, you will find that the results always ask for the most elementary topic of the ones listed. This is just because of the pyramid nature of learning mathematics, so there are always more people confused at a lower level than at a higher one, so following popular opinion will not always be the best, it will mean you will not cover the most advanced topics. Despite opinion polls, the highest level topics are the ones that are most valuable to cover, because these cannot be picked up by other means. Thank you again for your amazing lectures.
Yes, there will be more people confused about the lower-level topics but enough people may be more confused about the higher-level topics, so I agree it is valuable to cover the latter.
Dr. Borcherds, I was excited to see you starting a series on category theory. I'm glad to see a new video today and I greatly look forward to the rest of the series. Thanks so much for this.
I'm so glad to see you back again.
I am really glad that you came back! It’s the best math channel on youtube! At the same time I am kind of upset that you stopped your topology course. Look forward to see more of the topology lectures. Thank you for your work!
Wow I didn't expect the 2nd video of this series to come out the very next day, that was a nice surprise:) Thank you for doing this professor!
13:16 Forgetful functor
5:32 Fix a group G. Each element of G defines a morphism g: G --> G (which sends each element of x to gx ..it should be ignored?). If G acts on a set, say F(G), then each element of G defines a morphism F(g): F(G) --> F(G). This correspondence is a functor.
19:05 A pre-sheave can be regarded as a functor from category of posets to category of abelian groups.
5:04 we get a map F(a) from S to S corresponding to each morphism/element a of G. But don't we need F(a) to be a bijection for G to act on S ?
Goddamn, this is how to unwind. Thank you again professor!
Nothing puts me to sleep faster than math videos.
... and I am a mathematician lol
Wonderful explanations, as usual. I come from computer science (or actually software engineering with an interest in information modeling) and have been circling around category theory for some time while I try to get enough math to get through the usual introductory examples. This series is giving me the courage try reading Mac Lane, Categories for the Working Mathematician. I may not be able to do the exercises, but it may give me more a perspective as I review the other introductions I have been looking at (Leinster, Riehl, Spivak, Ed Morehouse's videos at OPLSS, Awodey, ...)
For a computer-science point of view, take a look at Category Theory for Programmers by Bartosz Milewski. You can find free pdf online. It's more mathematical than you might assume, but takes a Haskell centered point of view. It's a good companion to the pure mathematical texts like MacLane or Wells & Barr (Toposes Triples and Theories).
If you dont know Lawvere's book, Im sure youll like it
Conceptual Mathematics
This series is amazing thank you
Functors preserve or map structure from one category to another. In some cases the relationship of something in the first category has to be mapped to something that moves in the opposite direction in the second category. E.g., say there is a morphism representing "up" in the first category but in "opposite world" where "up" is "down" then one would have a contravariant functor. E.g., think of a sociopath that thinks good is bad and bad is good. In their world everything is "backwards" to yours(assuming you are not a sociopath). If you try and map the structure of your world(which has the same underlying objects) to theirs covariant then it won't make sense. You have to reverse the logic and their for have a contravariant functor. Alternatively you can think of having a covariant functor to their op-category. (a contravariant functor is a functor to the op-category)
E.g., we see how politicians have a logic where they say one thing but always do the other. They say things like "We will do X to get Y" such as "we will give universal healthcare to all citizens so we can have a healthy society"(this is what they say) but then it actually turns out to do the opposite "we will do not want a healthy society and so will not give universal healthcare"(this is what they do). So to set up a functor to understand what they are saying you map universal healthcare to no universal healthcare and healthy to unhealthy. The morphism is from universal healthcare to healthy and it is mapped to the politicians actions of unhealthy->no universal healthcare.
For one morphism one can't see any structure but the idea is that everything is reversed so if you had other morphisms attached it would be traced in the opposite order. For politicians they know how to lie and if you think of "truth" as your first category then mapping the entire structure backwards gives you "false". They speak the "language" of the common person but in their minds they revers the arrows(e.g., they say "I will help you" but in reality they are thinking "I will hurt you"). The common person, generally not being a sociopath, does not understand there is this "opposite world" that exists and so generally is easily manipulated(less so today, I guess but still relatively common).
One can see this logic, say U = universal healthcare and H = healthy. Then the logic they say: U->H while what they do is ~H->~U where ~ = not.
The idea is that functors map structure of A to structure of B and for some things the structure of B is sort of mirrored in some way of that of B and contravariant specifies that mirroring, although it is somewhat meaningless(it's just a short hand semantic to specify there is a sort of mirrored structure going on). One can always treat contravariant functors as functors(since they are functors) and as covariant functors(since they are covariant functors) but then they would have to specify the codomain as being the opposite category of the codomain of some functor.
So in some sense you need a reference frame for covariant and contravariant functors. It only makes sense to talk about them when you have defined the underlying categories. E.g., if you are talking to a sane person you might say you are using a covariant functor to map their logical structure to yours which will be close to the identity functor. But if you are talking with an insane person you might say you have to use a contravariant functor to map their logical structure to yours so you can make sense of what they really mean. The contravariant functor is in reference to a "normal" way something is being done somewhere else. It has no innate meaning in and of itself.
One has to have either of the two. Some part of the audience, the one equipped with great mathematical erudition, must be finding it satisfactory and comforting, while those of us, who lack broad mathematical horizons, see a day or two of hard work to go through the examples thoroughly. Would that be the "working mathematicians"? Complementarity with "idle" is not assumed.
yeeeeeeeeee
Thank you very much professor, Im complementing my abstract algebra course with your lectures. 😁
Does anyone have that original reference he mentioned in the beginning?
Thank you for your insights
A coconut is just a nut.
Christmas came wayyyy earlier this year baby
I am going through this now. He is talking of functors, but he means covariant functors.
Thank you, Richard, pointing out the contravariant nature of cohomology. Covariant times contravariant = contravariant. +1 times -1 = -1. Richard's lectures should be read before Mac Lane and Moedjick. Lovely comment on Alexander Grothendieck! I don't think he will mention Verdier's thesis. I think Richard would be remembered if he gave an account of mathematics starting from this!
Ben? Beeeeeennnn!
Ich Liebe dich
Abstract nonsense... and I love it
@@carlgauss1702 like I have studied math for like last 20 years ... I still don't know what category theory does ... still don't know what it tells us.
not at all :)
you can see Lawvere's book*
and learn the profound "practicality" of cat. theory
* Conceptual mathematics
(really great)