This is gonna be a great series. I don't know if the lectures are all recorded as of now, but in the lecture about natural transformations it'd be nice to see some intuition behind why natural transformations are "natural" in the non-technical sense of the word.
If I may to throw some of my intuition - natural transformations are like polymorphic functions. Natural transformation α:F -> G gives a morphism αC: FC->GC for each object C. But these morphisms are "defined in the same way", the definition of αC doesn't use any properties of C, it is just a placeholder. For example, function that takes a first element from a tuple is polymorphic: we have πC: C×C->C for each C, where C is just a placeholder, the definition works for any C. The naturality square says exactly that if we change C to some another D, then the result will be analoguous.
Really happy to catch this series as it begins. Got really into Category theory a few months back but got busy and haven't revisited. Perfect time to start following along with this
I won't go against MacLane's book (it is a classic one, no doubt), but I would also never recommend it for the first-time learner. I learned category theory for more than a year now, but yet I still won't say I feel comfortable while reading MacLane. (I think it has something to do with the lack of consistency of notations to the modern convention, like he uses lowercase for objects, while the modern convention is to use capital letters.) A more elementary (and short and free on arxiv) book for introduction I recommend would be Tom Leinster's "Basic Category Theory", which have some of the "nicest" treatment to (co)limits, relationship between (co)limits, representables, and adjunction, as well as an abundant amount of examples, and a complete guided proof of the (contravariant) Yoneda lemma. It also ends with an appendix proving the GAFT. The only downside is that it covers less than MacLane. Also, welcome back.
@@Gilh Yes, but you might find the beginning tediously boring if you are already advanced enough. (I think they designed the book for high school students, if I remember correctly; I certainly did not read the whole thing.)
@@shiina_mahiru_9067 it starts slow, then (subtly) blows up around session 10-11 IIRC. I found it to have a more of a creative/exploritory/artistic style, which I very much like.
Greetings from Mexico! I'm sending your videos to my classmates in my university, they have been incredibly helpful, thank you. Y aprovechando, un saludote a toda la raza de ESFM que la sigue cotorreando.
Welcome back Dr. Borcherds, glad to see you :) I would like to mention that, following myself watching your lectures in algebraic geometry, I am taking a course in algebraic geometry this semester at my institution (special topic), and it's great fun so far! Thank you for the introduction to this wonderful field!
Well I don't have the level to understand all of that yet, but I'm really are glad that you are on youtube, giving free access to knowledge to everyone, even at best level
I liked Steve Awodey's Category Theory for an intro. Did an individual instruction in undergrad on the topic, we found Awodey to be a bit more gentle and intro than MacLane.
I feel dumb for asking this, but I can't get my head around the definition of the morphism: I watched this video several times, and searched in the internet for examples, and even though I think I found a counterexample that is comprehensible to me, I still have trouble to process the fact that the class of morphism(X,Y) does not consist necessarily of functions. It may be because I have a poor background in math in general ( I only studied real analysis, linear algebra and some very basic set theory). Can anyone explain in a more intuitive way what a morphism is? I'm comfortable with defining the "set" of objects without using cardinals of classes, but rather as any "collection" of things that satisfy some properties. However, I really cannot wrap my head around the definition of a morphism. Maybe it would be easier if the notation used for indicating morphism wasn't the same used for functions (f:A--->B to indicate a morphism from A to B, is the same as the used for indicating a function from A to B, and the composition uses the same symbol). I really hope someone can help me because I feel like this subject could be very interesting, even for someone who doesn't have a large background in math. Thanks in advance for anyone answering to this comment
there is a great intro to categories by Lawvere and Schanuel (a book, Conceptual Mathematics). By the way, that book is a great and fresh kind of new intro to maths
21:25 when you're talking about this map from Z to Q, we are talking about a homomorphism right? Because else I don't see how this would be an epimorphism.
We are talking about the inclusion map i : Z -> Q in the category of rings. I'll now prove that it is an epimorphism. The ring homomorphism i is an epimorphism iff for any ring homomorphisms f : Q -> R and g : Q -> R we have f = g whenever f i = g i, where R is an arbitrary ring. Let x be any rational number. Now x = p/q for some integers p and q where q is non-zero. Since i(p) = p and i(q) = q we have f(p) = f(i(p)) = g(i(p)) = g(p) and f(q) = f(i(q)) = g(i(q)) = g(q). Now f(x) = f(p/q) = f(p)f(1/q) = f(p)/f(q) = g(p)/g(q) = g(p)g(1/q) = g(p/q) = g(x). Thus f and g agree on all rational numbers which is the whole domain of the functions which means that f = g. QED
Good to have more channels deal with this topic. Repetition alone in too few sources is not enough for me. Could have been better if you defined sections and retractions too, and then section&retraction does imply isomorphism always. credit to the book you chose to not mention: Abstract And Concrete Categories a.k.a The Joy Of Cats. This reminds me that the name Categories For The Working Mathematician always bothered me because it implies that it omits some topics because they are not considered practical enough by its author. Both books seem to contain topics omitted by each other, but ACC does seem more encyclopedic and thorough to me.
[ 03:16 ] Does this have something to do w the set of all sets being required by def to contain a set which does not contain itself, as well as a set which does? As well as *the set of all sets* itself...
Pretty much, yes; in standard set-theoretical foundations there is the Axiom of Comprehension, which says that if A is a set and P is a property, then the elements of A that satisfy P form a set, which we denote by {a in A : P(a)}. The axiom of comprehension is why the collection of all sets - denoted by U - cannot be a set; if it were a set then we could consider A = U, allowing us to form sets of the form {a in U : P(a)} = {a : P(a)} - this is called the "Axiom of Unrestricted Comprehension", and intuitively it says that anything we can comprehend forms a set (precisely, any property in e.g. first-order logic gives rise to a set), but the axiom of unrestricted comprehension leads to a contradiction via Russel's paradox, and hence U cannot be a set. (For those who aren't familiar with Russel's paradox, it goes as follows: using unrestricted comprehension, define the set R = {a : a is not an element of a}, or in set-theory notation, R = {a : a ∉ a}. Then the contradiction is that the answer to the question "is R an element of R?", is "yes and no"; if R ∉ R then, by definition, R ∈ R, and similarly if R ∈ R then, by definition, R ∉ R, and so R ∈ R if and only if R ∉ R.)
@@VideographerExperience collections that are too big to be sets are called proper classes, and cardinals simply do not (cannot) apply to them because they are too huge :D
@@VideographerExperience yes; for example consider the collection O of all sets with one element (which is a proper class) - any set A whatsoever can be packaged in curly braces as {A}, which is then an element of O. That is, the map A |-> {A} gives an injection from ALL SETS to O, so the class O is at least as big as the entire universe of sets
Whenever he says what the morphisms are there’s always a way to tell if they’re equal. For the example of the group, the morphisms are group elements, and group elements can be equal (e.g. in the group axiom of associativity). In the poset example the morphisms (corresponding to inequalities) are elements of sets with just one element, so if you end up with the same inequality they both correspond to an element of the same set of morphisms, but that set only had one element so they have to correspond to the same element of this one element set of morphisms, i.e. the same morphism. Hopefully there’s an easier way to explain this.
@@netrapture thanks! I understand this, but the point of defining a monomorphism and epimorphism is to avoid mentioning the underlying elements of the objects in consideration, right? But equality of morphisms is implicitly using the underlying elements itself, which is what prompted my comment
@@JPK314 I think you're taking the wrong viewpoint. Morphisms are part of the data that makes a category. They are given when the category is born. So, when you're not dealing with concrete structures like Sets, Groups..., morphisms are primitive and you always know whether two of them are the same.
@@yaeldillies I think this is the correct answer. Proving equality requires a definition of equality; we can state things involving that equality regardless of definition as long as it satisfies reflexivity and transitivity
A problem exists. How should we solve it? 1. Carefully restrict our actions so the problem doesn't manifest. 2. Create a new terminology, giving up centuries of difficult work in set theory. 3. Create a system of ever-expanding universes. 4. Ignore the problem.
I was worried that maybe you quit teaching, I'm happy you come back 24:28 Epimorphism is not surjective in general. i:Z -->Q is epimorphism. In fact, for any ring homomorphisms f,g:Q-->A with fi=gi and any rational n/m, f(n/m)=fi(n)fi(m)^{-1}=gi(n)gi(m)^{-1}=g(n/m) and thus f=g on rationals Q. 12:22 The forth example of category, I am not sure why the set of morphisms between two objects n,m is Mat(n,m). Theme of category ignores inner structure of sets...but a natural number mean something like a dimension of vector space on some field??
A quite well known professor once referred in a lecture to this subject as ``general nonsense". The very little I learned about it prompted the following question. Is there any use for category theory? I do not mean that in the sense that one might ask if there is any use for topology. I mean it in the following sense. What is to be gained by identifying groups and sets as different examples of categories? That is, what information does one get about a group or groups in general from identifying it as a member of the category of groups? It seems to me that, for example, algebraic topology makes explicit use of what is known about algebraic structures to extract information about topological spaces. Can any result of algebraic topology, for example, be stated in terms of a result from general category theory? I like your lectures, even with the mistakes. Perhaps you know which mathematician is credited with saying, ``I don't play chess, because in chess you are not allowed to take a move back. In mathematics, you are allowed to take back as many moves as you like."
If you see the Lawvere's book you can actually feel a new and profound view of maths. There is a huge field of applications: Works in theor. physics like that of Isham et al .. The Topos of Music... Software engineering... etc. It is kind of (r)evolution of math foundations
Try a book (Lawvere, Conceptual Mathematics) very basic about this theme; it is a new form to see maths... profound, and Lawvere explains it with specific and simple examples really useful
Welcome back to RUclips!
He is back!
Legends said he will be back someday
I'm glad you're back - I really enjoy your lectures.
I'm also glad his back
@@noellundstrom7447 Mine too
I'm happy about this as well!
@@noellundstrom7447 @Geoffrey Churchill Hahaha... Well played. I probably shouldn't type things late at night when I'm tired and miss stuff like that.
My heart missed a beat when I saw your new upload. The return of the king.
Life feels more complete with prof borcherds back
European Civil War? That's an interesting description.
Hamsterxxx
He must be very pro EU
This whole channel is like finding a gold mine in your back garden. Absolutely brilliant!
Good to see you again professor. Thank you for everything you have done for humanity
When the world needed him most! Or rather, when I was just pondering how I would represent the change of variables theorem with a commutative diagram.
Ahhhhhhhh! I am waiting every day for half a year! This channel helps me get a master degree in math!
This is gonna be a great series. I don't know if the lectures are all recorded as of now, but in the lecture about natural transformations it'd be nice to see some intuition behind why natural transformations are "natural" in the non-technical sense of the word.
If I may to throw some of my intuition - natural transformations are like polymorphic functions. Natural transformation α:F -> G gives a morphism αC: FC->GC for each object C. But these morphisms are "defined in the same way", the definition of αC doesn't use any properties of C, it is just a placeholder. For example, function that takes a first element from a tuple is polymorphic: we have πC: C×C->C for each C, where C is just a placeholder, the definition works for any C. The naturality square says exactly that if we change C to some another D, then the result will be analoguous.
"I'm not going to say whether these are the identities of A or B because I'll get them the wrong way round if I do." LOL.
Dear Prof., glad you are back!
Really happy to catch this series as it begins. Got really into Category theory a few months back but got busy and haven't revisited. Perfect time to start following along with this
I won't go against MacLane's book (it is a classic one, no doubt), but I would also never recommend it for the first-time learner. I learned category theory for more than a year now, but yet I still won't say I feel comfortable while reading MacLane. (I think it has something to do with the lack of consistency of notations to the modern convention, like he uses lowercase for objects, while the modern convention is to use capital letters.) A more elementary (and short and free on arxiv) book for introduction I recommend would be Tom Leinster's "Basic Category Theory", which have some of the "nicest" treatment to (co)limits, relationship between (co)limits, representables, and adjunction, as well as an abundant amount of examples, and a complete guided proof of the (contravariant) Yoneda lemma. It also ends with an appendix proving the GAFT. The only downside is that it covers less than MacLane.
Also, welcome back.
Lawvere and Schanuel's "Conceptual Mathematics" is also a beautiful exposition to the subject.
@@Gilh Yes, but you might find the beginning tediously boring if you are already advanced enough. (I think they designed the book for high school students, if I remember correctly; I certainly did not read the whole thing.)
@@shiina_mahiru_9067 it starts slow, then (subtly) blows up around session 10-11 IIRC. I found it to have a more of a creative/exploritory/artistic style, which I very much like.
Greetings from Mexico! I'm sending your videos to my classmates in my university, they have been incredibly helpful, thank you.
Y aprovechando, un saludote a toda la raza de ESFM que la sigue cotorreando.
Joder, claro que sí
I was just talking with my professor in my commutative algebra class about category theory, what perfect timing :)
I guess I'll learn category theory now :')
As a undergraduate student who is interested in category this is mind blowing for them .... lots of respect to you professor from India
Welcome back Dr. Borcherds, glad to see you :)
I would like to mention that, following myself watching your lectures in algebraic geometry, I am taking a course in algebraic geometry this semester at my institution (special topic), and it's great fun so far! Thank you for the introduction to this wonderful field!
We missed you!!! 🤗
My fav topic. Thank you very much Prof. Borcherds.
Nice, I just started to study topos theory for my SPA's and that involves categories a lot. Good to see prof Borcherds making videos again.
Woohoo. Seems my lunches for the next weeks will be improved again!
Well I don't have the level to understand all of that yet, but I'm really are glad that you are on youtube, giving free access to knowledge to everyone, even at best level
there is a great intro to categories by Lawvere and Schanuel (a book, Conceptual Mathematics)
!! Excellent that you have come back. The fact that university level or beyond teachers so freely teach on RUclips is amazing
So glad you're well and doing lectures again!!, and category theory... Thank you so much professor ☺️
*Category Theory!!!*
[I'm infinitely excited]
Glad to see you back Professor!
It's amazing to see you back! We really missed you !!
THE KING RETURNS
The king has returned!
These lectures are brilliant - thanks for teaching me
Glad to have you back Professor! Are you planning to finish the algebraic topology series?
Just got into your channel yesterday; what a nice surprise.
what a nice surprise! :)
You are back!
I liked Steve Awodey's Category Theory for an intro. Did an individual instruction in undergrad on the topic, we found Awodey to be a bit more gentle and intro than MacLane.
So excited for this course!!! Such a great teacher
9:42 is not anti-symmetry an axiom of posets? I.e. if a,b in S, then if a
The video that I was looking for!
Welcome back professor.
Welcome back :) I was wondering where you were
the beast returns
A wonder math and Professor. Interested persons in this type math are fortunate to have a chance on yt.
Hooray!!! He's back!
Welcome back professor
the man is back
This is amazing, I was just starting to explore more this specific subject. Thanks a lot Prof!
live long and prosper!
nice to see you again !
Thank you Professor!
Thank you Prof., for your Lectures on Umbral Moonshine!♤♡
Let's goooo!
nice lecture for me to review category theory!
I feel dumb for asking this, but I can't get my head around the definition of the morphism: I watched this video several times, and searched in the internet for examples, and even though I think I found a counterexample that is comprehensible to me, I still have trouble to process the fact that the class of morphism(X,Y) does not consist necessarily of functions. It may be because I have a poor background in math in general ( I only studied real analysis, linear algebra and some very basic set theory). Can anyone explain in a more intuitive way what a morphism is? I'm comfortable with defining the "set" of objects without using cardinals of classes, but rather as any "collection" of things that satisfy some properties. However, I really cannot wrap my head around the definition of a morphism. Maybe it would be easier if the notation used for indicating morphism wasn't the same used for functions (f:A--->B to indicate a morphism from A to B, is the same as the used for indicating a function from A to B, and the composition uses the same symbol). I really hope someone can help me because I feel like this subject could be very interesting, even for someone who doesn't have a large background in math. Thanks in advance for anyone answering to this comment
there is a great intro to categories by Lawvere and Schanuel (a book, Conceptual Mathematics).
By the way, that book is a great and fresh kind of new intro to maths
@@unplandivino Thanks for your answer, I will definitely try it
21:25 when you're talking about this map from Z to Q, we are talking about a homomorphism right? Because else I don't see how this would be an epimorphism.
We are talking about the inclusion map i : Z -> Q in the category of rings. I'll now prove that it is an epimorphism.
The ring homomorphism i is an epimorphism iff for any ring homomorphisms f : Q -> R and g : Q -> R we have f = g whenever f i = g i, where R is an arbitrary ring.
Let x be any rational number. Now x = p/q for some integers p and q where q is non-zero. Since i(p) = p and i(q) = q we have f(p) = f(i(p)) = g(i(p)) = g(p) and f(q) = f(i(q)) = g(i(q)) = g(q). Now f(x) = f(p/q) = f(p)f(1/q) = f(p)/f(q) = g(p)/g(q) = g(p)g(1/q) = g(p/q) = g(x). Thus f and g agree on all rational numbers which is the whole domain of the functions which means that f = g. QED
Nice lecture!
Yes! Borcherds returns!! :D
The best thing of my day!
Thanks for the video ! Did someone get why the injection from Z to Q is an epimorphism ?
Good to have more channels deal with this topic.
Repetition alone in too few sources is not enough for me.
Could have been better if you defined sections and retractions too, and then section&retraction does imply isomorphism always.
credit to the book you chose to not mention:
Abstract And Concrete Categories a.k.a The Joy Of Cats.
This reminds me that the name Categories For The Working Mathematician always bothered me because it implies that it omits some topics because they are not considered practical enough by its author.
Both books seem to contain topics omitted by each other, but ACC does seem more encyclopedic and thorough to me.
[ 03:16 ] Does this have something to do w the set of all sets being required by def to contain a set which does not contain itself, as well as a set which does? As well as *the set of all sets* itself...
Pretty much, yes; in standard set-theoretical foundations there is the Axiom of Comprehension, which says that if A is a set and P is a property, then the elements of A that satisfy P form a set, which we denote by {a in A : P(a)}. The axiom of comprehension is why the collection of all sets - denoted by U - cannot be a set; if it were a set then we could consider A = U, allowing us to form sets of the form {a in U : P(a)} = {a : P(a)} - this is called the "Axiom of Unrestricted Comprehension", and intuitively it says that anything we can comprehend forms a set (precisely, any property in e.g. first-order logic gives rise to a set), but the axiom of unrestricted comprehension leads to a contradiction via Russel's paradox, and hence U cannot be a set.
(For those who aren't familiar with Russel's paradox, it goes as follows: using unrestricted comprehension, define the set R = {a : a is not an element of a}, or in set-theory notation, R = {a : a ∉ a}. Then the contradiction is that the answer to the question "is R an element of R?", is "yes and no"; if R ∉ R then, by definition, R ∈ R, and similarly if R ∈ R then, by definition, R ∉ R, and so R ∈ R if and only if R ∉ R.)
@@schweinmachtbree1013 I wonder what the smallest collection of elements that is not a set is. The cardinality of the smallest "nonset"
@@VideographerExperience collections that are too big to be sets are called proper classes, and cardinals simply do not (cannot) apply to them because they are too huge :D
@@schweinmachtbree1013 Aha. That fully makes sense. Perhaps these beasts must be non-denumerably infinite.
@@VideographerExperience yes; for example consider the collection O of all sets with one element (which is a proper class) - any set A whatsoever can be packaged in curly braces as {A}, which is then an element of O. That is, the map A |-> {A} gives an injection from ALL SETS to O, so the class O is at least as big as the entire universe of sets
How do you define equality of morphisms without referring to the underlying elements of each object?
Whenever he says what the morphisms are there’s always a way to tell if they’re equal. For the example of the group, the morphisms are group elements, and group elements can be equal (e.g. in the group axiom of associativity). In the poset example the morphisms (corresponding to inequalities) are elements of sets with just one element, so if you end up with the same inequality they both correspond to an element of the same set of morphisms, but that set only had one element so they have to correspond to the same element of this one element set of morphisms, i.e. the same morphism. Hopefully there’s an easier way to explain this.
@@netrapture thanks! I understand this, but the point of defining a monomorphism and epimorphism is to avoid mentioning the underlying elements of the objects in consideration, right? But equality of morphisms is implicitly using the underlying elements itself, which is what prompted my comment
@@JPK314 I think you're taking the wrong viewpoint. Morphisms are part of the data that makes a category. They are given when the category is born. So, when you're not dealing with concrete structures like Sets, Groups..., morphisms are primitive and you always know whether two of them are the same.
@@yaeldillies I think this is the correct answer. Proving equality requires a definition of equality; we can state things involving that equality regardless of definition as long as it satisfies reflexivity and transitivity
A problem exists. How should we solve it?
1. Carefully restrict our actions so the problem doesn't manifest.
2. Create a new terminology, giving up centuries of difficult work in set theory.
3. Create a system of ever-expanding universes.
4. Ignore the problem.
(I'm certainly not complaining, I'm greatly enjoying this introduction!)
Wow long time!! Algebraic Topology though 🥵
Welcome back
I was worried that maybe you quit teaching, I'm happy you come back
24:28 Epimorphism is not surjective in general.
i:Z -->Q is epimorphism. In fact, for any ring homomorphisms f,g:Q-->A with fi=gi and any rational n/m, f(n/m)=fi(n)fi(m)^{-1}=gi(n)gi(m)^{-1}=g(n/m) and thus f=g on rationals Q.
12:22 The forth example of category, I am not sure why the set of morphisms between two objects n,m is Mat(n,m). Theme of category ignores inner structure of sets...but a natural number mean something like a dimension of vector space on some field??
Interesting, thank you.
1:30 this make me laugh more than i should have
I like the dry humour.
So when talk about f º Id = f, what does this "=" actually mean? they are obviously not the same arrow.
Or is this how "=" is defined?
Usual notion of equality -f . Id can be substituted into any true mathematical statement about f to still get a true mathematical statement
Well come, mister Borcherds.
Yay! Yippee! Hooray!
A quite well known professor once referred in a lecture to this subject as ``general nonsense". The very little I learned about it prompted the following question. Is there any use for category theory? I do not mean that in the sense that one might ask if there is any use for topology. I mean it in the following sense. What is to be gained by identifying groups and sets as different examples of categories? That is, what information does one get about a group or groups in general from identifying it as a member of the category of groups? It seems to me that, for example, algebraic topology makes explicit use of what is known about algebraic structures to extract information about topological spaces. Can any result of algebraic topology, for example, be stated in terms of a result from general category theory?
I like your lectures, even with the mistakes. Perhaps you know which mathematician is credited with saying, ``I don't play chess, because in chess you are not allowed to take a move back. In mathematics, you are allowed to take back as many moves as you like."
If you see the Lawvere's book you can actually feel a new and profound view of maths.
There is a huge field of applications:
Works in theor. physics like that of Isham et al ..
The Topos of Music...
Software engineering...
etc.
It is kind of (r)evolution of math foundations
>European civil war going on at the time
Didn't expect a Keynes reference in a category theory video.
Ye!!!!
大好き！
Instant like
YES!!!!!!!!!!!
Clearly ...
Why am I watch this when I don't know anything about it
Try a book (Lawvere, Conceptual Mathematics) very basic about this theme;
it is a new form to see maths... profound, and Lawvere explains it with specific and simple examples really useful
🙂
yeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee
Missed your yeeees
It's good to have ye back
ye
Yessiree
@@sambachhuber9419 who would have thought I’d see you here
😁
European civil war…aka: World war II? British humour at its best
"European civil war" lol
If you were in front of me, I would give you a big hug.