I've been curious about category theory for a few years, and I've begun reading / watching various introductions. This is by far the most lucid, most helpful, and most sensitive to the student's likely concerns. Please continue the series! Please publish your category theory lectures as a book!
I'm a retired Mathematician currently working with impoverished children at an Elementary School. I conduct some of their Interventions. I find your lecturing manner quite approachable and well paced. Please do continue in the same vein. Most enjoyable!
18:07 There's technical issue relating to requirement 3 (that distinct hom sets are disjoint). SET doesn't naturally have this property, a function doesn't know where it's co-domain is. For example, the identity on the natural numbers id: N-> N is the same (as a set) as the identity injection into the integers id: N -> Z. So, id ∈ Hom(N,N) and id∈Hom(N,Z), violating requirement 3. The issue is easily resolved, for instance by flagging each function with it's co-domain, which is what is done implicitly rather than explicitly. This (non)issue occurs in other example categories, and can be similarly resolved.
The way you explain it seems to me very simple, synthetic and right to the core of the theory at the tranquil pace which allowed me to follow your lecture without rewinding or stopping the video at any time, thanks.
Dr. Roman, your lectures and book on category theory are the best! I have been looking for weeks for a good introduction to category theory, and here you are!
For once at the end of a category theory lecture I feel like I've learned something more than just "did you notice isomorphisms are isomorphic to isomorphisms across different fields of math?"
Please continue this series. I have been a fan of your exposition since coming across a 1st edition copy of Intro to Mathematical Finance (I need to see what changes in the 2nd edition!), and this video as well as my own general fuzziness on Category Theoretic approaches is very helpful and illuminating. I think that you do a very good job of not being bogged down in the details but also being thorough in the procedure and nuances. I really like that you don't gloss over the fact that category theorists seemingly say the same thing in a number of different ways. I will be picking up the pdf of the book for certain now. Honestly, if you could find the time in your retirement (and the enjoyment or will!) to do a series for other topics such as the advanced algebra, I think that you may find yourself with another passion that is no so time consuming as authoring books.
+Steven Roman - Mathematics I've also been looking for lectures in abstract algebra. The best I've found so far are on TH-cam by Benedict Gross. Imho Gross is to abstract algebra (almost) what Gilbert Strang is to linear. Would you consider doing universal algebra and/or model theory?
Abstract algebra would be so Great! Especially with some Galois theory. I studied abstract algebra 8 years ago. I still enjoy it but it's getting a little rusty and some videos featuring your OutStanding lecture style would just be wonderful.
Please continue the series. A few more videos would help others forming an opinion. Personally I like your style. Also, There are only a few series on the subject so this new series would be more than welcome. Also, please when the time comes (ex: functors) use when possible also examples involving graphs and types. And if you ever see OOP connections please state them, usually CT is related to FP only.
'I think in all classes in mathematics, as a student one has to have a little bit of faith that what's coming will be important at some point even if the first few days or week of course seems maybe a little bit unmotivating' I love you. lol
Great lecture sir.... Respect from India..... Category theory was something I always wanted to kick start with..... Thank u very much for these lectures.....
Thank you very much, Mr. Steven Roman. I am major in computer science. I am very interesting in type theory and category theory. No other videos can explain category theory than your lectures!
What a great lecture! Thank you VERY much Dr. Roman! There are so few good videos targeting this level of mathematics. It seems all you can find is elementary math or research level math. I have a masters but I never studied category theory. This lecture was just perfect for me. I will now watch the second one immediately. :)
Jacob G. If you are interested in my book, it is available as an e-book through my web site www.sroman.com. If it is ever published by a commercial publisher, it will probably not appear for at least 9 months.
Hi Steven, I bought your book "advanced linear algebra" long ago ... great reference .... thanks ... I hope theses lectures help me to solve problems in molecular dynamics ...
Excellent! I'm reading Steve Awodey's book right now (in preparation for an algebraic geometry course in the Spring). I am very intrigued to read yours when it comes out. Looking forward to it :)
I got a little bit confused, what are the "elements" of Mor(sets), are they the sets Hom(A,B) for any A,B in Obj(sets) or are they the functions from one to another object in Obj(sets)? Great lecture btw, please don't stop making them!
Thank you very much for these videos and for your book "An Introduction to Category Theory". I've read a lot of such material and this is the most lucid introduction to the subject I've yet come across. It's up there with Bill Lavere's "Conceptual Mathematics". I believe, though, you've missed a couple of typos in the book: p. 10, Example 11 - the second and third paragraphs are near duplicates. I believe the 3rd is a correction of the 2nd, which you've omitted deleting; and, p.14 point 2) of the definition of left -invertible has fL: A->B which should be fL:B ->A. Thanks again.
Hi sir, Quick question. Because of 3) (Any 2 Hom Sets in the Category are disjoint), isn't Mor(C) = \XOR Hom(c_{i},c_{j}) , where , c_{i} , c_{j} belong to Obj(C) , and \XOR represents the Disjoint Union ?
Hello Professor Roman, I just wanted to say I enjoy your lectures very much -- and I am very interested in Category Theory, and they may persuade me to purchase your book. If you are able, please keep these lectures going, and thank you!
(Potentially silly) Question : After you clearly explained the difference between class and set, and then went on to clarify that in C.T. the collections of all objects in a Category are considered to be a class and the homsets are considered to be a set, why do you write with no hesitation "*class* of all morphisms" at 22:04? Is it because in total the class of all objects times the sets of all their homsets gives again a class? Or to put it in a very simplistic way: class x set = class ?? I apologize if this is a silly question
For Matr_F, I would have expected obj(Matr_F) to consist of all n-tuples of integers. How is a matrix going to map integers? I guess this is an example of a hom(A,B) that does not consist of functions on the objects.
12:34 Wait??? This can't be right. He probably meant a proper class can't be an element of a class. Is that it? Because a class can definitely be an element of another class, considering a set is a class and sets can be elements of sets ...
Keep in mind that the term "object" has a specific technical meaning in category theory. So it would be better to say that "If you don't assume that morphisms are functions, then they can be the elements of any set." In this case, composition is any binary operation you wish as long as it has the defining properties of a category--existence of identity element and associativity. Check the example of a poset P. Here the morphisms look like a
Would have been nice to have some more discussion of the required background for those new to the broader concept. There were a tremendous number of examples from many areas of higher math that many viewers wouldn't have previously had. I think it's important for them to know that if they don't understand a particular example, they can move on without much loss as long as they can attempt to apply the ideas to an area of math they are familiar with. Having at least a background in linear algebra and/or group theory are a reasonable start here. In some of the intro examples it would have been nice to have seen at least one more fully fleshed out to better demonstrate the point before heading on to the multiple others which encourage the viewer to prove some of the others on their own. Thanks for these Steven, I hope you keep making more! There's such a dearth of good advanced math lectures on the web, I hope these encourage others to make some of their own as well.
a bijection is the mapping from a set A to set B such that each individual element of A corresponds exactly to one element in B, for all of B. So it's one-to-one and onto. Ex: A = {a,b,c}, B = {1,2,3} f(a) = 1, f^-1(1) = a f(b) = 2, f^-1(2) = b f(c) = 3, f^-1(3) = c Notice the function maps each element of A to a particular element of B and the inverse of the function applied to B maps back to A, "one-to-one", and that every element of B is someone output to A "onto".
This really expects me to know a lot more mathematics than I do. It would be nice if there were a list of prerequisites. I'm just a lowly programmer looking to be better at what I do.
I am sorry that these lectures do not suit your background. Please understand that I am mathematician and that my primary purpose is to introduce category theory to students of mathematics. I hope that my lectures and my book will help students of computer science and other disciplines as well but it is not possible to escape the fact that category theory is mathematics and its primary origins, use and applications are in mathematics. Good luck to you. I hope you find a more suitable avenue of study for category theory.
+Steven Roman - Mathematics No worries! I was just disappointed because I was really enjoying your teaching style. Then you started diving deeper, into subjects I know little to nothing about. Hopefully my journey will eventually loop back here. See you then!
+gfixler I think that you may find that the prerequisites are simply a good and thorough stepping through a book on Modern or Abstract Algebra (there are many to choose from, and I personally think that if one has the time, Artin's Algrebra with Abstract Algebra by Dummit and Foote is the foolproof way) and a supplementary overview of linear algebra and perhaps some topology, more so for the understanding of the examples. As a programmer, I very much think that you will want to come to a greater understanding of abstract algebra, as there are some many principles and abstractions that lend greatly from concepts, which lead seemingly into even greater abstractions and methods in Category theory. I imagine that I am not one to make claims on the purposes or uses of the theory but it is my understanding that this branch of mathematical thought has underpinning uses in cross-correlating all types of other branches of mathematics and sciences, not least of all computation. I guess I'm trying to just say don't give up and I hope you do end up coming back to it.
Yes, but you’re ASSUMING: (1) the existence of the Identity and (2) that the Associative Property holds. Without those 2 axioms/given assumptions you have no proof. Likewise, you’ll eventually run into serious foundational problems if you attempt to develop the Category Theory based on the traditional Axiomatic Set Theory approach.
I would really like it if you continued doing these videos, you are a great lecturer!
Thank you very much for the nice comment.
@@stevenromanmath an extremely amazing lecturer 👨🏫. Love your lectures my friend .
@@johnlopez5777 Thank you. so much.
Excellent. Your definition of product at the end was very motivational.
Glad you liked it
I've been curious about category theory for a few years, and I've begun reading / watching various introductions. This is by far the most lucid, most helpful, and most sensitive to the student's likely concerns. Please continue the series! Please publish your category theory lectures as a book!
I'm a retired Mathematician currently working with impoverished children at an Elementary School. I conduct some of their Interventions.
I find your lecturing manner quite approachable and well paced. Please do continue in the same vein. Most enjoyable!
Thank you. I plan on continuing to do TH-cam lectures.
I have been looking for an approachable but thorough intro into Category Theory for some time. This is it.
18:07 There's technical issue relating to requirement 3 (that distinct hom sets are disjoint). SET doesn't naturally have this property, a function doesn't know where it's co-domain is. For example, the identity on the natural numbers id: N-> N is the same (as a set) as the identity injection into the integers id: N -> Z. So, id ∈ Hom(N,N) and id∈Hom(N,Z), violating requirement 3. The issue is easily resolved, for instance by flagging each function with it's co-domain, which is what is done implicitly rather than explicitly. This (non)issue occurs in other example categories, and can be similarly resolved.
Oooooh, this is good observation ... oh my Euler, thanks! I am writing about Category Theory and your observation is new to me. Thank you again.
The way you explain it seems to me very simple, synthetic and right to the core of the theory at the tranquil pace which allowed me to follow your lecture without rewinding or stopping the video at any time, thanks.
Very informative! You are a great expositor and I would love to see you continue this series.
Thank you for your nice comments. I appreciate them very much.
+Steven Roman - Mathematics xxxmoye
I agree. Thank you so much Sir!
I agree. Thank you so much Sir!
Thank you alot professor Roman
Really outstanding. Clear and focused. If you continue the series I'll certainly watch.
Dr. Roman, your lectures and book on category theory are the best! I have been looking for weeks for a good introduction to category theory, and here you are!
You lecture is very clear and informative! It will be great if you would like to continue the lectures!
For once at the end of a category theory lecture I feel like I've learned something more than just "did you notice isomorphisms are isomorphic to isomorphisms across different fields of math?"
HAHAHAHAHAHAHAHAHAHA.
Please continue this series. I have been a fan of your exposition since coming across a 1st edition copy of Intro to Mathematical Finance (I need to see what changes in the 2nd edition!), and this video as well as my own general fuzziness on Category Theoretic approaches is very helpful and illuminating. I think that you do a very good job of not being bogged down in the details but also being thorough in the procedure and nuances. I really like that you don't gloss over the fact that category theorists seemingly say the same thing in a number of different ways. I will be picking up the pdf of the book for certain now. Honestly, if you could find the time in your retirement (and the enjoyment or will!) to do a series for other topics such as the advanced algebra, I think that you may find yourself with another passion that is no so time consuming as authoring books.
+jon, thanks for the very nice comments. In fact, I was pondering the idea of doing lectures on abstract algebra if there is demand.
+Steven Roman - Mathematics I've also been looking for lectures in abstract algebra. The best I've found so far are on TH-cam by Benedict Gross. Imho Gross is to abstract algebra (almost) what Gilbert Strang is to linear. Would you consider doing universal algebra and/or model theory?
Abstract algebra would be so Great! Especially with some Galois theory. I studied abstract algebra 8 years ago. I still enjoy it but it's getting a little rusty and some videos featuring your OutStanding lecture style would just be wonderful.
Please continue the series. A few more videos would help others forming an opinion. Personally I like your style. Also, There are only a few series on the subject so this new series would be more than welcome. Also, please when the time comes (ex: functors) use when possible also examples involving graphs and types. And if you ever see OOP connections please state them, usually CT is related to FP only.
Thank you Aux. I will keep your comments in mind.
'I think in all classes in mathematics, as a student one has to have a little bit of faith that what's coming will be important at some point even if the first few days or week of course seems maybe a little bit unmotivating'
I love you. lol
That's a very bad statement.
@@samueldeandrade8535 That's a very bad comment.
@@AndrewStanish why?
Mr. Roman I am currently reading your advanced linear algebra text and I must say that I'm enjoying it immensely. Many Thanks!
This is awsome! The level of explanation is very suited for people like me that have just some basic undergraduate mathematics under their belt.
Thank you for the introduction Professor. Been trying to wrap my mind about this theory for quite a while now.
Great lecture sir.... Respect from India..... Category theory was something I always wanted to kick start with..... Thank u very much for these lectures.....
Thank you very much, Mr. Steven Roman. I am major in computer science. I am very interesting in type theory and category theory. No other videos can explain category theory than your lectures!
Oh wow. I had no clue professor Roman had online videos. I've used your linear algebra text! Will definitely watch this series.
Great. Thanks.
What a great lecture! Thank you VERY much Dr. Roman!
There are so few good videos targeting this level of mathematics. It seems all you can find is elementary math or research level math. I have a masters but I never studied category theory. This lecture was just perfect for me. I will now watch the second one immediately. :)
Jacob G. If you are interested in my book, it is available as an e-book through my web site www.sroman.com. If it is ever published by a commercial publisher, it will probably not appear for at least 9 months.
I am really satisfied with your teaching skills.
Thank you sir
Thanks Sir. Great lecture. Please continue your lecture content.🙏
Thank you. I am working on the fourth lecture in the algebra series--field theory.
Love you sir. You have done our work easy.
Respect from China. You are really a great scholar writing so many books of so many fields.
Hi Steven, I bought your book "advanced linear algebra" long ago ... great reference .... thanks ... I hope theses lectures help me to solve problems in molecular dynamics ...
Excellent! I'm reading Steve Awodey's book right now (in preparation for an algebraic geometry course in the Spring). I am very intrigued to read yours when it comes out. Looking forward to it :)
It didn't feel like 57 minutes to me, it felt like 5 minutes. It was fascinating.
I got a little bit confused, what are the "elements" of Mor(sets), are they the sets Hom(A,B) for any A,B in Obj(sets) or are they the functions from one to another object in Obj(sets)? Great lecture btw, please don't stop making them!
Mor(C) refers to the class of all morphisms in C. Thus, Mor(Set) is the class of all individaul set functions.
Amazing! I enjoyed the book as a companion to Riehl. Would you consider making videos on sheaves, monads, kan extensions?
Muchas gracias por sus videos profesor Roman, son muy útiles porque complementan la lectura del libro.
Thank you very much for these videos and for your book "An Introduction to Category Theory". I've read a lot of such material and this is the most lucid introduction to the subject I've yet come across. It's up there with Bill Lavere's "Conceptual Mathematics". I believe, though, you've missed a couple of typos in the book: p. 10, Example 11 - the second and third paragraphs are near duplicates. I believe the 3rd is a correction of the 2nd, which you've omitted deleting; and, p.14 point 2) of the definition of left -invertible has fL: A->B which should be fL:B ->A. Thanks again.
Hi sir,
Quick question. Because of 3) (Any 2 Hom Sets in the Category are disjoint), isn't Mor(C) = \XOR Hom(c_{i},c_{j}) , where , c_{i} , c_{j} belong to Obj(C) , and \XOR represents the Disjoint Union ?
Very good lecture. Thank you so much.
Please continue these lectures.
excellent lectures, show us more please.
Hello Professor Roman,
I just wanted to say I enjoy your lectures very much -- and I am very interested in Category Theory, and they may persuade me to purchase your book. If you are able, please keep these lectures going, and thank you!
Great video. Very helpful. Thank you.
Thank you so much for this class. I love it.
Hope I am not too late to wish you well and hope you find the time to make more videos. Have 'liked' and subscribed. Good videos.
Thank you very much for this really clear introduction
those lessons are really good, thanks for sharing it
Does the adjoint, listed as one of the 5 main principles of category theory at 11:49, have anything to do with the adjoint of an invertible matrix?
Wow. This is exciting!!
Interesting, continue Prof.
Nice lecture. Keep them coming. Why the four minutes of silence, though, at the end?
thx for making this series of videos!
You are indeed a good lecturer Professor. Can you upload your lecture on umbral algebra ?
Thank you for your lectures.
Thank you prof. A student From South korea.
You are welcome.
What’s the story of that clipboard? It looks like it’s been through the war.
It has been. I have been using this clipboard for about 30 years. hate to part with it.
Thank you sir for these gems🎉🎉
Thank you!
18:28 What is the meaning of disjoint between hom-sets? I think that the notion 'disjoint' is unclear.
Great lecture! Thank you.
If "all sets are classes" and "the class of all sets" exists. Then that is a class that contains classes. Could you clarify this?
No class can contain a PROPER class, that is, a class that is not a set. I apologize if i didn't say that properly in the lecture!
Steven Roman? Like THE Steven Roman?! I love your texts.
Thank you!
(Potentially silly) Question : After you clearly explained the difference between class and set, and then went on to clarify that in C.T. the collections of all objects in a Category are considered to be a class and the homsets are considered to be a set, why do you write with no hesitation "*class* of all morphisms" at 22:04?
Is it because in total the class of all objects times the sets of all their homsets gives again a class?
Or to put it in a very simplistic way: class x set = class ??
I apologize if this is a silly question
Hugo Nava Kopp It is not a silly question and the answer is"yes"
Thanks!
For Matr_F, I would have expected obj(Matr_F) to consist of all n-tuples of integers. How is a matrix going to map integers? I guess this is an example of a hom(A,B) that does not consist of functions on the objects.
Thank you, this is really helpful.
Great Introduction. Though more diagrams would would make facts more transparent.
Thank you professor.
12:34 Wait??? This can't be right. He probably meant a proper class can't be an element of a class. Is that it? Because a class can definitely be an element of another class, considering a set is a class and sets can be elements of sets ...
Wonderful!
Nice, thanks Steven.
if you don't assume that morphisms are functions, they can be any objects, so in that case what would a composition of objects mean?
Keep in mind that the term "object" has a specific technical meaning in category theory. So it would be better to say that "If you don't assume that morphisms are functions, then they can be the elements of any set." In this case, composition is any binary operation you wish as long as it has the defining properties of a category--existence of identity element and associativity. Check the example of a poset P. Here the morphisms look like a
The rule for composition has to be specified along with the objects and morphisms while defining the category.
Would have been nice to have some more discussion of the required background for those new to the broader concept. There were a tremendous number of examples from many areas of higher math that many viewers wouldn't have previously had. I think it's important for them to know that if they don't understand a particular example, they can move on without much loss as long as they can attempt to apply the ideas to an area of math they are familiar with. Having at least a background in linear algebra and/or group theory are a reasonable start here.
In some of the intro examples it would have been nice to have seen at least one more fully fleshed out to better demonstrate the point before heading on to the multiple others which encourage the viewer to prove some of the others on their own.
Thanks for these Steven, I hope you keep making more! There's such a dearth of good advanced math lectures on the web, I hope these encourage others to make some of their own as well.
What are bijections of the integers?
Are they functions from the integers to the integers that are bijective? So permutations of the integers?
a bijection is the mapping from a set A to set B such that each individual element of A corresponds exactly to one element in B, for all of B. So it's one-to-one and onto.
Ex: A = {a,b,c}, B = {1,2,3}
f(a) = 1, f^-1(1) = a
f(b) = 2, f^-1(2) = b
f(c) = 3, f^-1(3) = c
Notice the function maps each element of A to a particular element of B and the inverse of the function applied to B maps back to A, "one-to-one", and that every element of B is someone output to A "onto".
Thanks
Extremely abstract. Anything that is extremely abstract, although beneficial, alienates the mind
This really expects me to know a lot more mathematics than I do. It would be nice if there were a list of prerequisites. I'm just a lowly programmer looking to be better at what I do.
I am sorry that these lectures do not suit your background. Please understand that I am mathematician and that my primary purpose is to introduce category theory to students of mathematics. I hope that my lectures and my book will help students of computer science and other disciplines as well but it is not possible to escape the fact that category theory is mathematics and its primary origins, use and applications are in mathematics. Good luck to you. I hope you find a more suitable avenue of study for category theory.
+Steven Roman - Mathematics No worries! I was just disappointed because I was really enjoying your teaching style. Then you started diving deeper, into subjects I know little to nothing about. Hopefully my journey will eventually loop back here. See you then!
+gfixler I think that you may find that the prerequisites are simply a good and thorough stepping through a book on Modern or Abstract Algebra (there are many to choose from, and I personally think that if one has the time, Artin's Algrebra with Abstract Algebra by Dummit and Foote is the foolproof way) and a supplementary overview of linear algebra and perhaps some topology, more so for the understanding of the examples. As a programmer, I very much think that you will want to come to a greater understanding of abstract algebra, as there are some many principles and abstractions that lend greatly from concepts, which lead seemingly into even greater abstractions and methods in Category theory. I imagine that I am not one to make claims on the purposes or uses of the theory but it is my understanding that this branch of mathematical thought has underpinning uses in cross-correlating all types of other branches of mathematics and sciences, not least of all computation. I guess I'm trying to just say don't give up and I hope you do end up coming back to it.
Try to read Walters' Categories and Computer Science.
The first proof came out of nowhere for a novice. Could have gone a bit slower on that one and explained it. On identity.
I'm Like #666. This is the second time this has happened to me this week. Guess I'm the one who has to do the thing. Just kidding.
Pax ave et vale
Go to 5:09 to skip the introduction and go directly to the discussion of the theory
ه
Yes, but you’re ASSUMING: (1) the existence of the Identity and (2) that the Associative Property holds. Without those 2 axioms/given assumptions you have no proof. Likewise, you’ll eventually run into serious foundational problems if you attempt to develop the Category Theory based on the traditional Axiomatic Set Theory approach.
WHO needs that ? It's impractical.
1videoshow computer programmers need that.
please write clearly your hand writing on the board is not tangible or never been seen please.