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Basic Category Theory

arxiv.org

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Re: Basic Category Theory

#31

Post by hilbertseries »

keyboardwriter wrote: post #3
petercooper wrote: post #2

"for readers with relatively little mathematical background." A well put together guide, but note that 'relatively little' here means you're OK with some abstract algebra, at least, as the second example of the introduction begins: This example involves rings, which in this book are always taken to have a multiplicative identity, called 1. Similarly, homomorphisms of rings are understood to preserve multiplicative id…

Why do mathematicians use phrases such as "relatively little mathematical background" or "introduction to.." when they assume previous knowledge? I find that really annoying and a turn off from reading most math textbooks that are suppose to be "introductions". Is there an actual good book on category theory for someone that didn't complete a math degree? Additionally, what are applications of category theory to comp…

The target audience of this book isn't the layman, it's people studying mathematics. So a phrase like "relatively little mathematical background" is meant in the context of academia.

I don't think there's a basic and gentle approach to learning category theory that doesn't remove the rigor. You can learn at a high level what some of the stuff in category theory and get a basic feel and intuition for it, if that's what you want. But, if you want to learn category with the rigor, you're going to have to first learn how to write a proof in mathematics. Really though, you're going to want to have a decent grasp on some basic abstract algebra. Otherwise you're never going to be able to understand the examples for category theory.

Re: Basic Category Theory

#34

Post by Smaug123 »

cokernel wrote: post #6
petercooper wrote: post #2

"for readers with relatively little mathematical background." A well put together guide, but note that 'relatively little' here means you're OK with some abstract algebra, at least, as the second example of the introduction begins: This example involves rings, which in this book are always taken to have a multiplicative identity, called 1. Similarly, homomorphisms of rings are understood to preserve multiplicative id…

The "note to the reader" clarifies things a bit -- he's aiming for requiring "no more mathematical knowledge than might be acquired from an undergraduate degree at an ordinary British university". Though he does not specify whether he has in mind a mathematics degree, I think this can be deduced from the fact that he indicates that the text developed out of a master's-level course.

The text developed from approximately six lectures' worth of the MMath-level 24-lecture Part III Introduction to Category Theory at Cambridge, I believe. (Source: I took that course last year, and was part of a small reading group studying Leinster's Basic Category Theory at that time. We found that book really, really helpful.)

Re: Basic Category Theory

#35

Post by Smaug123 »

petercooper wrote: post #10
keyboardwriter wrote: post #3

Earlier quoted context omitted.

Why do mathematicians use phrases such as "relatively little mathematical background" or "introduction to.." when they assume previous knowledge? I find that really annoying and a turn off from reading most math textbooks that are suppose to be "introductions". Is there an actual good book on category theory for someone that didn't complete a math degree? Additionally, what are applications of category theory to comp…

I think it's easily done when someone knows a topic too well to teach it to beginners from outside the field. Rings, fields and identities may just be math's equivalents of string, operator, or variable, when similarly used by introductory programming books without any explanation.

This is correct: "ring" is a concept that will be taught to all maths undergraduates by the end of the second year at latest.

Re: Basic Category Theory

#36

Post by Smaug123 »

Bahamut wrote: post #17

This article definitely assumes more knowledge than basic abstract algebra - there are also some key facts used from set theory assumed (i.e. In a proof, it is assumed to be known that if a compositiom of two maps is the identity map on one set, and the reverse composition is the identity on the other set, then the map is a bijection - it also assumes knowledge from abstract algebra that the composition of two homomo…

I'm in the middle of writing https://arbital.com/p/universal_property_outline/ which is precisely an Intro to the Universal Property - comments, feedback and assistance welcome, if you care to help :)

Re: Basic Category Theory

#37

Post by Smaug123 »

This book also has the [backing of Peter Smith][1], who is basically the king of logic-related reading materials (logicmatters.net). I was part of a student reading group studying BCT last year with him, and I found it extremely helpful in the context of people studying for the MMath. One of the most important parts of basic category theory is seeing how adjunctions, limits and universal properties play with each other; this book is about exactly that, and goes into it in a depth I haven't seen very well elsewhere.

[1]: http://www.logicmatters.net/2017/01/02/tom-leinsters-basic-c...

Re: Basic Category Theory

#38

Post by Smaug123 »

danharaj wrote: post #20
decasteve wrote: post #16

For an even more basic introduction I'd recommend starting with Lawvere and Schanuel's Conceptual Mathematics: A First Introduction to Categories . Requires less abstract algebra. If you have some mathematical intuition then this is the book to start with. I'm surprised he cites Sets for Mathematics but doesn't include it in his further reading. If you have some university level abstract algebra (groups, rings, vecto…

> For an even more basic introduction I'd recommend starting with Lawvere and Schanuel's Conceptual Mathematics: A First Introduction to Categories. Requires less abstract algebra. If you have some mathematical intuition then this is the book to start with. It's a great book! Don't forget to do the exercises. It starts off very easy but if you don't do the exercises all of a sudden you'll be lost.

This is generally good advice for all maths study, of course: "mathematics is not a spectator sport".

Re: Basic Category Theory

#39

Post by maxiepoo »

Koshkin wrote: post #22
adamnemecek wrote: post #5

Earlier quoted context omitted.

You might be interested in "Algebra: Chapter 0" http://amzn.to/2iYpAn5 "The primary distinguishing feature of the book, compared to standard textbooks in algebra, is the early introduction of categories, used as a unifying theme in the presentation of the main topics."

> categories, used as a unifying theme in the presentation of the main topics Categories are not a good way to be introduced to the "main topics" of abstract algebra. Yes, Category Theory can be seen as the "Chapter 0" of a course on abstract algebra, but for beginners a traditional presentation based on sets would be much easier to digest. There is a similarity between using categories and using functional programmi…

Do you have experience teaching both or are you just speculating?

Re: Basic Category Theory

#40

Post by j2kun »

Koshkin wrote: post #22
adamnemecek wrote: post #5

Earlier quoted context omitted.

You might be interested in "Algebra: Chapter 0" http://amzn.to/2iYpAn5 "The primary distinguishing feature of the book, compared to standard textbooks in algebra, is the early introduction of categories, used as a unifying theme in the presentation of the main topics."

> categories, used as a unifying theme in the presentation of the main topics Categories are not a good way to be introduced to the "main topics" of abstract algebra. Yes, Category Theory can be seen as the "Chapter 0" of a course on abstract algebra, but for beginners a traditional presentation based on sets would be much easier to digest. There is a similarity between using categories and using functional programmi…

> for beginners a traditional presentation based on sets would be much easier to digest.

If you read the referenced book, you'd know the author does exactly this.

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