Categories for the Working Mathematician

In summary: I'm studying it for a directed independent study, in order to gain a better foundational understanding of category theory. I don't think reading many different books on this subject will help much in this respect - they might be more for entertainment than anything else. However, I welcome your input if you have more insights.
  • #1
SrVishi
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Hello. I am about to start learning category theory. I keep hearing mixed opinions on the book Categories for the Working Mathematician, by Sanders MacLane (I am aware he is one of the founders of the theory). Some say it's a "must read", and others have called it "outdated." What would seem outdated about this book? What would be the pros and cons of using it? Is there a book or a collection of books that you feel cover the same (or more) material but better? IF any of my background is needed, I have a fair amount of mathematical maturity. I can read Rudin, Lang (Grad algebra), and other such terse books and fill in or construct my own such proof fairly well when things are missing. I might just be shaky in terms of knowledge, such as definitions etc. For example, I haven't learned algebraic topology yet.
 
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  • #2
At least I have the impression that you can read it. The question is, why do you wish to read it? Do you have a special goal, e.g. to learn some tools for cosmology, in which case I would recommend a book that is more (co-)homological and less categorical. If you are interested in anything in the sphere of logic, I'd say read it. So the answer to your question depends pretty much on where you want to arrive at.

The fact it is some decades old doesn't matter a second. It's a book on fundamental conceptions and these haven't changed. (And I doubt they will at any time.)
 
  • #3
i would suggest that you mainly want to know what a functor is, and a natural transformation, a representable functor, and then Yoneda's lemma. That's about it, as far as I am concerned. Oh and I guess you want to know the categorical definitions of isomorphisms, products and sums (coproducts), as well as inverse and direct limits.

edit: My viewpoint is that of an algebraic geometer who is not a category theorist. So to me most of the stuff in the free book you linked is totally unnecessary verbiage. As a youngster I recall thinking category theory was a lot of fun, but as a practicing mathematician, it seemed like (to quote one somewhat cranky and opinionated algebraic geometer, Miles Reid, p.116, Undergraduate Algebraic Geometry) "surely one of the most sterile of all intellectual pursuits".

This in the vein of the earlier question of what are your goals. I.e. if your goal is to be a mathematician in a field other than category theory, you will not need all this technical terminology. But if you enjoy this pursuit, then wonderful. Go for it.

I would actually recommend reading the original paper that started the theory, at least the introduction:

http://www.ams.org/journals/tran/1945-058-00/S0002-9947-1945-0013131-6/S0002-9947-1945-0013131-6.pdf
 
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  • #4
Well, here are more details on my current situation. I am taking a directed independent study in Category Theory. My professor (he is one of the few calling it outdated) wants me to use the book Category Theory: An Introduction by Herrlich and Strecker. I am aware that these authors have another free online popular book Abstract and Concrete Categories. So, my plan so far for this independent study course is to use Herrlich and Strecker's introduction book, supplement it with Awodey's book, and then Abstract and Concrete Categories. I guess a better question would be, if I read all of those books, would it cover everything MacLane does? Would there be any incentive to do so after the independent study, say on my own next semester?
 
  • #5
Sorry, but your details left me with even more questions.
SrVishi said:
I am taking a directed independent study in Category Theory. My professor (he is one of the few calling it outdated) wants me to use the book Category Theory: An Introduction by Herrlich and Strecker.
What is directed + independent? Directed by whom and what for? Independent from whom and what for? How is your professor related to these questions? In general, I would simply recommend to follow his advice, for he knows best about your situation and has probably more insights as well in the subject as in your personal development and goals than anyone here on PF.
SrVishi said:
Would there be any incentive to do so after the independent study, say on my own next semester?
Which leaves us again at the starting point of my previous post: How do you measure incentive and what's going on in your next semester?
Why do you wish to read so many different books on this subject?

However, I'm probably not the one to ask anyway. But what you call details aren't any (IMO). I thought, you might want to know.
 
  • #6
What's your background? What are you studying category theory for? Have you taken homological algebra?
As far as I know, Categories for the working mathematician draws example from algebraic topology (where the subject naturally arose). If you have no idea what algebraic topology is, the book might go over your head. If your interest lies in applying category theory to other mathematics subjects (or CS, as a matter of fact) you most likely won't need to go through a whole book. Without more information, we won't be able to help you much.
 

Related to Categories for the Working Mathematician

1. What is "Categories for the Working Mathematician"?

"Categories for the Working Mathematician" is a book written by Saunders Mac Lane that serves as a comprehensive guide to category theory. It covers the fundamental concepts and applications of category theory, a branch of mathematics that studies the general properties of mathematical structures and their relationships.

2. Who is the target audience for this book?

This book is primarily aimed at working mathematicians and graduate students in mathematics who are interested in learning about category theory and its applications in other areas of mathematics.

3. What are some key topics covered in "Categories for the Working Mathematician"?

Some key topics covered in this book include the basic definitions and properties of categories, functors, natural transformations, universal properties, adjoint functors, and limits and colimits.

4. How is this book organized?

This book is organized into five parts: Categories and Functors, Universal Constructions, Representability, Limits and Colimits, and Adjoints and Monads. Each part is further divided into chapters that cover specific topics in detail.

5. Is "Categories for the Working Mathematician" suitable for self-study?

Yes, this book can be used for self-study. It is written in a clear and concise manner, with many examples and exercises included to help readers grasp the concepts. However, some familiarity with abstract algebra and basic set theory is recommended for a better understanding of the material.

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