As I mentioned here, back on the old home of goodmath, I was taking a poll of what good math topic to cover next. In that poll, graph theory and topology were far away the most popular topics, tying for most votes (8 each), compared to no more than 2 votes for any other subject.

So, the next topic I’m going to talk about is: category theory.

There is actually a reason for that. I’m not just ignoring what people voted for. Based on the poll, I was planning on writing about topology, so I started doing some background reading on toplogy. What came up in the first chapter of the book I picked up? Explanations of how to interpret category-theory diagrams, because the author of the text found cat theory to be useful for explaining some of the concepts of topology.

I also looked up some articles on graph theory – in particular, graph isomorphisms, because that’s something I’ve done some work on. And what do I find when I start to read them? Again, category theory diagrams.

And what do I find when I look at wikipedia, to see if I missed anything in my recent series of posts on the semantics of lambda calculus? Category theory.

Category theory is a wierd subject, which has an amazing way of generating incredibly polarizing attitudes among mathematicians. But it’s cropping up more and more in numerous fields of mathematics, and it’s widely used in computer science. There seem to be significant doubts among many people as to whether or not category theory represents anything dramatically new, whether or not it provides new insights that couldn’t have been gained by any other fields of mathematics. But whether or not it’s a source of new knowledge, it seems to be undeniable that it is extremely useful as a tool for understanding and explaining other mathematical fields.

So I *will* definitely write about topology and graph theory soon. But first, it’s going to be category theory.

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IthikaW00t. Since I started learning Haskell I’ve been reading the occasional introduction to category theory to try and see where the monads thing came from. So far it has been too confusing for me. I’m looking to you, Mark, to sort it all out. No pressure! 😉

JPMy graduate Algebra prof liked to call category theory “Abstract Nonsense”. He also pointed out, that it can be very useful nonsense. Despite its usefulness however, he steered me away from doing a reading course in category theory in favor of one in non-communtative ring theory (the right choice). In the end, for me, studying a system of abstractions is just not as interesting as studying more concrete things (or slightly less abstract things anyway).

QalmleaWe got to catetories late in my graduate Abstract Algebra class. At the time, I saw no use for them whatsoever (however, that particular professor nearly always loses my interest halfway through the course). So I’m curious to see them presented with more of an emphasis on applications.

terpiscoreiCategory theory is a great subject! It’s a great way to see how universal kinds of mathematical objects can be, and it’s pretty amazing to see how effective these completely abstract notions are when they’re applied to concrete situations.

One of the great things about math is how interconnected it is. Just as you’ve found references to categories in researching other subjects, there are connections that arise at even a really superficial level. As you probably already know, categories have nice, natural depictions as directed graphs. Going backwards, every partially ordered set has a natural interpretation as a category and the resulting category can actually turn out to be very useful.

And just to tie topology in, the language of category theory has unbelievable utility in algebraic topology. If you’re comfortable with categories, you get a lot of nice results in algebraic topology virtually for free that you probably wouldn’t want to prove using more basic methods.

Anyway, best of luck with your readings! I’m looking forward to seeing your further thoughts about the subject.

SharonCI’ve just found this blog and love it already!

Any mathematics/computing topic you want to talk about it fine by me! (including category theory)

DavisCategory theory has a fairly big role in one wing of algebraic geometry, as well. These days you’ll hear lots of people talk about

stacks; no one ever seems to define them, but they rely heavily on category theory. Stacks seem to be the right thing to use if you’re doing work on moduli spaces (which is a huge area in algebraic geometry).