Category Theory.

[swestrup]

I have been making another attempt to understand Category Theory. This time I've managed to get as far as figuring out that what I simply don't understand is the basic definition of a category and its properties. I've now read the first few chapters of about a dozen different books which claim to be gentle introductions to category theory from various perspectives.

The trouble is that they ALL use almost the same language in their definitions and its very clear that the words they use don't mean what I think they mean. They say 'assign' in contexts where no assignment is occurring (or possible). They say a 'diagram commutes iff X' and then give a condition for X that, depending on meanings I assigned to undefined terms earlier on, is either a tautology or a contradiction. In other words, as far as I can tell based on the definition of a category, either all diagrams commute, or no diagram commutes.

This happens time and time again as I try to figure out what the heck the author MEANS when they write their definitions and before long I have this huge tree of possible interpretations, all of which are contradicted by the examples. Its extremely frustrating.

I don't suppose anyone on my friends list has gotten further in this pursuit and would be willing to help translate this stuff into language I can follow?

Comments

[sps.livejournal.com]

Goodness, I hope I can explain that much! Perhaps even in English.

PS

[sps.livejournal.com]

I suspect they're using 'assign' in the pre-computational sense. (Side effects screwed everything up, I don't know why we have them.)

[hendrikboom.livejournal.com]

is either a tautology or a contradiction

Well. I guess you understood that part, then.

A lot of category theory expresses the patterns in stuff that is indeed tautological. But it can really express content. The first application in which it was used nontrivially was algebraic topology. And there, the content was decidedly nontrivial. But heavily artificial;

Pleas, let's chat. I should at least be able to explain what kind of thing it can be about. (it's abstract enough that I have to talk in terms of what it can be about, rather than what it is about.)

[swestrup]

We may well need to get together for a chat about this. Some things sps told me may have helped, so I'm going to reread some of the stuff that didn't make sense before, and see if I can work it out now.

[swestrup]

Oh, and I don't think I have any trouble with the concept of what kind of thing it can be about. If I hadn't worked that out already, I wouldn't have had any use for it.

Personally, I think there are major things being left out of the definitions that a trained mathematician would simply assume, and which I don't.

[moomlyn.livejournal.com]

Probably best to start with some concrete examples, such as the category of groups, or the category of finite dimensional vector spaces, or the category of topological spaces. Then try to work out what the definitions mean for your examples.

The secret to abstraction is a handful of concrete examples mixed with a large dose of analogical reasoning.

[swestrup]

I don't have any trouble with abstraction, I work with it every day. There just seems to be something in how they define categories that I can't wrap my head around. As I was explaining to sps on the phone today, I can't even get the examples to work out if I use a concrete category with 1, 2, 3 and addition modulo 3. I always end up either with tautologies or contradictions.

[moomlyn.livejournal.com]

All of mathematics is a tautology.

Addition modulo 3 can't be a morphism on a three object category, because addition is a binary operation. It has two inputs and one output. A morphism has one input and one output.

You could take the objects 1,2,3 together with the morphism (plus 1) mod 3. In which case the resulting category is simply a triangle.

A finite category, is of course, just a multigraph with loops.

[moomlyn.livejournal.com]

btw, its still very much an open question as to whether or not mathematics is in fact, all a contradiction.

Most mathematicians don't really like to talk about that, however.

[swestrup]

Yeah, I meant increment, not addition. Its been a long day. I shall probably take another stab at the thing tomorrow.

[moomlyn.livejournal.com]

what I said is also not quite correct. You need to have the identity morphism, so need to also have (plus 0) mod 3.