category theory
goodmath | July 10, 2006Today's contribution on category theory is going to be short and sweet. It's an example of why we really care about [natural transformations][nt]. Remember the trouble we went through working up to define [cartesian categories and cartesian closed categories][ccc]?
As a reminder: a [functor][functor] is a structure preserving mapping between categories. (Functors are the morphisms of the category of small categories); natural transformations are structure-preserving mappings between functors (and are morphisms in the category of functors).
Since we know that the natural transformation can…
goodmath | July 6, 2006We're almost at the end of this run of category definitions. We need to get to the point of talking about something called a *pullback*. A pullback is a way of describing a kind of equivalence of arrows, which gets used a lot in things like interesting natural transformations. But, before we get to pullbacks, it helps to understand the equalizer of a pair of morphisms, which is a weaker notion of arrow equivalence.
We'll start with sets and functions again to get an intuition; and then we'll work our way back to categories and categorical equalizers.
Suppose we have two functions mapping from…
goodmath | July 4, 2006What's a subset? That's easy: if we have two sets A and B, A is a subset of B if every member of A is also a member of B.
What's a subgroup? If we have two groups A and B, and the values in group A are a subset of the values in group B, then A is a subgroup of B.
For any kind of thing **X**, what does it mean to be a sub-X? Category theory gives us a way of answering that in a generic way. It's a bit hard to grasp at first, so let's start by looking at the basic construction in terms of sets and subsets.
The most generic way of defining subsets is using functions. Suppose we have a set, A.…
goodmath | June 27, 2006Before I dive into the depths of todays post, I want to clarify something. Last time, I defined categorical products. Alas, I neglected to mention one important point, which led to a bit of confusion in the comments, so I'll restate the important omission here.
The definition of categorical product defines what the product looks like *if it's in the category*. There is no requirement that a category include the products for all, or indeed for any, of its members. Categories are *not closed* with respect to categorical product.
That point leads up to the main topic of this post. There's a…
goodmath | June 22, 2006 Sorry, but I actually jumped the gun a bit on Yoneda's lemma.
As I've mentioned, one of the things that I don't like about category theory is how definition-heavy it is. So I've been trying to minimize the number of definitions at any time, and show interesting results of using the techniques of category theory as soon as I can.
Well, there are some important definitions which I haven't talked about yet. And for Yoneda's lemma to make sense, you really need to see some more examples of how categories express structure. And to be able to show how category theory lets you talk about…
goodmath | June 19, 2006The thing that I think is most interesting about category theory is that what it's really fundamentally about is structure. The abstractions of category theory let you talk about structures in an elegant way; and category diagrams let you illustrate structures in a simple visual way. Morphisms express the structure of a category; functors are higher level morphisms that express the structure of relationships between categories.
In my last category theory post, one of the things I mentioned was how category theory lets you explain the idea of symmetry and group actions - which are a kind of…
goodmath | June 12, 2006 Let's talk a bit about functors. Functors are fun!
What's a functor? I already gave the short definition: a structure-preserving mapping between categories. Let's be a bit more formal. What does the structure-preserving property mean?
A functor F from category C to category D is a mapping from C to D that:
Maps each member m ∈ Obj(C) to an object F(m) ∈ Obj(D).
Maps each arrow a : x → y ∈ Mor(C) to an arrow F(a) : F(x) → F(y), where:
(∀ o ∈ Obj(C)) F(1o) = 1F(o). (Identity is preserved by the functor mapping of morphisms.)
(&forall m,n ∈ Mor(C)) F(n º o) = F(o) º F(n). (…
goodmath | June 9, 2006 For me, the frustrating thing about learning category theory was that
it seemed to be full of definitions, but that I couldn't see why I should care.
What were these category things, and what could I really talk about using this
strange new mathematical language of categories?
To avoid that in my presentation, I'm going to show you a couple of examples up front of things we can talk about using the language of category theory: sets, partially ordered sets, and groups.
Sets as a Category
We can talk about sets using category theory. The objects in the category of sets are, obviously, sets…
goodmath | June 8, 2006One of the things that I find niftiest about category theory is category diagrams. A lot of things that normally turn into complex equations or long-winded logical statements can be expressed in diagrams by capturing the things that you're talking about in a category, and then using category diagrams to express the idea that you want to get accross.
A category diagram is a directed graph, where the nodes are objects from a category, and the edges are morphisms. Category theorists say that a graph commutes if, for any two paths through arrows in the diagram from node A to node B, the…
goodmath | June 7, 2006To get started, what is category theory?
Back in grad school, I spent some time working with a thoroughly insane guy named John Case who was the new department chair. When he came to the university, he brought a couple of people with him, to take temporary positions. One of them was a category theorist whose name I have unfortunately forgotten. That was the first I'd ever heard of cat theory. So I asked John what the heck this category theory stuff was. His response was "abstract nonsense". I was astonished; a guy as wacky and out of touch with reality as John called something abstract…
goodmath | June 6, 2006As 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…