category theory:
Since I mentioned the idea of monoids as a formal models of computations, John Armstrong made the natural leap ahead, to the connection between monoids and monads - which are a common feature in programming language semantics, and a...
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Posted on March 11, 2008 10:57 AM • 5 Comments •
By now, we've seen the simple algebraic monoid, which is essentially an abstract construction of a category. We've also seen the more complicated, but interesting monoidal category - which is, sort of, a meta-category - a category built using...
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Posted on February 28, 2008 1:28 PM • 1 Comments •
In the last post on groups and related stuff, I talked about the algebraic construction of monoids. A monoid is, basically, the algebraic construction of a category - it's based on the same ideas, and has the same properties;...
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Posted on February 18, 2008 6:34 PM • 4 Comments •
This post started out as a response to a question in the comments of my last post on groupoids. Answering those questions, and thinking more about the answers while sitting on the train during my commute, I realized that...
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Posted on February 7, 2008 10:03 PM • 1 Comments •
In my introduction to groupoids, I mentioned that if you have a groupoid, you can find groups within it. Given a groupoid in categorical form, if you take any object in the groupoid, and collect up the paths through...
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Posted on February 5, 2008 12:17 PM • 23 Comments •
Today's entry is short, but sweet. I wanted to write something longer, but I'm very busy at work, so this is what you get. I think it's worth posting despite its brevity. When we look at groups, one of...
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Posted on January 24, 2008 4:12 PM • 16 Comments •
So far, I've spent some time talking about groups and what they mean. I've also given a brief look at the structures that can be built by adding properties and operations to groups - specifically rings and fields. Now,...
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Posted on January 20, 2008 7:18 PM • 16 Comments •
As promised, I'm finally going to get to the theory behind monads. As a quick review, the basic idea of the monad in Haskell is a hidden transition function - a monad is, basically, a state transition function. The...
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Posted on January 31, 2007 2:02 PM • 8 Comments •
Suppose we've got a topological space. So far, in our discussion of topology, we've tended to focus either very narrowly on local properties of T (as in manifolds, where locally, the space appears euclidean), or on global properties of T....
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Posted on December 12, 2006 5:19 PM • 13 Comments •
This is going to be a short but sweet post on topology. Remember way back when I started writing about category theory? I said that the reason for doing that was because it's such a useful tool for talking about...
Posted on September 25, 2006 4:43 PM • 2 Comments •
This is one of the last posts in my series on category theory; and it's a two parter. What I'm going to do in these two posts is show the correspondence between lambda calculus and the cartesian closed categories. If...
Posted on August 10, 2006 2:33 PM • 5 Comments •
Sorry for the delay in the category theory articles. I've been busy with work, and haven't had time to do the research to be able to properly write up the last major topic that I plan to cover in cat...
Posted on August 9, 2006 11:51 AM • 9 Comments •
Today we'll finally get to building the categories that provide the model for the multiplicative linear logic. Before we jump into that, I want to explain why it is that we separate out the multiplicative part. Remember from the simply...
Posted on July 31, 2006 2:25 PM • 0 Comments •
Things are a bit busy at work on my real job lately, and I don't have time to put together as detailed a post for today as I'd like. Frankly, looking at it, my cat theory post yesterday was half-baked...
Posted on July 26, 2006 1:57 PM • 3 Comments •
So, we're still working towards showing the relationship between linear logic and category theory. As I've already hinted, linear logic has something to do with certain monoidal categories. So today, we'll get one step closer, by talking about just what...
Posted on July 25, 2006 3:42 PM • 3 Comments •
Time to come back to category theory from out side-trip. Category theory provides a good framework for defining linear logic - and for building a Curry-Howard style type system for describing computations with state that evolves over time. Linear logic...
Posted on July 19, 2006 4:13 PM • 3 Comments •
Monday, I said that I needed to introduce the sequent calculus, because it would be useful for describing things like linear logic. Today we're going to take a quick look at linear logic - in particular, at propositional linear logic;...
Posted on July 18, 2006 4:16 PM • 11 Comments •
(This post has been modified to correct some errors and add some clarifications in response to comments from alert readers. Thanks for the corrections!) Today, we're going to take a brief diversion from category theory to play with some logic....
Posted on July 17, 2006 4:52 PM • 8 Comments •
One of the questions that a ton of people sent me when I said I was going to write about category theory was "Oh, good, can you please explain what the heck a monad is?" The short version is: a...
Posted on July 12, 2006 10:55 AM • 4 Comments •
So, at last, we can get to Yoneda's lemma, as I promised earlier. What Yoneda's lemma does is show us how for many categories (in fact, most of the ones that are interesting) we can take the category C, and...
Posted on July 11, 2006 11:43 AM • 4 Comments •
Today's contribution on category theory is going to be short and sweet. It's an example of why we really care about natural transformations. Remember the trouble we went through working up to define cartesian categories and cartesian closed categories? As...
Posted on July 10, 2006 3:10 PM • 0 Comments •
We'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...
Posted on July 6, 2006 7:26 PM • 4 Comments •
What'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...
Posted on July 4, 2006 11:30 AM • 9 Comments •
Before 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...
Posted on June 27, 2006 4:19 PM • 6 Comments •
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...
Posted on June 22, 2006 3:31 PM • 3 Comments •
The 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...
Posted on June 19, 2006 10:49 AM • 1 Comments •
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...
Posted on June 12, 2006 8:40 AM • 7 Comments •
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...
Posted on June 9, 2006 3:59 PM • 9 Comments •
One 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...
Posted on June 8, 2006 11:25 AM • 17 Comments • 1 TrackBacks
To 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...
Posted on June 7, 2006 10:02 AM • 19 Comments •
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...
Posted on June 6, 2006 11:52 AM • 6 Comments •
