2012-05-10 § 14 Comments
Editing text is the opposite of handling exceptions; or, to put it another way, editing text is like exception handling but backwards in time. I realise this is an unexpected claim, so I hope you will permit me to explain. Although it has the ring of nonsense, there is a perfectly good sense in which it is just straightforwardly true.
Ah yes, category theory. Our old friend. Elucidating structural connections between apparently disconnected topics since 1945. Let me tell you a story.
2007-07-05 § 1 Comment
2007-07-03 § 1 Comment
When I started doing research, I mostly worked on categories of games. I even went so far as to write a first-year report that suggested — even confidently claimed — that I would write a thesis on the subject. Well, I’m writing that thesis now, and games appear only in passing in a single paragraph. But recently, one or two people have expressed some interest in some of the ideas in that first-year report, so I’d like to flesh out one of idea that is only sketched very vaguely in what I wrote.
The starting point is the idea that a simple category of games is (equivalent to) the initial model of a certain simple theory. « Read the rest of this entry »
2006-06-22 § Leave a comment
In my previous entry, I deferred the problem of defining lax monoidal functors between radical monoidal categories. But yesterday evening on the train I realised that there is a cute way to think about lax monoidal functors, which makes it possible to simply calculate the answer.
Suppose we have monoidal categories C and D, and a functor F: C → D. To give a lax monoidal structure on F is to give a monoidal structure on the comma category C↓F, such that the projections C↓F → C and C↓F → D are strict monoidal.
I wasn’t going to explain that, because it’s fun to figure it out and rather dull to explain, but in the end I decided to put an explanation at the bottom of this post (appendix B). « Read the rest of this entry »
2006-06-20 § 8 Comments
As you can probably tell, I’m hugely excited about Joachim Kock’s paper. I apologize to those of you who read it a year ago, and think I’m a bit late to the party.
Most late-stage PhD students, I imagine, have written an imaginary textbook entitled Things I wish I’d known three years ago. My own contribution to this genre has a snappier title – it’s called Monoidal categories – and since yesterday it’s been frantically rewriting itself. (The advantage of imaginary books over real ones is that they can do that.)
2006-06-19 § Leave a comment
This morning’s crop of arxiv updates included a new version of Joachim Kock’s Elementary remarks on units in monoidal categories. Somehow I hadn’t noticed the earlier version; it’s a beautiful result, and it implies the lemma of mine that I mentioned in Paré’s observation.
2006-06-15 § 8 Comments
It’s a curious (though well known) phenomenon that an equivalence in a bicategory can always be converted into an adjoint equivalence by tweaking one of the 2-cells. There are two ways to prove it, that I know of. The elementary way is just to write down a (rather complicated) definition and laboriously prove that it has the desired properties. I always found this rather baffling, so I preferred the more abstract approach: in essence, it’s easy to show that equivalences in Cat have this property by using the Yoneda lemma, then one can use the bicategorical Yoneda lemma to transfer the result to an arbitrary bicategory. But recently I learnt something that makes the direct approach a lot easier for me to understand.
2006-06-14 § 6 Comments
One of the great joys of category theory is the way you can so often watch familiar structures emerge unexpectedly from general constructions. It’s particularly amusing to apply the formal theory of monads to the 2-category Rel of sets and relations. (I mean the 2-category whose objects are sets, whose 1-cells are relations, and where there is a unique 2-cell from R to S just when R is a subset of S.) This is an easy example to study because all diagrams of 2-cells commute, since by definition there is at most one 2-cell between any pair of relations.
« Read the rest of this entry »
2006-06-14 § 5 Comments
It gives a recipe for constructing models of linear logic that have specified properties, by starting from a poset that has the required properties and constructing a category based on that. There is something rather mystifying about the construction as it stands, so I wondered whether there’s some way of thinking about it that sheds more light on it. I think there is, and I may as well record it here in case anyone is interested. « Read the rest of this entry »
2006-06-06 § 1 Comment
I was chatting to Craig “Cap. Astro” Pastro yesterday evening over a beer, and he mentioned an interesting question: if you take the monoidal centre of a braided monoidal category, do you get back the thing you started with? We figured out the answer at breakfast this morning, and I’ll explain it in a bit. But first maybe I should explain what the question means.