March 4th, 2007


What I do for my day job

I was out in the hills yesterday (and we had snow! Actual snow!) with an architect friend, and he asked me what my thesis was about. "Category theory," I said, "It's the theory of connections between different branches of mathematics." "Ah," he said, "so you're building bridges?". I thought about this for a bit, and replied "Not really; it's more like I'm studying the principles of bridge design. The bridges turn out to be quite interesting in their own right, you see."

I'm quite pleased with this metaphor, so I'm going to try pushing it until it breaks.

Imagine an archipelago, criss-crossed by bridges. Most of the islands have settlements on them: some of them are very large, and have thousands of inhabitants, and some are very small. Most people live permanently on one or other of the islands, and only occasionally go to visit some other island if they need something that they can't get on their home island. To get to another island, of course, they need a bridge, but bridges aren't terribly interesting: the interesting things are all on the islands. Some of them are commuters, who spend roughly equal amounts of time on each of a few islands, and it was for people like that that the bridge network was set up about sixty years ago.

But there are a few people who don't stay on one or two islands: they like to move around. They're interested in the bridges, and the connections between the islands. The island-dwellers think that this is slightly strange behaviour: really, bridges are just things you need to use when you have to go to another island, but who'd devote their life to them? The island-dwellers who use bridges a lot might be grateful that there are a few bridge-geeks around maintaining the bridges, but there are still a large number of Old Testament types, who think that all this messing around with bridges is against Nature - if Man was meant to travel from island to island, he would have flippers! The bridge geeks, for their part, think it's a bit strange to stay on one or two islands all your life - properly considered, islands are just attaching-points for bridges, and anyway, one island is much like another. They try to keep this view to themselves so as not to ruffle feathers. Their lofty perspective occasionally allows them insights about the islands that the island-dwellers aren't privy to, however: occasionally, an island-dweller will say "Hey, look at this cool feature of my island!" to a bridge-geek friend, only to get the response "Well, of course. All islands with underground systems and palm trees have that feature - it's obvious, if you think about it". This tends not to make the bridge geeks too many friends. Of course, each island has features that are special and peculiar to that one island, and here the bridge-geeks' perspective isn't very helpful (though even then, the view of an island from a bridge can sometimes tell you something new). The bridge geeks accept this as a price worth paying for their greater mobility, and anyway aren't too bothered about things that only exist on one or two islands.

In this metaphor, the islands are categories, and the bridges are functors (the term "category theory" is arguably itself an artefact of pre-categorical thinking: "functor theory", or even "adjunction theory" might be a better name). The island-dwellers and commuters are the majority of mathematicians, and the bridge geeks are of course the category theorists.