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pozorvlak

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What I do for my day job [Mar. 4th, 2007|06:01 pm]
pozorvlak
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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.
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Comments:
[User Picture]From: mi_guida
2007-03-04 07:06 pm (UTC)
For someone who needs video games explained in terms of moths and "blue shiny things" that was great. Does every island have bridge to every other island, or do you sometimes have to go via one or more intermediary islands?

Please, try to answer in terms of islands and such? I won'r understand technical stuff. Also, a monkey somewhere in the theory (to go with the palm tree) would be fun.
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From: michiexile
2007-03-05 07:56 am (UTC)
Very many of the islands are connected via at least one bridge to one very specific island - mathematically, very many interesting categories have a forgetful functor to Set - and so you might be able to go between all those by bouncing through this the bridge main central.

Then again, some of the islands are extremely weird. I don't know to what extent bridges always exist but I'd expect some of the bridges to be very rickety and unreliable, requiring you to leave your entire load behind so as to not raze that bridge.
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[User Picture]From: pozorvlak
2007-03-05 10:24 am (UTC)
I like the "rickety bridges" idea! In almost all cases, there are very rickety bridges (mathematically, constant functors) from one island to another, but they're mostly of interest to bridge geeks. Of course, if you're coming from a small enough island, then you won't have much stuff to carry with you, and a rickety bridge will do. Bridges which can carry heavy traffic are rarer, though heavily-populated islands (in particular Set, as michiexile observes) tend to be served by many large bridges.

There are some curiosities: for instance, there's one (mostly barren - perhaps empty apart from a monkey and a pine tree?) island which has a bridge that goes there from every other island, but no bridges that leave it (bridges are one-way: I should have mentioned that, sorry). There's also an (entirely barren) island which has a bridge going to every other island, and none arriving :-)
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[User Picture]From: susannahf
2007-03-06 04:45 pm (UTC)
I now have a vision of bridges composed of escalators and travelators (occasionally made of wood with big holes between the slats). And I think I actually have a pretty good idea of what you're doing now.
You're a mathematical civil engineer!
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[User Picture]From: wholepint
2007-03-04 07:48 pm (UTC)
I liked that explanation a lot... it even made sense to me at the end of my shift! :)
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[User Picture]From: azrelle
2007-03-04 11:06 pm (UTC)
oi

it's the weekend

no in depth thinking allowed
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[User Picture]From: antoniabaker
2007-03-04 11:35 pm (UTC)
oh so thats what your doing...nods in understanding...gives up...no sorry angel still don't get it.
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[User Picture]From: mi_guida
2007-03-05 12:30 am (UTC)
You know some bridges have really pretty nuts and bolts and shaped metalwork? He looks at them, and how they all fit together, and while he's on the bridge he looks at the funny people who are scared of heights and stay on the land.

I think.
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[User Picture]From: pozorvlak
2007-03-05 10:26 am (UTC)
That's more or less it, yes :-)
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[User Picture]From: antoniabaker
2007-03-06 05:12 pm (UTC)
I understand the metaphor, I just haven't got a clue what the metaphor is for.
There are other types of maths, and his maths is the way the other types lead in to each other?
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[User Picture]From: pozorvlak
2007-03-07 01:13 am (UTC)
Yes, exactly.
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[User Picture]From: stronae
2007-03-05 07:49 pm (UTC)
That really is an awesome metaphor. I can totally appreciate it even if we haven't gotten to the formal definition of a category yet in my Homological Algebra class. :)
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[User Picture]From: pozorvlak
2007-03-06 01:41 pm (UTC)
Glad you liked it :-)

Without wishing to undermine your course too much, the definition of a category is actually really simple: a category is a directed graph, with an operation called composition on chains of arrows. So a chain of arrows a1 ->^f a2 ->^g a3 gives an arrow g.f : a1 -> a3. Composition is associative, and there's an "identity arrow" for every vertex, such that 1.f = f = f.1 for all arrows f. And that's it. A functor is a graph map that preserves composition and identities. A natural transformation is like a homotopy between functors: formally, if F and G are functors C -> D, a natural transformation \alpha : F -> G is a functor \alpha : 2 x C -> D, where 2 is the category (0 -> 1), and \alpha(0,-) = F, \alpha(1,-) = G. That isn't the usual definition, but it's equivalent to it.

Some examples:
  • Every partially ordered set is a category: put a single arrow a -> b iff a ≤ b.
  • Every group (in fact, every monoid) gives rise to a category: take one object, and an arrow for every element of the group, and compose as in the group.
  • For every directed graph G, there's a category called the "free graph on G", which has an object for every vertex of G, and an arrow a -> b for every path in G from a to b. The identity arrows are the zero-length paths.
  • There's a category Set, whose objects are sets and whose arrows are functions (we don't worry too much about set-theoretical size considerations, though it's often a good idea to assume that the arrows a -> b form a set for any objects a and b).
  • Similarly, there are categories Grp, Mon, Rng, Top, Mfld, etc, of groups, monoids, rings, topological spaces, manifolds etc and the appropriate kind of map
  • Recursively, there's a category Cat of categories (ignoring set theoretic hackery to avoid Russell's Paradox)
  • Take any two categories C and D. There's a "functor category" [C,D} whose objects are functors C -> D and whose morphisms are natural transformations.
  • Take the disjoint union of the categories formed from each symmetric group. The resulting category is equivalent (read "homotopic to") the category of finite sets and bijections.
  • And so on...

Examples of functors:
  • Take Grp, Top, etc, and forget the structure on the objects to give their underlying sets. Forget about any special properties of the morphisms and take the underlying function. This gives the "forgetful functor" to Set.
  • To each directed graph, assign its free category. To each graph map f: G -> H, assign the obvious functor that it generates between the free categories on G and H. This gives the "free category" functor Digraph -> Cat. This is in a sense dual to the forgetful functor Cat -> Digraph.
  • Similarly, there's a "free group" functor Set -> Grp, a "free ring" functor Set -> Rng, and so on. They're all dual to the appropriate forgetful functors.
I hope that helped :-)
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From: ext_41350
2007-04-12 06:45 pm (UTC)

the monkey

... and, finally, the monkey performs in the free ring circus.
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