More mathematics
Oct. 25th, 2006 09:08 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
michiexileIt was waaaaaaay too long ago I wrote mathematics here. Most of it ends up at my blog. But I thought I'd share a morsel from the last Oberseminar here: important since it is a comparatively hands-on fact, which however came as news for almost everyone involved - including my advisor.
Homotopy is one of the key tools in my branch of mathematics. The very easiest idea is that of putting rubber bands on thingies. Say you have an object - a coffee cup, or a ball, or something else. This object of yours we shall call a 2-manifold in order to put a name on it. (this immediately implies that we don't care for anything but the actual surface of your thingie)
And you are equipped with funky rubber bands. The rubber bands start out as just bands, and you can connect a band once to form a loop. Once it forms a loop, it sticks at your object as tight as it can go, and stays as tight as it can around whatever it ends up against. Once there, you can start nudging it around, but you are not allowed to break it off again.
For the ball, any way you start, you can always nudge the band until it pops off the ball again, having drawn itself together down to a small clot of rubber. For the coffee cup, the number of times you wind it around the handle decides what you can do with it: no times -> you can pull it off, n times -> you can make it look like anything else that's wound n times around, but not like anything else. And then there are some funky things going on with various ways you could wind it around the handle.
Now, this precise observation is what topologists call homotopy, and algebraists use as a really, really funky tool. The rubber bands get formalized as "maps f from the interval [0,1] to the object, with f(0)=f(1)", nudging between two rubber band positions f and g gets formalized as "a family of maps ft(x) such that f0(x)=f(x), f1(x)=g(x), and the nudging is continuous as a function", and we turn out to be interested in what remains if we put everything as equal that can be nudged into each other.
This all gets far more mangled if you do algebra - so you'll have families of vector spaces, organised into chains of vector space, linear map, vector space, linear map, et.c. Between a pair of such spaces, you can have maps that take all vector spaces in the first chain into corresponding spaces in the second chain, and which don't care about the linear maps in the chains. Two such maps correspond to two rubber bands, and we call the maps f,g between chains C and D with chain-internal maps d and d' homotopic precisely if we can find another map h from the chain C to the chain D so that f-g = d'h+hd.
Now, these things don't really look as if they have much to do with each other, now do they?
What happened at the seminar was that another homotopy for chains was introduced: there is one particular chain thingie called I that corresponds precisely to the [0,1] interval: it looks like
0 -> k -> k2 -> 0
with the single map k -> k2 given as a matrix by
/ 1 \
\ -1 /
and this one gets combined with the chain C to form a new thingie C*I. Then this h is the same thing as a map t from C*I to D such that it takes on the values of f and g for some specified values. This, all of a sudden, looks a lot more like the rubber bands: a function from this C*I to the D is precisely a function in two variables: the t and the x in ft(x), and the conditions we put on the map correspond precisely to that f0(x)=f(x), f1(x)=g(x).
And the funky thing in the seminar was proving that these two homotopies: the h such that f-g=d'h+hd and the map C*I -> D are really the very same thing.
Homotopy is one of the key tools in my branch of mathematics. The very easiest idea is that of putting rubber bands on thingies. Say you have an object - a coffee cup, or a ball, or something else. This object of yours we shall call a 2-manifold in order to put a name on it. (this immediately implies that we don't care for anything but the actual surface of your thingie)
And you are equipped with funky rubber bands. The rubber bands start out as just bands, and you can connect a band once to form a loop. Once it forms a loop, it sticks at your object as tight as it can go, and stays as tight as it can around whatever it ends up against. Once there, you can start nudging it around, but you are not allowed to break it off again.
For the ball, any way you start, you can always nudge the band until it pops off the ball again, having drawn itself together down to a small clot of rubber. For the coffee cup, the number of times you wind it around the handle decides what you can do with it: no times -> you can pull it off, n times -> you can make it look like anything else that's wound n times around, but not like anything else. And then there are some funky things going on with various ways you could wind it around the handle.
Now, this precise observation is what topologists call homotopy, and algebraists use as a really, really funky tool. The rubber bands get formalized as "maps f from the interval [0,1] to the object, with f(0)=f(1)", nudging between two rubber band positions f and g gets formalized as "a family of maps ft(x) such that f0(x)=f(x), f1(x)=g(x), and the nudging is continuous as a function", and we turn out to be interested in what remains if we put everything as equal that can be nudged into each other.
This all gets far more mangled if you do algebra - so you'll have families of vector spaces, organised into chains of vector space, linear map, vector space, linear map, et.c. Between a pair of such spaces, you can have maps that take all vector spaces in the first chain into corresponding spaces in the second chain, and which don't care about the linear maps in the chains. Two such maps correspond to two rubber bands, and we call the maps f,g between chains C and D with chain-internal maps d and d' homotopic precisely if we can find another map h from the chain C to the chain D so that f-g = d'h+hd.
Now, these things don't really look as if they have much to do with each other, now do they?
What happened at the seminar was that another homotopy for chains was introduced: there is one particular chain thingie called I that corresponds precisely to the [0,1] interval: it looks like
0 -> k -> k2 -> 0
with the single map k -> k2 given as a matrix by
/ 1 \
\ -1 /
and this one gets combined with the chain C to form a new thingie C*I. Then this h is the same thing as a map t from C*I to D such that it takes on the values of f and g for some specified values. This, all of a sudden, looks a lot more like the rubber bands: a function from this C*I to the D is precisely a function in two variables: the t and the x in ft(x), and the conditions we put on the map correspond precisely to that f0(x)=f(x), f1(x)=g(x).
And the funky thing in the seminar was proving that these two homotopies: the h such that f-g=d'h+hd and the map C*I -> D are really the very same thing.
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Re: Hmm ...
Date: 2006-10-28 03:43 pm (UTC)By the way, here's a similar (but simpler) result: let I be the category with two objects, 0 and 1, and one non-identity arrow 0->1. If C and D are categories, and F,G: C -> D are functors, then a natural transformation F -> G is exactly a functor alpha: CxI -> D for which alpha(_,0) = F and alpha(_,1) = G.
(*) Actually, is that true? I know Top is a model category, but are the weak equivalences given by homotopies?
Re: Hmm ...
Date: 2006-10-28 07:52 pm (UTC)Weak equivalences X -> Y are maps such that the induced maps on the pointed homotopy groups end up being isomorphisms.
Fibrations are Serre fibrations
Cofibrations are what they need to be to fit with these two.
And every homotopy equivalence is a weak equivalence according to this definition. There may, I think, however be more weak equivalences.
***
So, in other words, a natural transformation between two functors really is "just" a homotopy between these two functors, eh?