On Numbers and God

Arbtirary thoughts on nearly everything from a modernist poet, structural mathematician and functional programmer.

Saturday, January 30, 2010

The fundamental group functor

This is something I've always (read: since I learned about it less than 6 months ago) found pretty neat. There's nothing terribly original here-- everything can be found in any algebraic topology book, and in most general topology books, but I don't think categorical language makes its way in there all the time...

The point of this "little" post is to point out that the operation taking a (pointed) topological space $(X,x_0)$ to it's fundamental group, $\pi_1(X,x_0)$ is a functor which preserves products and coproducts... (Did that sentence have a point? Sorry... I'm done.)

First, as a technical point: we need to work int he category of pointed spaces: Top$_*$. (A pointed topological space is just a pair $(X,x_0)$ where $x_0\in X$. The morphisms are continuous functions $f:(X,x_0)\rightarrow (Y,y_0)$ such that $f(x_0)=y_0$. The idea is we are distinguishing a point, just as we do to get the fundamental group.) The reason for this is that it gives a nice way of distinguishing between base points (for our fundamental group) in different path-components-- every selection of base point gives us a new space-- Some are isomorphic. This allows us the avoid the technical nightmare of what to do with non-path-connected spaces. (I.e., we don't get a functor if we're only working in Top) There's another reason for this: Wedge products give Top$_*$ a sensible notion of coproduct-- or at least, one which is actually preserved by the functor.

So, first of all, what does it mean for us to have a functor? A functor is a map between categores which preserves identities and composition of morphisms. In other words, for categories $C$ and $D$, $F:C\rightarrow D$ is a functor if $F(id_c)=id_{F(c)}$ for every object $c\in C$, and for every pair of morphisms
$c_0\stackrel{f}{\rightarrow}c_1\stackrel{g}{\rightarrow}c_2$
In C, we have that $F(g)\circ F(f) = F(g\circ f)$.

Given a function $f:(X,x_0)\rightarrow(Y,y_0)$, $f$ induces a homomorphism $f_* : \pi_1(X,x_0)\rightarrow \pi_1(Y,y_0)$-- Any path in $X$, when fed through $f$ becomes a path in $Y$. Since the map preserves basepoints, a loop at $x_0$ becomes a loop at $y_0$-- seeing that this is compatible with homotopy isn't too difficult.

To say that $\pi_1(-)$ is a functor means that $f_*(\pi_1(X,x_0)) = \pi_1(\operatorname{Im} f,y_0)$ and that $(id_X)_* = id_{\pi_1(X)}$ (Sorry, commutative diagrams are not working so hot in this $\LaTeX$ package... I'll need to do something about that.) A quick diagram chase shows that this is the case.

Now is where things finally get interesting... and... I'm tired, and will finish this later today.

Tuesday, January 19, 2010

A few words on Balaam's Error

I'm not sure why I'm writing this down now, but: I don't agree that "Balaam's error" has anything to do with money. Based on his actions, monetary reward seems to be a small concern for him. His error comes from this: He is afraid to contradict the Moabites. He is too polite, too unwilling to offend.
Sometimes things need to be said which are offensive-- causing offense is rarely good in its own right, but offensive things are important. Balaam was too afraid (either socially, or for his life) to tell the Moabites something offense, "God says 'no!'"

We should learn from this.
Cory Knapp.