I talked briefly about paths in that post. Let's talk about them a little more. We want a way to talk about a path from a to b in an arbitrary topological space. In $\mathbb{R}$, this is easy enough: take any portion of a curve which starts at a and ends at b. While that's easy to understand, it's a bit unwieldy to work with directly. But we do know that a curve is some sort of a function (think back to 8th grade algebra). So, let's decide this curve is a function. And let's say $f(0)=a$, and $f(1)=b$, and for every $c\in [0,1]$, $f(c)$ lies on the curve we just talked about.
There are a two important things to notice here:
- We now know the domain of our function: the unit interval (let's call it $I$).
- This curve should be continuous, or else we have to jump... and what kind of a path is that?
There's something great about these two facts: Where did I mention that our path is a path on $\mathbb{R}$? Nowhere. This means that we can replace $\mathbb{R}$ with any topological space.
So: A path on a topological space $X$ is a continuous function $p:I\rightarrow X$, and we call $p(0)$ the starting point, and $p(1)$ the ending point.
I mentioned categories in the titles, so you may be wondering about now where the category comes from. Let our objects be the points of $X$, and let a morphism from $a$ to $b$ be a path with end-points $a$ and $b$. Is this actually a category? let $id_a$ be constant function $\forall c\in I, p(c)=a$, and let the composition be convolution: $q\circ p = p*q$, where $p*q (c) = p(2c)$ if c<1/2 and $q(2(c-1))$ otherwise.
We run into one big problem here: composition isn't associative, and composition with the identity isn't quite right... but they're both close. You hit all the same points in the right order, but the "speeds" aren't quite right.
We can rectify this with homotopies, but if we use just any homotopy, we'll get pretty boring spaces... so we do what topologists always do to their homotopies when they need to restrict them: fix the end points. So, two paths equivalent up to homotopy with fixed endpoints are now the same.
This gives us a topological space as a category. If this is a category, we should be able to get continuous functions as functors. Yep!
If $F:X\rightarrow Y$, a path $p$ will be sent to the path $F\circ p$. (Exercise: Check that this is indeed a functor, with our wishy washy paths.)
I'm not sure if I've talked about natural transformations. A natural transformation intuitively is a way to transform one functor into another, at the objects.
For two functors $F,G:C\irghtarrow D$, a natural transformation $\eta:F\rightarrow G$ is a collection of morphisms $\eta_x : Fx \rightarrow Gx$, one for each object of $C$. These morphisms need to interact properly with $F$ and $G$. Namely, if $f:x\rightarrow y$ in $C$, then
$\eta_y\circ Ff = Gf\circ\eta_x$. In other words-- starting from $Fx$, following $Ff$ and then a component takes you to the same place as following a component, and then following $Gf$. This must happen everywhere.
One way to understand what's going on is this: There are 3 worlds, $C$, and two other worlds living inside of $D$: $F$-world, and $G$-world. The components of a natural transformation allow us to travel from $F$-world to $G$-world, and if $\eta$ is truly a natural transformation, than we can travel from $F$-world to $G$-world, and then around $G$-world, or you can make the same trip in $F$-world, and then cross over to $G$-world, and either way, you end up in the same place.
What do natural transformations look like when our categories are these path-spaces? Let's see if we can't figure out what is going on. First, let $F,G:X\rightarrow Y$, where X and Y are topological spaces. Now let's look at a path p with endpoints a and b. This will give us two paths in Y, $Fp:Fa\rightarrow Fb$, and $Gp:Ga\rightarrow Gb$. We need a way to turn the first path into the second... How do we do this? Take a homotopy $H:X\times I\rightarrow Y$, where $H_0=F,\; H_1=G$. If we take $H\circ (p\times id)$ (I.e., we take our path, and then apply the homotopy), we get a homotopy from $Fp$ to $Gp$. If we further restrict ourselves to $H_0$, we get... a path $Fa\rightarrow Ga$, and likewise, if we look at $H_1$ we get a path $Fb\rightarrow Gb$. So if $H$ is truly a homotopy, it defines a natural transformation from $F$ to $G$. Likewise, if we have such a natural transformation, we can define a homotopy.
So natural transformations are homotopies. I'm going to stop here, but a fun remark: natural transformations turn Cat what's called a 2-category. So our "2-dimensional" homotopy space (i.e., paths as morphisms and homotopies as natural transformations) turn Top into a 2-category. We can keep going: homotopies between homotopies make 3-morphisms, homotopies between homotopies between ... form n-morphisms, and suddenly we have some notion of $\infty$-category. Moreover, homotopies (and paths) are invertible; which means we actually have an $\infty$-groupoid.
And hopefully that helps motivate some of (higher) category theory. Cheers.
(Disclaimer: There may be gross inaccuracies in this post... please let me know if you find any)
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