Ehresmann's fibration theorem

Okay, kiddo, let's talk about Ehresmann's fibration theorem!

Imagine you have a ball of yarn that you want to unwind. If you slowly unravel the yarn, you will see that it forms a long, thin thread that stretches out in a straight line.

Now, let's say that instead of unraveling it straight, you wind the yarn around a circular object, like a donut. If you pull the thread from the donut, you'll see that it looks different from the straight line of yarn we saw before. This time, the thread is curved, and it wraps around the donut.

Ehresmann's fibration theorem is a mathematical concept that says something similar can happen with more complicated shapes. In math, these shapes are called manifolds, and they can be very complicated and have lots of different dimensions.

But no matter how complicated the manifold is, Ehresmann's theorem says that you can always imagine it as a bunch of circles, each one wrapped around the manifold in a different way. In other words, the manifold can be "fibred" by circles.

Why is this useful? Well, it helps mathematicians study these complicated shapes in a simpler way. Just like you can understand a donut by looking at how the yarn wraps around it, mathematicians can understand a manifold by looking at how it's fibred by circles.

So there you have it, kiddo! Ehresmann's fibration theorem is just a fancy way of saying that complicated shapes can be made up of simpler shapes that are wrapped around them like yarn around a donut. Cool, huh?