Homotopy

Okay kiddo, let's talk about homotopy. Imagine you have a piece of dough and you want to shape it into a circle. You can squish it and reshape it until it looks like a perfect circle. This is kind of like what homotopy is!

In math, homotopy is a way to see if two things are the same shape, even if they look different. We use a special kind of map called a continuous map to help us see this. Imagine you have two different shapes, like a square and a circle. A continuous map is a way to stretch and squish the square until it looks like a circle.

But wait, there's more! It's not quite that simple. Sometimes we can't just stretch and squish one shape into another. For example, imagine you have a donut and a coffee cup. They both have a hole in the middle but they definitely are not the same shape. However, if we look at them in a special way and use the "squish and stretch" technique with a continuous map, we can actually show that the donut and the coffee cup are the same shape! This is called homotopy equivalence.

So homotopy helps us understand shapes and how they can be changed into each other. It may seem complicated, but just like squishing and stretching your dough, with a little bit of math, we can understand how shapes can change and be the same at the same time!