Simplicial homotopy

Okay, so let's pretend you have some shapes made out of really simple lines and dots, like triangles and squares. We call these shapes "simplicial complexes".

Now, let's say you have two of these shapes, and you want to see if you can turn one into the other by stretching and bending the lines around, without ripping or tearing anything. If you can do that, we say the two shapes are "homotopy equivalent".

To figure out if two shapes are homotopy equivalent, we can use something called "simplicial homotopy". Basically, we look at all the different ways we can stretch and bend the lines in one shape to make it look like the other shape. But we have to follow some rules:

1. We can only move the vertices (the dots) of the shape, and we have to keep them on the lines of the shape. So we can't just move a dot anywhere we want.
2. We can't add or remove any lines or dots, or change the way they connect to each other. We have to keep the shapes the way they are.
3. We can't rip or tear any of the lines.

If we can find a way to stretch and bend the lines around following these rules, and make one shape look like the other, then we know they are homotopy equivalent.

This might seem like a lot of complicated rules, but it's actually really useful for studying shapes and figuring out how they're related to each other. Plus, it's kind of fun to try to stretch and bend things around and see if you can make them look like something else!