Simplicial homotopy theory

Okay, kiddo. So imagine you have a bunch of shapes that you can build with little triangles. You can put these triangles together in different ways to make different shapes.

Now, let's say you made two shapes, but they look a little different. Maybe one has a little dip in the middle and the other doesn't. But you know that they are pretty similar because they are both made with the same little triangles.

In simplicial homotopy theory, we want to figure out when two shapes made with these little triangles are basically the same, even if they look a little different. We do this by moving the triangles around, like shifting a puzzle piece, and seeing if we can turn one shape into the other. If we can do this without breaking any of the triangles, we say the two shapes are homotopic.

Now, this might sound a little complicated, but it's actually really important in math because we use these shapes to understand things like curves and surfaces. So by figuring out when two shapes are homotopic, we can better understand how different mathematical objects are related to each other.