Homotopy colimit

Okay, imagine you have a lot of toy blocks. You want to stack them up to make a really tall tower, but you don't want to stack them directly on top of each other. Instead, you want to stack them in a way that looks smooth and continuous.

Now imagine that each block represents something called a "space" in math. These spaces can be really weird and complicated, like a wiggly, twisty piece of spaghetti or a bumpy, lumpy piece of clay.

When you stack the blocks on top of each other, it's like connecting these spaces in a certain way. But instead of just stacking them directly on top of each other, you're allowed to move them around, stretch them, and squish them a little bit. This is called a "homotopy".

So when you connect these spaces using homotopy, you get something called a "homotopy colimit". It's kind of like the final shape you get after you've stacked all the blocks and made all the necessary moves to make the connecting of the spaces look smooth and continuous.

It may sound complicated, but it's just a way of combining lots of weird and complicated spaces in a way that looks nice and smooth.