Derived algebraic geometry

Okay kiddo, so you know what algebra is, right? It's like math with letters instead of numbers, and sometimes those letters can stand for things like shapes or lines or numbers we don't know yet.

Well, in derived algebraic geometry, we're using algebra in a special way to talk about shapes that are really complicated or hard to describe. Imagine you're looking at a picture of a balloon, and you want to figure out what it's made of, but you can't touch it or take it apart. That's kind of like what it's like to study shapes in derived algebraic geometry.

So instead of just talking about regular shapes like circles or squares, we use something called a "category" to describe the shape. A category is kind of like a list of things that are related to each other in a certain way, and we can use algebra to study how those things interact.

But the really cool thing is that we don't just have one category for a shape - we have a whole bunch of them, each one describing the shape in a different way. And by looking at all those different categories and how they relate to each other, we can learn a lot about the shape itself!

So in short, derived algebraic geometry is a way to use fancy math to study really complicated shapes and learn more about them, even when we can't touch them or take them apart. Pretty neat, huh?