Semiorthogonal decomposition

So imagine you have a bunch of toys that you want to organize. However, you don't just want to throw them all in a big pile, you want to group them together in a special way. This way, you can find the toys you want more easily.

When you do a semiorthogonal decomposition, it's like organizing the toys. You take a big group of objects (these are called "sheaves" in math), and you break them down into smaller groups.

But you can't just group them any old way - you have to make sure that the groups are different enough from each other that they don't overlap too much. That way, each group is unique and special. It's like having a group of toys that are all dinosaurs, another group that are all cars, and another group that are all princesses.

Sometimes, you can't make the groups completely different - they might have some overlap. But that's okay! The important thing is that they're still useful and make it easier to find the sheaves you're looking for.

Overall, a semiorthogonal decomposition is just a fancy way of organizing a bunch of sheaves so that they're easier to understand and use.