Groupoid scheme

Okay, imagine you have some toys, and you want to organize them into groups based on certain similarities between them. Maybe you put all the animals in one group, all the cars in another group, and all the dolls in yet another group.

A groupoid scheme is kind of like that, but with mathematical objects called schemes. These schemes are like collections of points with certain properties. Instead of toys, we're organizing these schemes into groups based on how they're related to each other.

But how do we decide how schemes are related to each other? Well, we use something called a groupoid. A groupoid is like a way of comparing different schemes and seeing how they're related.

Let's say you have two schemes, A and B. Maybe there's a morphism (a fancy name for a relationship between schemes) that goes from A to B, which we can write as A → B. And maybe there's another morphism that goes from B to A, which we can write as B → A.

If we imagine A, B, and the morphisms between them as a little diagram, it might look something like this:

A → B
↓ ↓
B → A

This diagram is telling us that there are two ways to get from A to B: either by going directly from A to B, or by going from A to B and then back to A. And it's telling us that there are two ways to get from B to A: either by going directly from B to A, or by going from B to A and then back to B.

If we have a groupoid scheme, it means we have a whole bunch of these little diagrams, with different schemes and morphisms between them. And we can do all kinds of cool things with these diagrams, like compare different morphisms and look for patterns.

So that's what a groupoid scheme is: it's a way of organizing schemes into groups based on how they're related to each other, using these little diagrams called groupoids.