Equivariant sheaf

Let's pretend you have a toy box with different types of toys in it, like cars, dolls and blocks. Now, let's imagine you have a magic wand that can change the way you play with your toys. Sometimes you might want to play with all your cars first and then your dolls, and other times you might want to play with your dolls and then your blocks. Your magic wand helps you switch things up and play with your toys in different orders.

In math, we have something called a sheaf, which is like a collection of toys. But instead of different types of toys, we have mathematical objects like numbers, functions, and geometric shapes. Sometimes, we want to play with these objects in different ways, just like we want to play with our toys in different orders. This is where the idea of an equivariant sheaf comes in.

An equivariant sheaf is like a sheaf with a magic wand. It's a collection of mathematical objects that can change the way they behave depending on how we use the magic wand. The magic wand is something called a symmetry, which is like a rule that tells us how to transform our objects. For example, we might have a symmetry that flips a shape over, or a symmetry that rotates a shape.

When we have an equivariant sheaf, we can use our symmetries to change the way our objects behave. Just like our magic wand can help us change the order we play with our toys, our symmetries can help us change the way we play with our mathematical objects. So, an equivariant sheaf is a special kind of collection of mathematical objects that can transform themselves in different ways depending on the symmetries we use.