Cohomology of a stack

Imagine you have a big pile of blocks. Each block has a different shape, size, and color. Now, let's say you want to know how many blocks are in your pile that are blue and square-shaped. You could start taking blocks out of the pile one by one and checking if they are blue and square-shaped, but that would take a long time. Instead, you could organize your blocks by grouping together all the blue blocks and then grouping all the square-shaped blocks. Then, you could look at the blue and square-shaped group and see which blocks are in both groups. This is kind of like what cohomology of a stack does.

A stack is like a big pile of blocks, but instead of different shapes and colors, it's made up of different mathematical objects. These objects might be manifolds, schemes, or something else. Cohomology is a way to measure how these objects fit together, kind of like counting the number of blocks that are blue and square-shaped. But cohomology of a stack is a bit more complicated than just counting blocks.

In cohomology of a stack, you group together the mathematical objects that have something in common, like being a certain shape or having a certain property. This creates a bunch of groups, called cohomology groups. These groups tell you about the ways that the objects fit together and how they interact with each other.

So, cohomology of a stack is like organizing a big pile of blocks by grouping together the ones that are similar and seeing how they fit together. It helps us understand the structure of the mathematical objects that make up the stack and how they behave.