Motivic cohomology

Motivic cohomology is like a big playground where we play with shapes and patterns. But the shapes and patterns are not just regular ones like squares and circles, they come from special things called algebraic varieties.

You know how we have colors for painting, like red, blue, yellow, and so on? Well, these shapes and patterns also have colors. These colors are called motives, and we use them to keep track of which shapes and patterns we are playing with.

The way we play with these shapes and patterns is by doing something called cohomology. Co-what-now? Basically, we count how many holes a shape has. If a shape has no holes, it gets a zero. If it has one hole, it gets a one, and so on.

Now, let's go back to our algebraic varieties. These are shapes and patterns that people who like math have made up, and they are really special because we can do a lot of cool things with them. For example, we can talk about how curved they are, or how many points they have.

But we can also do cohomology on them, just like we did with regular shapes. And since they come in different colors (motives), we can separate them and play with each one separately.

Why do we care about all of this? Well, it turns out that by looking at these shapes and patterns and counting their holes (doing cohomology), we can learn a lot about them. We can even solve really hard math problems using these tools!

So, motivic cohomology is just a fancy way of playing with special shapes and patterns (algebraic varieties), using colors (motives) to keep track of them, and counting their holes (doing cohomology) to learn more about them.