Formally smooth map

Okay, kiddo! Have you ever seen a map like the one we use to figure out how to get from one place to another? Well, sometimes we want to talk about different kinds of maps, but instead of using them to find directions, we use them to understand how different things connect to each other.

One type of map is called a formally smooth map. A formally smooth map is like a special kind of bridge between two things that we call "commutative rings." These rings are like special boxes full of numbers that we can use to do math, like adding and multiplying.

Now, a smooth map between two boxes of numbers, or rings, is like a bridge that connects them so that we can travel from one box to the other without any bumps or rough spots. But a formally smooth map is an even smoother bridge! It's like a slippery slide that helps us travel from one box to the other without any twists or turns.

When we say a map is formally smooth, what we mean is that it helps us slide or smoothly transition from one box of numbers to another, even if there's some complicated math happening in between. This is really helpful for mathematicians who want to study how different types of math are related, because it means we can move smoothly between different boxes of numbers without getting stuck or lost along the way.