Fibration

Okay kiddo, imagine you are playing with a bunch of toys on a big table. You have a car toy, a plane toy, and a boat toy. And you also have a big map of the world on the table.

Now, you decide to play a game with your toys where you want to move them around on the map. But, instead of just moving them around randomly, you want to follow some rules. You want to take your car toy and move it only on the roads on the map, your plane toy can fly only in the air, and your boat toy can go only on water bodies.

This game you just created is actually an example of a fibration. The big map is called the total space and each of the toys has its own small space i.e., the roads, the air, and the water, called the fiber. And the rules or restrictions on each toy's movement are called the projection maps.

So, in the game, when you move your car toy, you are actually projecting it onto the roads on the map, and when you move your plane toy, you are projecting it onto the airspace on the map. And this projection is nothing but a way of transferring the information from the total space (the map) to the fiber (the road, air, or water) and vice versa.

This concept of sets of spaces (fibers) that are "glued" together to form a bigger space (total space) where the projection maps relate them to one another, is fundamental in mathematics and is called a fibration. And just like in the game you created, the idea of fibrations pops up all over mathematics, from topology to algebraic geometry, and is an essential tool for understanding many mathematical problems.