Fiber functor

Okay kiddo, imagine you have a basket full of toys and you love playing with them. Now let's say you have a friend who lives far away and also loves playing with toys. You want to share your toys with your friend, but you can't bring your whole basket with you.

Instead, what you can do is choose some of your favorite toys and put them in a smaller basket. Then, you can send that smaller basket to your friend so they can play with the same toys you love. That smaller basket is like a sample of all the toys in your bigger basket.

That's kind of what a fiber functor does, but with math instead of toys. A fiber functor takes a big, complicated math structure called a "topos" and picks out some smaller structures inside it to focus on. Those smaller structures are like the toys in the smaller basket.

Just like how you sent the smaller basket of toys to your friend, the fiber functor "sends" the smaller structures to a different math structure called a "category". This category is like your friend who loves toys and wants to play with them just like you do.

So, to sum it up, a fiber functor is like choosing a smaller basket of toys from a bigger basket, and then sending that smaller basket to a friend who loves playing with toys. It's a way of focusing on specific parts of a complicated math structure and using them in a different, simpler structure. How's that for an explanation, kiddo?