Small set (category theory)

Okay, so let's pretend that we have a box of toys, and each toy in the box is a special kind of toy. These toys are so special that we have a special name for them - we call them "sets."

Now, some of these sets might have a lot of toys in them, like the set of all the toys that are blue. Other sets might only have a few toys in them, like the set of all the toys that have superhero capes.

When we talk about a "small set" in category theory, we're talking about one of these sets that has only a finite number of toys in it. That means we can count each and every toy in the set, and we won't run out of numbers.

For example, let's say we have a small set that contains just three toys: a red car, a yellow banana, and a green frog. This set is small because we can count those three toys, and we know that there aren't any more toys in the set.

Now, in category theory, we use small sets (and bigger sets, too!) to help us understand how things are related to each other. We might say that the set of all the toys that are green is related to the set of all the toys that hop, because the green frog is the only toy that's in both sets.

And that's the idea behind small sets in category theory! They're just special sets that have a limited number of toys, but they're still really useful for helping us see how things are connected to each other.