Awodey:

Of course, the objects of a category do not have to be sets, either. Here are some very simple examples:

- The category
1looks like this:

It has one object and its identity arrow, which we do not draw.- The category
2looks like this:

It has two objects, their required identity arrows, and exactly one arrow between the objects.- The category
3looks like this:

It has three objects, their required identity arrows, exactly one arrow from the first to the second object, exactly one arrow from the second to the third object, and exactly one arrow from the first to the third object (which is therefore the composite of the other two).- The category
0looks like this:

It has no objects or arrows.

We don’t have a direct application for these categories, but it’s good to keep them in mind to expand our notion of categories, and also so we can use them to form other concepts.

herding cats — Finite categories