Nodes do not really represent

sets; this is again the

overspecification that we are trying to

rebel against. Nodes represent nodes.

Arrows do not represent

functions, either.

And a category isn't really a graph, because it's usually too big! So the collection of all nodes is not a set. At best, you could say a category is a Graph (the vertices form a class, not a set), but even that would limit you away from some categories. Best not to think about it!

But for most categories, nodes will be a *type* of set, and arrows a *type* of function.

Examples of categories:

- Sets
- Nodes are sets; we write
*A* -> *B* if there is a function from *A* to *B*.
- Groups
- Nodes are groups; we write
*A* -> *B* if here is a homomorphism (of groups) from *A* to *B*.
- Fields, Rings, ...
- Fill in the blanks for yourself
- Topological spaces
- Nodes are topological spaces; we write
*A* -> *B* if there is a continuous function from *A* to *B*.
- pointed categories
- If
**X** is a category, Pointed-**X** is the category with nodes (*A*,a), where *A* is a node of **X** and a a "point" of *A*; we write (*A*,a) -> (*B*,b) if *A* -> *B* by an arrow that takes a to b. If we take **X**=topological spaces, this lets us talk about continuous functions copying some given point to another given point, which is a useful concept in topology, too.

The other important concept in category theory is the functor, which see.