category theory

In category theory, the primitive notions are nodes and arrows. A category is a set of nodes connected with arrows such that if a path leads from one node to another, so does a direct arrow; in other words, a transitive directed graph.

Nodes represent sets; arrows represent functions; categories describe particular types of sets entirely in terms of the functions that operate on them, and on related sets.

This avoids the overspecification you often get when describing mathematical objects in set theoretic terms. Take a look at the definition of tuples in terms of sets, for instance: < x, y, z > is defined as { x, {y, {z}}}. Clearly this construct 'behaves like' a tuple but it cannot really be said to be 'the real definition' of a tuple. Category theory tries to do away with this nonsense.

That's as far as I understand it. Remind me to add to this when I finish the textbook.

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.

Category theory is the study of dots (objects) and directed arrows (morphisms). Every morphism has a destination and source object (or domain and codomain); if a morphism f goes from A to B this is usually written f:A->B. Morphisms must have a composition rule: two morphisms f:A->B and g:B->C specify a particular morphism gf:A->C. Composition must be associative and have an identity for each object. Everything else is just definitions layered on top.

In addition to the familiar categories like Set (the category of sets and functions), there are a lot of cases where morphisms aren't functions at all. One common example is that any ordered set is a category, with the elements of the set forming the objects and a morphism between A and B if and only if A is less than B.

From the perspective of category theory, similarities of structure become more easily evident. My favorite example is the notion of product; cartesian products in Set are exactly the same category-theory construction as greatest lower bounds in an ordered set.