Thinking about Category Theory

I've been trying to teach myself category theory for over a year now. Its affectionately referred to as "abstract nonsense" at times, and I've been wading into the abstraction with little direction, guided only by my love of computation and my desire to understand how we can use category theory in computer science.

At first I thought of categories as domains of mathematical discourse. Set, the category of sets, is set theory, Mon, the category of Monoids, is monoid theory, etc. This is a pretty abstract notion, and most of elementary category theory seemed difficult and complex in this interpretation.

Then I realized that categories are really just dots and lines. This helps with understanding the importance of commutative diagrams, as they just specify things about whats dots are connected by which lines.

Then I realized that categories as just another mathematical objects, like lattices, algebras, sets, groups, etc. I don't know why it took me so long, but I was thinking of categories as being so deep and complex I didn't see the obvious. Just like other mathematical objects, it is interesting to see which objects are categories and which are in categories- there is a category of posets, and each poset is a category, there is a category of monoids, and each monoid is a category.

Then I was thinking about the algebraic structure of categories and wondering if it related to any other structure. The lack of closure for composition (not all morphisms are composable) confused me for a little while- all other algebraic structures start with closure as a basic axiom. Well, if no other algebraic structure is like a category, then that is the definition of a category- an algebraic structure that is not closed, but rather its operator can only act on pairs of composable objects (and a series of other axioms).

All of these interpretations are useful at different times, and its been fun to see this concept from different angles.

I plan on posting more about category theory pretty soon- I've been making huge breakthroughs in understanding categorical function type theory and category theory is starting to be a useful way of thinking and understanding new material for me.