Duality

Take any category $\mathcal C$. Define a new category, called the opposite category, written as $\mathcal C^{\textrm{op}}$, which has the same objects, but has all the arrows reversed. For the opposite of a morphism $f$, we’ll write $f^{\textrm{op}}$. For instance, check that $(f\circ g)^{\textrm{op}}=g^{\textrm{op}}\circ f^{\textrm{op}}$.

Can we just do this? Well, look at what we require from a category. We need arrows to have composition and an identity arrow. I just showed you how composition works. The identity arrows are the same (but reversed). They compose with the other arrows in the obvious way. There’s really nothing more to it than reversing the arrows.

The first thing to notice is that $\mathcal (C^{\textrm{op}})^{\textrm{op}}=\mathcal C$. There’s really no content to this statement, but it’s worth noting, because duality is an amazingly important idea in category theory.

We’ve defined some interesting things in a category, so we can ask what happens when we pass to the opposite category. If we have a  “thing” in one category, a co-“thing” is what we get when we pass to the opposite category. This brings us a truly awful mathematical pun:

A co-co-nut is just a nut.

If you don’t think that’s funny, get off my blog.

Anyway, let’s do an example. What is a co-initial object? An initial object is something that has a unique arrow out of it going to every other object, so a co-initial object is an object that has a unique arrow coming into it from every object. We have a name for that. We call it a terminal object. Similarly, co-terminal is another word for initial.