## Archive for February, 2011

### This is cool (Freyd’s initial object theorem)

February 7, 2011

I blogged about it here, and added it to the CRing project here. Basically, the point is that complete categories are prone to having initial objects (of course, cocomplete categories are!) if they have a “weak” initial object: one that homs into every other object, but not necessarily uniquely.

I don’t think we’ll ever use it in the project, but it is neat. Covering a bit of basic category theory for its own sake seems like a reasonable thing to do, moreover.

To see this, one can use the following argument of McCabe. Let $k'/k$ be an extension finitely generated as an algebra. There is a subring $k[x_1, \dots, x_n]$ over which $k’$ is algebraic. The finite generation hypothesis implies that we can find a localization $k[x_1, \dots, x_n]_y$ over which $k’$ is even integral. But if one has an integral extension and the top ring is a field, then so is the bottom one; this implies $k[x_1, \dots,x_n]_y$ is a field. So $y \in k[x_1, \dots, x_n]$ must lie in every maximal ideal, meaning it is zero, contradiction.