Freyd’s initial object theorem

It’s been a couple of weeks since I’ve posted anything here. I’ve been trying to understand homotopy theory, especially the modern kind with model categories. The second semester of my algebraic topology course is slated to cover that, to which I am looking forward. Right now, we are learning about spectral sequences. I have also been trying to understand Tate’s thesis, unsuccessfully.

Today, I’d like to prove a fairly nontrivial result, due to Freyd, following MacLane; this is a post that, actually, I take from a recent change I made to the CRing project. This gives a sufficient condition for the existence of initial objects.

Let be a category. Then we recall that if for each , there is a unique . Let us consider the weaker condition that for each , there exists a map .

Definition 1 Suppose has equalizers. If is such that for each , then is called weakly initial.

We now want to get an initial object from a weakly initial object. To do this, note first that if is weakly initial and is any object with a morphism , then is weakly initial too. So we are going to take our initial object to be a very small subobject of . It is going to be so small as to guarantee the uniqueness condition of an initial object. To make it small, we equalize all endomorphisms.

Proposition 2 If is a weakly initial object in , then the equalizer of all endomorphisms is initial for .

 

CRing project blog

The CRing project now has a blog. As a result, I’ll be able to return to normal posting here, while discussion about CRing will be able to occur freely there.

 

Git server for CRing now up

Adeel Khan has set up a git server for the CRing project. In particular, you can follow how open source commutative algebra evolves in real time. More practically, you can download source files from there; they’re also on the main website, of course, but the ones there are likely to be slightly newer: I can’t update the website on my college account instantly. Plus, you can see who contributed what in what is really an intuitive and transparent manner. You can also use the git server to submit contributions, though for that you’ll need the password. For this, you can write to cring.project(at)gmail.  Again, if you don’t want to use git, we’re happy to receive contributions by email.

So why and how should you contribute? Johan deJong has explained it here (for the Stacks Project); the same applies to the CRing project. After all, the source code to your old homework sets or class notes isn’t doing anything on your hard drive. We’d be thrilled to receive it and to list you as a contributor. And we’ll work out how to edit it in (unless you want to, which you are welcome to).

 

The CRing project: open source, massively collaborative commutative algebra?

[The present post is an announcement of the CRing project, whose official webpage is here.]

Like most mathematics students, I spend a lot of time writing stuff, for instance homework assignments and (of course) blog posts. So I have a lot of random, unorganized write-ups littered around my hard drive, which might be useful to others if organized properly, but which currently slumber idly.

Last semester, I took a fairly large amount of notes for my commutative algebra class (about 160 pages). I made the notes available on my webpage, and was pleased with the reception that they received from my classmates. After seeing Theo-Johnson Freyd’s projects, I decided that it might be a productive exercise to edit the notes I had taken into a mini-textbook. I quickly made progress, since the basic structure of the book was already set by the lectures. I decided early on that the work was going to be open source: to me, it seemed the best way to ensure that anyone who wanted could freely access and modify it.

But I think the project is bigger now. Namely, instead of an open source textbook, I want a massively collaborative open source textbook. This is to say that I don’t want it to be my work anymore, but my work as well as, and more importantly, the work of enthusiastic professors, procrastinating graduate students, nerdy high-schoolers,  or whoever else wishes to contribute. The goal is to end with an openly available textbook suitable for a beginner familiar only with elementary abstract algebra, but which will provide adequate preparation for the serious study of algebraic geometry.

So, I present you the CRing project.