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 ${\mathcal{C}}$ be a category. Then we recall that ${A \in \mathcal{C}}$ if for each ${X \in \mathcal{C}}$, there is a unique ${A \rightarrow X}$. Let us consider the weaker condition that for each ${ X \in \mathcal{C}}$, there exists a map ${A \rightarrow X}$.

Definition 1 Suppose ${\mathcal{C}}$ has equalizers. If ${A \in \mathcal{C}}$ is such that ${\hom_{\mathcal{C}}(A, X) \neq \emptyset}$ for each ${X \in \mathcal{C}}$, then ${X}$ is called weakly initial.

We now want to get an initial object from a weakly initial object. To do this, note first that if ${A}$ is weakly initial and ${B}$ is any object with a morphism ${B \rightarrow A}$, then ${B}$ is weakly initial too. So we are going to take our initial object to be a very small subobject of ${A}$. 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 ${A}$ is a weakly initial object in ${\mathcal{C}}$, then the equalizer of all endomorphisms ${A \rightarrow A}$ is initial for ${\mathcal{C}}$. (more…)