On conglomerates' axiom of choice(Category theory)

Question

There is a requirement of conglomerate(collection of classes) which demands the following property. Axiom of choice for conglomerates: For each surjection between congomerates $f:X\to Y$, there is an injection $g:Y\to X$ with $f\circ g=Id_Y$. I do not see why this must be required. And is this implying that X and Y has the same cardinality? Can someone illuminate this a bit? And why this is called axiom of choice?

Answer 1

The axiom of choice is a set theoretic axiom which states that if $X$ is a family of non-empty sets, then there is a function $f$ whose domain is $X$, such that $f(x)\in x$ for each $x\in X$.

We call these choice functions because they choose an element from each $x\in X$.

It is equivalent, however, to say that if $f\colon X\to Y$ is a surjection, then there is an injective function $g\colon Y\to X$ such that $f\circ g=\operatorname{Id}_Y$. To see why these two are equivalent, define $X_y=\{x\in X\mid f(x)=y\}$, then $\{X_y\mid y\in Y\}$ is a family of non-empty sets, and therefore it has a choice function, which we can easily prove to be injective with the wanted property; the other direction is similar.

When talking about categories, the formulation using splitting surjections is more natural than talking about choice functions and so on. So we formulate it by saying that every two surjections split.

Note that while the axiom of choice implies that given any two sets $|X|\leq|Y|$ or $|Y|\leq|X|$, and using the Cantor-Bernstein theorem we can say that if there are surjections from $X$ onto $Y$ and vice versa then the cardinals are equal; but this is certainly not true. It might be that every surjection from $X$ to $Y$ splits, but there is no bijection between them. For example if $X$ has two elements and $Y$ has just one.

The axiom, in general, has many many consequences. There are books which include consequences and equivalents of the axiom of choice, and I'm not sure which you'd like to hear about.

Finally, note that I was talking about sets, but the same can be said about classes or conglomerates.