Split exact sequences.

Question

Edit: The setting is some abelian category.

The splitting lemma says that the following conditions are equivalent for a short exact sequence $$0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$$

1. The left arrow has a left inverse
2. The right arrow has a right inverse
3. Both the above conditions hold in such a way that $B$ is the biproduct of $A$ and $C$

A short exact sequence is said to split if it satisfies any of the above conditions. Given a biproduct, the sequence $$0\rightarrow A\overset{\iota _1}{\rightarrow} A\oplus C\overset{\pi _2}{\rightarrow}C\rightarrow 0$$ is always exact, and in fact always split. Now, Borceux says that the splitting lemma asserts that all split exact sequences are of this type up to isomorphism.

But why the "up to isomorphism"? "The" biproduct is not really unique, so it seems that for any split sequence, $B$ is just a "version" of $A\oplus C$. What isomorphism is Borceux talking about? Is he picking a particular "version" of the biproduct and merely saying all the others are isomorphic to it?

Answer 1

One point to make is that the author isn't simply saying that $B$ is isomorphic to $A\oplus C$. Rather, he's saying that the two short exact sequences are isomorphic (in the category of short exact sequences). This is a stronger statement.

However, I think your point is correct, that he could simply say "all short exact sequences are of this form", relying on the fact $A\oplus C$ (and the whole sequence containing it) is only defined up to isomorphism anyhow. This relies on taking $A\oplus C$ to be defined by its universal property. In some familiar abelian categories, though, we might define $A\oplus C$ to consist of pairs $(a,c)$ with $a \in A$ and $c\in C$. In that case, $A\oplus C$ is defined as a set, and therefore we need to point out that $B$ is not the same set as $A\oplus C$, i.e., that it is only isomorphic to it. (This is the set-theoretic issue which Hoot referred to.)