Confused with the notation about category of abelian groups

Question

The following text is from an MIT open course.

Let the category $\mathcal{C}=\text{Ab}$. Let $f:A \to B$ be a split epi and $g:B \to A$ be a section of $f$. Let $\iota: \text{ker}A \to A$ be the inclusion map. Then we have the map $$[g \quad \iota]: B \oplus \text{ker}A \to A$$

What does it mean by the map $[g \quad \iota]$? Thank you.

Answer 1

I guess they mean $[g \;\;\; \iota] = g + \iota$ defined pointwise, i.e a tuple $(b,a)$ is send to $g(b) + a$.

Answer 2

These notes were produced by students taking notes during lectures. (They seem not too useful without the context of the lecture.)

At any rate, this is most likely the copairing which in the case of an additive category is simply the sum of the morphisms $g\circ\pi_1$ and $\iota\circ\pi_2$ where the $\pi_i$ are projections. The copairing is often written $[g,\iota]$ and that might be what led to what was in the notes. As the nLab article states, often this is written vertically $\begin{bmatrix}g\\\iota\end{bmatrix}$.

That said, they are using a more matrix-like notation. We can express maps between finite biproducts generally in a matrix form. In this context, $\begin{pmatrix}h\\k\end{pmatrix}:A\to B\oplus C$ would mean the pairing map for $h:A\to B$ and $k:A\to C$, i.e. $a\mapsto (h(a),k(a))$, and $\begin{pmatrix}g&\iota\end{pmatrix}:B\oplus\ker A\to A$ would be the copairing.