Example 12.5.6. Let $\mathcal{A}$ be an abelian category. Pushouts and fibre products in $\mathcal{A}$ have the following simple descriptions:

1. If $a : x \to y$, $b : z \to y$ are morphisms in $\mathcal{A}$, then we have the fibre product: $x \times _ y z = \mathop{\mathrm{Ker}}((a, -b) : x \oplus z \to y)$.

2. If $a : y \to x$, $b : y \to z$ are morphisms in $\mathcal{A}$, then we have the pushout: $x \amalg _ y z = \mathop{\mathrm{Coker}}((a, -b) : y \to x \oplus z)$.

There are also:

• 7 comment(s) on Section 12.5: Abelian categories

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).