Definition 12.5.3. Let $f : x \to y$ be a morphism in an abelian category.

1. We say $f$ is injective if $\mathop{\mathrm{Ker}}(f) = 0$.

2. We say $f$ is surjective if $\mathop{\mathrm{Coker}}(f) = 0$.

If $x \to y$ is injective, then we say that $x$ is a subobject of $y$ and we use the notation $x \subset y$. If $x \to y$ is surjective, then we say that $y$ is a quotient of $x$.

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