The Stacks project

4.13 Monomorphisms and Epimorphisms

Definition 4.13.1. Let $\mathcal{C}$ be a category and let $f : X \to Y$ be a morphism of $\mathcal{C}$.

  1. We say that $f$ is a monomorphism if for every object $W$ and every pair of morphisms $a, b : W \to X$ such that $f \circ a = f \circ b$ we have $a = b$.

  2. We say that $f$ is an epimorphism if for every object $W$ and every pair of morphisms $a, b : Y \to W$ such that $a \circ f = b \circ f$ we have $a = b$.

Example 4.13.2. In the category of sets the monomorphisms correspond to injective maps and the epimorphisms correspond to surjective maps.

Lemma 4.13.3. Let $\mathcal{C}$ be a category, and let $f : X \to Y$ be a morphism of $\mathcal{C}$. Then

  1. $f$ is a monomorphism if and only if $X$ is the fibre product $X \times _ Y X$, and

  2. $f$ is an epimorphism if and only if $Y$ is the pushout $Y \amalg _ X Y$.

Proof. Omitted. $\square$

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