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. Let suppose that $f$ is a monomorphism. Let $W$ be an object of $\mathcal C$ and $\alpha , \beta \in \mathop{\mathrm{Mor}}\nolimits _\mathcal C(W,X)$ such that $f\circ \alpha = f\circ \beta $. Therefore $\alpha = \beta $ as $f$ is monic. In addition, we have the commutative diagram

\[ \xymatrix{ X \ar[r]^{\text{id}_ X} \ar[d]_{\text{id}_ X} & X \ar[d]^{f} \\ X \ar[r]^{f} & Y } \]

which verify the universal property with $\gamma := \alpha = \beta $. Thus $X$ is indeed the fibre product $X\times _ Y X$.

Suppose that $X \times _ Y X \cong X $. The diagram

\[ \xymatrix{ X \ar[r]^{\text{id}_ X} \ar[d]_{\text{id}_ X} & X \ar[d]^{f} \\ X \ar[r]^{f} & Y } \]

commutes and if $W \in \mathop{\mathrm{Ob}}\nolimits (\mathcal C)$ and $\alpha , \beta : W \to X$ such that $f \circ \alpha = f \circ \beta $, we have a unique $\gamma $ verifying

\[ \gamma = \text{id}_ X\circ \gamma = \alpha = \beta \]

which proves that $\alpha = \beta $.

The proof is exactly the same for the second point, but with the pushout $Y\amalg _ X Y = Y$. $\square$


Comments (2)

Comment #8860 by Francisco Gallardo on

Proof of Lemma 08LR, last part of the proof. It says ''commutes and if and such that..." It should say . Also, maybe I'm not seeing it, but saying that isn't enough? Should it be ? Thanks in advance.

Comment #9229 by on

Thanks for the typo which is fixed here. Yes, strictly speaking we need to tell the reader the maps.


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