Section 4.13 (003A): Monomorphisms and Epimorphisms

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 $\mathcal{C}$ .

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 $a = b$ .
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

$f$ is a monomorphism if and only if $X$ is the fibre product $X \times _ Y X$ , and
$f$ 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 March 13, 2024 at 23:04

Proof of Lemma 08LR, last part of the proof. It says ''commutes and if $W\in \operatorname{Ob}(\mathcal{C})$ and $\alpha,\beta\colon X\to Y$ such that..." It should say $\alpha,\beta\colon W\to X$ . Also, maybe I'm not seeing it, but saying that $X\cong X \times_Y X$ isn't enough? Should it be $(X,\operatorname{id}_X,\operatorname{id})\cong (X\times_Y X,\text{pr_1},\text{pr_2})$ ? Thanks in advance.

Comment #9229 by Stacks project on May 16, 2024 at 10:13

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

The Stacks project

4.13 Monomorphisms and Epimorphisms

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