3
$\begingroup$

In the question Do pushouts preserve monic? it is said that monics in a topos are stable under push out. I would like a precise reference or a nice proof of this fact for elementary topoi (for Grothendieck follows since it holds in Sets, then in presheaves and then in sheaves (pointwise computation + associated sheaf).

$\endgroup$

2 Answers 2

3
$\begingroup$

This is half of Corollary 2.4.3 in [Sketches of an elephant, Part A]. Here is (a paraphrase of) the proof:

Let $\mathcal{E}$ be an elementary topos, let $f : A \rightarrowtail B$ be a monomorphism in $\mathcal{E}$, and let $g : A \to C$ be any morphism in $\mathcal{E}$. Then the pushout of $f$ along $g$ is monic.

Proof. Let $d : C \to D$ and $e : B \to D$ constitute the colimiting cocone. We wish to show that $d$ is monic. Consider the morphism $\langle f, g \rangle : A \to B \times C$: this is monic because $f$ is. Let $h : B \to P C$ be the name of this binary relation, and let $\{ \cdot \} : C \to P C$ be the name of the equality relation on $C$. The universal property of pushouts then yields a factorisation of $d : C \to D$ through $\{ \cdot \} : C \to P C$. But $\{ \cdot \}$ is monic, so this implies $d$ is monic.

$\endgroup$
3
$\begingroup$

Just for reference, this is also the first half of corollary 4, ch. IV-10 in MacLane-Moerdijk "Sheaves in geometry and logic". The proof follows the same idea as the proof in Zhen Lin answer, except that the map $h: B \to PC$ is obtained using the previously proved property that $PC$ is injective, so that $\{.\}g$ extends to $B$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.