Do pull backs preserve epi morphisms in a topos?

Question

Do pull backs preserve epi morphisms in a topos? I know epi morphisms are not always preserved by pull back, but what if the category is a topos?

Answer 1

Yes, this is true. Let $f \colon X \to Z$, $g \colon Y \to Z$. Assume that f is an epimorphism. We want to prove that the projection $X \times_Z Y \to Y$ is an epimorphism.

We'll work in the internal language of the topos. In terms of this internal language, a map $h \colon A \to B$ is an epimorphism if for every $b \in B$, there exists an $a \in A$ such that $h(a) = b$. The pullback $X \times_Z Y$ is $\{(x, y) \in X \times Y | f(x) = g(y)\}$.

Now let $y_0 \in Y$. We want to show that there exists an element of $X \times_Z Y$ such that the projection of that element is $y_0$. Since $f$ is epic, there exists $x_0 \in X$ such that $f(x_0) = g(y_0)$. Hence, $(x_0, y_0)$ is in $\{(x, y) \in X \times Y | f(x) = g(y)\}$ and the projection into $Y$ is $y_0$.

Since this is true for any $y_0 \in Y$, the projection is an epimorphism.

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For a proof avoiding the internal language, see section 6 here, where it's proved that every epimorphism in an elementary topos is a coequalizer (Corollary 6.14). Since toposes are locally cartesian closed, pullbacks preserve coequalizers (this is an instance of left adjoints preserving colimits) and so pullbacks preserve all epimorphisms.