Two questions about monomorphisms and pullbacks in a Grothendieck topos Let $\mathbf{C}$ be a Grothendieck-topos and $f:Y\to X$ a morphism.


*

*Is the pullback $W\times_X Y\to Y$ of a split monomorphism $g:W\to
    X$ along $f$ again a split monomorphism? I don't think this is true
but can't think of a counterexample.

*Suppose that the morphism $g:X\to {*}$ from $X$ into the terminal
object ${*}$ is a monomorphism. Does it follow that $X$ is a
terminal too? If not, what is a counterexample?

 A: *

*It's not quite true that a pullback of a split monomorphism is a split monomorphism, but something very close is true. Let $\mathcal{C}$ be a category with pullbacks, let $f : Y \to X$ and $g : W \to X$ be morphisms and suppose $r : X \to W$ is a morphism such that $r \circ g = \mathrm{id}_W$. We may consider $f$, $g$, and $g \circ r$ as objects in the slice category $\mathcal{C}_{/ X}$, and we may consider $g$ as a morphism $(g) \to (g \circ r)$ and   $r$ as a morphism $(g \circ r) \to (g)$, of course, then $r \circ g = \mathrm{id}_{(g)}$. Pulling back along $f$ is a functor $f^* : \mathcal{C}_{/X} \to \mathcal{C}_{/Y}$, and any functor whatsoever preserves split monomorphisms, so $f^* g : f^* (g) \to f^* (g \circ r)$ is a split monomorphism; the only problem is that $f^* (g \circ r)$ need not be $Y$.

*Most Grothendieck toposes have objects $X$ such that the unique morphism $X \to 1$ is a monomorphism but not an isomorphism. For example, $X$ could be the initial object. (There is precisely one Grothendieck topos in which the initial object is isomorphic to the terminal object: the degenerate topos.)
