Projective Objects in a Topos I'm working a problem in MacLane and Moerdijk, and am not sure how to proceed. The end goal is to show that if $Sh(X)$ has enough projectives and $X$ is $T_1$, then $X$ has a basis of clopen sets. I know that if the terminal object is projective, then this is true. The exercise seems to suggest going from what I know to the goal should be something easy.

Question: If a topos has enough projectives, is $1$ projective? What if the topos is sheaves on a $T_1$ space?


Every presheaf topos has enough projective objects (because representable objects are projective). But consider presheaves on $\bullet \leftarrow \bullet \rightarrow \bullet$, i.e. cospans in $\mathbf{Set}$. This category has no terminal object, so the terminal presheaf is not representable. In fact, it is not even projective: consider the presheaf $X$ defined by the diagram $\{ 0 \} \hookrightarrow \{ 0, 1 \} \hookleftarrow \{ 1 \}$ in $\mathbf{Set}$; the unique presheaf morphism $X \to 1$ is an epimorphism, but $\mathrm{Hom}(1, X) \to \mathrm{Hom}(1, 1)$ is not surjective.
It's worth pointing out that this presheaf topos is equivalent to the topos of sheaves on a $T_0$ topological space, via Alexandrov duality, but it is not $T_1$. I do not know any interesting $T_1$ space for which the topos of sheaves has enough projectives – the only ones I can think of right now are the discrete spaces.
