Pullback-stability of internally projective objects

Question

An object $X$ of a category $C$ is said to be projective if the hom-functor $C(X,-)$ preserves epimorphisms (or, in general, some restricted class of epimorphisms such as the regular or effective ones). The axiom of choice is equivalent to the assertion that all objects of Set are projective. In general, a "projective set" $X$ is one such that we can make $X$-indexed families of choices.

In section D4.5 of Sketches of an Elephant, an object $X$ of a topos $C$ is said to be internally projective if the right adjoint $\Pi_X : C/X \to C$ preserves epimorphisms, and $C$ is said to satisfy the internal axiom of choice if all objects are internally projective. The latter definition is found in many other places; I haven't seen the former elsewhere, but I don't know its actual origin.

My question is, if $X$ is internally projective in this sense in $C$, is $X\times Y$ internally projective in $C/Y$ for another object $Y$? This seems to be a necessary condition for the definition to be sufficiently "internal", but I haven't been able to prove it yet.

A similar question is, if $C$ satisfies the internal axiom of choice, does the functor $\Pi_f : C/X \to C/Y$ preserve epimorphisms for any morphism $f:X\to Y$?

Answer 1

For toposes, the stability property of your first question does hold. Suppose $X$ is internally projective in $\mathbb{C}$. And suppose $q: B \to A$ is an epimorphism from $v : B \to I$ to $u: A \to I$ in $\mathbb{C}/I$, hence an epimorphism in $\mathbb{C}$. Write $X^*$ for the object $X \times I$ of $\mathbb{C}/I$. The exponential $u^{X^*}$ in $\mathbb{C}/I$ has underlying object: $$ \{(i,f) : I \times A^X \mid \forall x:X.\, u(f(x))=i\}$$ And the map $q^{X^*}$ is a pullback in $\mathbb{C}$ of the map $$I \times q^X: I \times B^X \to I \times A^X$$ along the embedding of $u^{X^*}$ in $I \times A^X$. Since the displayed map above is an epi by internal projectivity of $X$, so is its pullback $q^{X^*}$.

It is also true that the internal axiom of choice (IAC) is preserved by slicing. (I haven't managed to see that this follows as a direct consequence of the previous.) Suppose IAC holds in a topos $\mathbb{C}$. Consider an arbitrary object $w: Z \to I$ in $\mathbb{C}/I$, and an epimorphism $q$ as above. The exponential $u^w$ in $\mathbb{C}/I$ has underlying object $$\{(i,f): I \times (1+A)^Z \mid \forall z:Z.\, (w(z)= i \to \exists a:A. f(z) = \text{inr}(a) \wedge u(a) = i) \wedge (w(z) \neq i \to f(z) = \text{inl}(*)) \}$$ This uses a standard representation of slice-category exponentials in toposes in terms of partial map classifiers, with the simplification that, since $\mathbb{C}$ is boolean (using Diaconescu's theorem that IAC implies boolean), the partial map classifier for $A$ is $1 + A$. Similar to the previous argument, one now observes that $q^w$ is a pullback in $\mathbb{C}$ of the map $$I \times (1+q)^Z : I \times (1+B)^Z \to I \times (1+A)^Z$$ along the subobject inclusion of $u^w$. And once again the map displayed above is an epi, by internal projectivity of $Z$.