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Let $D$ and $E$ be toposes and let $f_{\ast}\colon D\to E$ be the direct image part of a geometric morphism $(f^{\ast},f_{\ast})$ between them. Considered as categories, we have (covariant) power-object endofunctors on each: $$P_D\colon D\to D \hspace{.5in} P_E\colon E\to E$$ where, for a morphism $\phi$ in $D$ we have $P_D(\phi)=\exists_\phi$, sending a sub-object of the domain to its image under $\phi$.

I'm trying to construct a natural transformation $$A_f\colon\ f_{\ast}\circ P_D\to P_E\circ f_{\ast}$$ of functors $D\to E\ $.

Question: For what geometric morphisms $(f^\ast,f_\ast)$ is such a natural transformation $A_f$ guaranteed to exist?

For example, such a thing exists in the case of change-of-base morphisms between slice toposes of ${\bf Set}$. If $q\colon X\to Y$ is a function, it induces a logical morphism $\Pi_q\colon {\bf Set}/X\to {\bf Set}/Y$. In this case the natural transformation $$A^~_{\Pi^~_q}\colon\Pi_q\circ P_{{\bf Set}/X}\to P_{{\bf Set}/Y}\circ \Pi_q$$ exists. It acts fiberwise on $Y$; for each $y\in Y$ it sends a $q^{-1}(y)\ $-indexed collection of subsets to their product.

How to construct this map $A_f$ in general? I wanted to use what would generalize to the morphism $f_{\ast}f^{\ast}\Omega_E\to\Omega_E\ $ induced by the mono-part of the epi-mono factorization for $\Omega_E\to f_{\ast}f^{\ast}\Omega_E$, where $\Omega_E$ is the subobject classifier in $E$. But while such a map does exist for all geometric morphisms, and can be used to construct the components of my desired $A_f$, I couldn't see how to prove that the naturality squares for $A_f$ commute. I showed it in the ${\bf Set}$ case using a basic set-theoretic argument.

So what made it work for slice toposes of ${\bf Set}$? Was it very specific to that case? Was it that these change-of-base functors are logical morphisms, or that they're essential geometric, or does such an $A_f$ always exist?

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Since every power object is an internal Heyting algebra, and $f_*$ preserves the structure of internal Heyting algebras, there are trivial examples of such natural transformations corresponding to the constants $\top$ and $\bot$. Of course, this is uninteresting.

Let me write $P^{\mathcal{D}}$ and $P^{\mathcal{E}}$ for the respective contravariant power object functors. Since $f_*$ preserves monomorphisms, there is a canonical comparison morphism $f_* \Omega_{\mathcal{D}} \to \Omega_{\mathcal{E}}$; since $f_*$ preserves products, there is a canonical natural morphism $f_* (Y^X) \to (f_* Y)^{f_* X}$; and so there is a canonical natural morphism $f_* P^{\mathcal{D}} X \to (f_* \Omega_{\mathcal{D}})^{f_* X} \to P^{\mathcal{E}} f_* X$. So there is an interesting canonical natural transformation $\theta : f_* P^{\mathcal{D}} \Rightarrow P^{\mathcal{E}} f_*$.

Now allow me to argue using generalised elements. Let $T$ be an arbitrary object of $\mathcal{E}$, and let $p : X \to Y$ be a morphism in $\mathcal{D}$. Given a generalised element $t : T \to f_* P_\mathcal{D} X$, what is $\theta_X \circ t : T \to P_{\mathcal{E}} f_{\ast} X$, and what is $f_* \exists_p \circ t : T \to f_* P_{\mathcal{D}} Y$? Let $t' : f^* T \to P_\mathcal{D} X$ be the left adjoint transpose of $t$, and let $A' \rightarrowtail X \times f^* T$ be the subobject classified by $t'$. It is clear that $\theta_X \circ t$ is just the classifying morphism for the pullback of $f_* A' \rightarrowtail f_* X \times f_* f^* T$ along $f_* X \times T \to f_* X \times f_* f^* T$. Also, by naturality, $f_* \exists_p \circ t$ must be the right adjoint transpose of $\exists_p \circ t' : f^* T \to P_\mathcal{D} Y$, which is none other than the classifying morphism for the image of the composite $A' \rightarrowtail X \times f^* T \to Y \times f^*T$.

This suggests the crucial criterion is that $f_*$ preserve epimorphisms (and hence, epi–mono factorisations) – and this automatic for all base change morphisms for slices over $\textbf{Set}$ because $\textbf{Set}$ and its slices have the axiom of choice. So assume $f_*$ preserves epimorphisms. If we write $A \rightarrowtail f_* X \times T$ for the subobject classified by $\theta_X \circ t$, $B' \rightarrowtail Y \times f^* T$ for the image of $A' \rightarrowtail X \times f^* T \to Y \times f^* T$, and $B \rightarrowtail f_* Y \times T$ for the subobject classified by $\theta_Y \circ f_* \exists_p \circ t$, then the preservation of epi–mono factorisations implies that $f_* B'$ remains the image of $f_* A'$ under $f_* X \times f_* f^* T \to f_* Y \times f_* f^* T\ $; but epi–mono factorisations are stable under pullback in a topos, hence $B$ is the image of $A$ under $f_* X \times T \to f_* Y \times T$. Thus, we have $$\theta_Y \circ f_* \exists_p \circ t = \exists_{f_{\ast} p} \circ \theta_X \circ t$$ for all generalised elements $t : T \to f_* P_{\mathcal{D}} X$, and thus $\theta$ is also a natural transformation $f_* P_{\mathcal{D}} \Rightarrow f_* P_{\mathcal{E}}$.

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That $f_\ast$ should preserve epis was the conclusion of my analysis as well (but when I came here to write it out, your answer was here already (-:). –  Todd Trimble Mar 17 '13 at 1:38
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For posterity, let $D$ be the topos of cospans in ${\bf Set}$ and let $E$ be the topos ${\bf Set}$ of sets. The unique geometric morphism $f_{\ast}\colon D\to E\ $ sends each cospan to its fiber product. It does not preserve epis, indeed let $X$ be the cospan $\{1\}\to\{1,2\}\leftarrow\{2\}\ $ and let $Y$ be the terminal cospan. The unique morphism $p\colon X\to Y\ $ is epi but $f_{\ast}(X)=\emptyset$ whereas $f_{\ast}(Y)=1\ $. One can check that the components given by $\theta$ constructed above do not form a naturality square for $p$. –  David Spivak Mar 18 '13 at 4:07
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A different way to describe the same answer that Zhen and Todd arrived at is to work in the internal logic of $E$. That way we may pretend that $f_*$ is the global sections functor $\mathrm{Hom}(1,-) : D \to \mathrm{Set}$, as long as we treat $\mathrm{Set}$ constructively. Then we have the components of a putative natural transformation

$$ \mathrm{Hom}(1,P A) \to P(\mathrm{Hom}(1,A)) $$

which, under the universal property of power objects $ \mathrm{Hom}(1,P A) \cong \mathrm{Sub}(A)$, sends a subobject $S\rightarrowtail A$ in $D$ to the set of all global sections $1 \to A$ which factor through it. The naturality square for $p:A\to B$ requires that if we take the direct image subobject $p_!(S)$, then a global section of $B$ factors through $p_!(S)$ just when it lifts to some global section of $A$ factoring through $S$. It's easy to see that this is the same as asking that $1\in D$ be projective, which is equivalent to saying that the global sections functor $f_* = \mathrm{Hom}(1,-)$ preserves epimorphisms.

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Looks neat, but how do we formalize our ability to "pretend that $f_{\ast}$ is the global sections functor..."? Maybe a reference for understanding a morphism $f_{\ast}\colon D\to E$ of topoi in terms of the internal logic of $E$ would help me understand this. –  David Spivak Mar 17 '13 at 14:39
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If we have a geometric morphism $f : \mathcal{D} \to \mathcal{E}$, then $f^*$ makes $\mathcal{D}$ into an $\mathcal{E}$-indexed topos $\mathbb{D}$, whose fibre over an object $E$ is the slice $\mathcal{D}_{/ f^* E}$. Johnstone explains this in detail in Chapter B3 of Sketches of an elephant. A less high-tech version of this is to simply consider $\mathcal{D}$ as an $\mathcal{E}$-enriched category, with $\underline{\mathcal{D}}(X, Y) = f_*(Y^X)$; then $f_* : \mathcal{D} \to \mathcal{E}$ becomes a enriched-representable functor in an obvious way. –  Zhen Lin Mar 17 '13 at 16:00
    
Now that I finally understand Zhen's answer, I realize how cool the perspective of Mike's answer is. I don't think I could have easily started there, but it looks like a very powerful approach, and I'm about to sink my teeth into Johnstone B.3 to get a piece of it. Thanks guys! –  David Spivak Mar 18 '13 at 17:43
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