Given a geometric morphism $$f:\mathscr{F}\to\mathscr{E}$$ where $\mathscr{F},\mathscr{E}$ are toposes, we know that $f^*$ does not preserve internal homs, i.e. $f^*[X,Y]\ncong[f^*X,f^*Y]$. We do have a morphism $\Phi:f^*[X,Y]\to[f^*X,f^*Y]$, though no extra conditions on this morphism (SGA4, Tome 1, Exposé iv, 10.5). Similarly, if $(\mathscr{E},A)$ is a ringed topos, $f^*$ does not preserve internal Ext, i.e. $$f^*\underline{\text{Ext}}^n_A(M,N)\ncong\underline{\text{Ext}}^n_{f^*A}(f^*M,f^*N),$$ where $\underline{\text{Ext}}^n_A(M,N)$ is the right derived functor of the internal $A$-hom $[M,N]_A$. Does there exist a correction to this, e.g. a spectral sequence that relates the two?
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$\begingroup$ The context of your question is very much unclear --- what are $\mathcal{E}$, $\mathcal{F}$, $X$, $Y$? What do you mean by a geometric morphism? $\endgroup$– SashaCommented Apr 3 at 10:08
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1$\begingroup$ Thanks, I meant a geometric morphism between toposes, as in ncatlab.org/nlab/show/geometric+morphism. I've edited the question $\endgroup$– CameronCommented Apr 3 at 10:10
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My position would be: It is a fundamental feature of pullback along geometric morphisms to not preserve exponentials.
But, given objects $X$ and $Y$ in a topos, we can do more than just form the exponential $[X,Y]$ of morphisms from $X$ to $Y$, which is again just an ordinary object of the topos.
Instead, we can also form the (internal) space of morphisms from $X$ to $Y$. By "space" I could mean "internal locale" or "internal site" or "internal topos" (in a suitable sense). The points of this space will just be the morphisms from $X$ to $Y$, but the space is not determined by just its points.
We then have the recovering observation that pullback along geometric morphisms does preserve function spaces.
Most likely you'll find an exposition to this circle of ideas in papers by Steve Vickers. So far, I have never seen this being used in an "algebraic context" (like you are alluding to with the Ext), but perhaps the time is ripe to explore that now. If anything in this answer is unclear I'm happy to supply more details.
answered Apr 25 at 20:25
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$\begingroup$ Late reply to this but thank you Ingo! This isn't something I'm familiar with so I've been reading up various expositions. Hopefully this will help but in the meantime I've marked as useful. Gavin Wraith's Localic Groups gives the simplest way of forming the "function space" of two plain objects. The elephant seems to say these aren't preserved by pullback on the nose, but that there is a pullback operation that preserves the frames, if not the objects $\endgroup$– CameronCommented Jun 11 at 16:04
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$\begingroup$ Yes indeed, there is a well-defined pullback operation, but this does NOT proceed by applying the inverse image functor $f^*$ to the frame object. (Applying $f^*$ to a poset will yield a poset again, but even if it was complete before after applying $f^*$ it usually won't be.) Instead, locales need to be pulled back as follows: Pick a generating system for its frame, pull back that generating system using $f^*$, and then complete this generating system into a full-fledged frame again. Details are on page 25 of Steve's paper $\endgroup$ Commented Jun 17 at 13:48
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$\begingroup$ $f:\mathcal{E}\rightarrow\mathcal{F}$ gives rise to an adjunction $\Sigma_f \dashv f^*: \mathbf{Loc}_{\mathcal{E}} \rightarrow \mathbf{Loc}_{\mathcal{F}}$. this adjunction stably satisfies Frobenius reciprocity and $f^*$ preserves the Sierp\{'}nski locale. And any adjunction with these properties comes from a unique (up to iso.) geometric morphism $f$. So $f^*:\mathbf{Loc}_{\mathcal{F}} \rightarrow \mathbf{Loc}_{\mathcal{E}}$ preserves any exponential objects that exist. I.e. you can represent geometric morphisms so that the desired property holds; but the exponentiation is not in the topos. $\endgroup$ Commented Oct 20 at 17:53