TITLE: Sheaf-theoretic approach to forcing
QUESTION [47 upvotes]: Inspired by the question here, I have been trying to understand the sheaf-theoretic approach to forcing, as in MacLane–Moerdijk's book "Sheaves in geometry and logic", Chapter VI.
A general comment is that sheaf-theoretic methods do not a priori produce "material set theories". Here "material set theory" refers to set theory axiomatized on the element-of relation $\in$, as usually done, in ZFC. Rather, they produce "structural set theories", where "structural set theory" refers to set theory axiomatized on sets and morphisms between them, as in the elementary theory of the category of sets ETCS. I will always add a collection (equivalently, replacement) axiom to ETCS; let's denote it ETCSR for brevity. Then Shulman in Comparing material and structural set theories shows that the theories ZFC and ETCSR are "equivalent" (see Corollary 9.5) in the sense that one can go back and forth between models of these theories. From ZFC to ETCSR, one simply takes the category of sets; in the converse direction, one builds the sets of ZFC in terms of well-founded extensional trees (modeling the "element-of" relation) labeled by (structural) sets.
So for this question, I will work in the setting of structural set theory throughout.
There are different ways to formulate the data required to build a forcing extension. One economic way is to start with an extremally disconnected profinite set $S$, and a point $s\in S$. (The partially ordered set is then given by the open and closed subsets of $S$, ordered by inclusion.) One can endow the category of open and closed subsets $U\subset S$ with the "double-negation topology", where a cover is given by a family $\{U_i\subset U\}_i$ such that $\bigcup_i U_i\subset U$ is dense. Let $\mathrm{Sh}_{\neg\neg}(S)$ denote the category of sheaves on the poset of open and closed subsets of $S$ with respect to this topology.
Then $\mathrm{Sh}_{\neg\neg}(S)$ is a boolean (Grothendieck) topos satisfying the axiom of choice, but it is not yet a model of ETCSR. But with our choice of $s\in S$, we can form the ($2$-categorical) colimit
$$\varinjlim_{U\ni s} \mathrm{Sh}_{\neg\neg}(U)$$
called the filter-quotient construction by MacLane–Moerdijk. I'm highly tempted to believe that this is a model of ETCSR — something like this seems to be suggested by the discussions of forcing in terms of sheaf theory — but have not checked it. (See my answer here for a sketch that it is well-pointed. Edit: I see that well-pointedness is also Exercise 7 of Chapter VI in MacLane–Moerdijk.)
Questions:

Is it true that $\varinjlim_{U\ni s} \mathrm{Sh}_{\neg\neg}(U)$ is a model of ETCSR?
If the answer to 1) is Yes, how does this relate to forcing?

Note that in usual presentations of forcing, if one wants to actually build a new model of ZFC, one has to first choose a countable base model $M$. This does not seem to be necessary here, but maybe this is just a sign that all of this does not really work this way.
Here is another confusion, again on the premise that the answer to 1) is Yes (so probably premature). An example of an extremally disconnected profinite set $S$ is the Stone-Cech compactification of a discrete set $S_0$. In that case, forcing is not supposed to produce new models. On the other hand, $\mathrm{Sh}_{\neg\neg}(S)=\mathrm{Sh}(S_0)=\prod_{S_0} \mathrm{Set}$, and if $s$ is a non-principal ultrafilter on $S_0$, then $\varinjlim_{U\ni s} \mathrm{Sh}_{\neg\neg}(U)$ is exactly an ultraproduct of $\mathrm{Set}$ – which may have very similar properties to $\mathrm{Set}$, but is not $\mathrm{Set}$ itself. What is going on?

REPLY [17 votes]: Thanks for all the enlightening answers! Let me summarize my understanding now. (Please correct me if I'm saying something stupid!)
First, as explained by Mike Shulman in his answer, the answer to Question 1) is Yes: $\varinjlim_{U\ni s} \mathrm{Sh}_{\neg\neg}(U)$ is a model of ETCSR. So if you, like me, always wanted to have a hands-on understanding of what forcing does, I think you can go with this explicit model (and do away with all meta-mathematical baggage surrounding either boolean models, or countable base models, and generic filters).
It remains to answer Question 2): How is this related to forcing? The answer seems to be that it is a generic extension $\mathrm{Set}_s[G]$ of $\mathrm{Set}_s$ where $\mathrm{Set}_s$ is an elementary extension of $\mathrm{Set}$.
More precisely, recall the concept of "boolean ultrapowers", which is "ultrapowers with Stone-Cech compactifications replaced by a general extremally disconnected profinite set $S$". Namely, given $S\ni s$ as above, and any model $M$ of some first-order theory, we can look at the constant sheaf with value $M$ on $S$ with respect to the double-negation topology, and take its stalk at $s\in S$. (Warning: This is not really a stalk! I.e., $s$ does not define a point of the topos of double-negation sheaves. By "stalk" I just mean the colimit $\varinjlim_{U\ni s}$; equivalently, push forward to usual sheaves on the topological space $S$, and take the stalk there.) This defines a new model $M_s$ of the first-order theory. (If $S$ is the Stone-Cech compactification of $S_0$ and $s\in S$ is an ultrafilter on $S_0$, then the sheafification is taking any open and closed subset $U\subset S$, corresponding to some subset $U_0\subset S_0$, to $\prod_{U_0} M$, and then the stalk is $\varinjlim_{s\in \beta U_0} \prod_{U_0} M$, i.e. an ultraproduct.) This procedure can be applied in particular to a model of ZFC, or a model of ETCSR, with the procedures being equivalent. This construction features prominently in this beautiful article of Hamkins–Seabold.
The Boolean ultrapower $\mathrm{Set}_s$ of $\mathrm{Set}$ (at $s\in S$) is thus given by the stalk at $s\in S$ of the constant sheaf of categories with value $\mathrm{Set}$. The constant sheaf of categories with value $\mathrm{Set}$ is different from set-valued sheaves! However, there is a natural functor, in particular giving a functor
$$\mathrm{Set}_s\to \varinjlim_{U\ni s}\mathrm{Sh}_{\neg\neg}(U).$$
As in Zhen Lin's answer, the topos $\mathrm{Sh}_{\neg\neg}(S)$ classifies generic filters $G$. In particular, it contains such a generic filter itself, given by
$$ G=\bigsqcup_{U\subset S} U.$$
(Here, $U$ is identified with the sheaf it represents.)
Now I believe that the discussion around Theorem 22 in Hamkins-Seabold should translate into the following statement.

$\mathrm{Set}_s$ is an elementary extension of $\mathrm{Set}$, $G$ is a generic filter over $\mathrm{Set}_s$, and $\varinjlim_{U\ni s} \mathrm{Sh}_{\neg\neg}(U)$ identifies with the generic extension $\mathrm{Set}_s[G]$ of $\mathrm{Set}_s$.

(What is $\mathrm{Set}_s[G]$ here? It corresponds to the (material) forcing extension by $G$ under translating back-and-forth between structural and material set theory.)
Finally, how is all of this related to usual forcing? In usual forcing, one first makes an auxiliary enlargement $V\subset \tilde{V}$ (i.e. $\mathrm{Set}\to \tilde{\mathrm{Set}}$) so that after this enlargement, "there are new elements of $S$" in the sense that the corresponding complete Boolean algebra $B$ admits new homomorphisms $B\to \{0,1\}$. One can then redo the previous construction, but take the stalk at a "new point" $s\in \tilde{S}$ (where $\tilde{S}$ is the spectrum of $B$ in $\tilde{V}$). All of the above should still work in this setting, giving us $\mathrm{Set}\to \mathrm{Set}_s\to \mathrm{Set}_s[G]$. However, if $s$ corresponds to a $V$-generic filter, then it turns out that $\mathrm{Set}_s=\mathrm{Set}$. So in this case the forcing extension is really one of $\mathrm{Set}$ itself, not of an elementary extension.
This also answers my final confusion in the OP: If $S$ is a Stone-Cech compactification, then $\mathrm{Sh}_{\neg\neg}(S)$ is in fact the constant sheaf of categories with value $\mathrm{Set}$, so in particular $G$ already lies in $\mathrm{Set}_s$ for all $s\in S$, and $\mathrm{Set}_s[G]=\mathrm{Set}_s$. Thus, the forcing extension is trivial, but the elementary extension $\mathrm{Set}\to \mathrm{Set}_s$ to the ultrapower remains.
Why does one usually want generic filters? I think the point is that the extension $\mathrm{Set}_s\subset \mathrm{Set}_s[G]$ always preserves ordinals (in its material incarnation). By contrast, $\mathrm{Set}\subset \mathrm{Set}_s$ very much does not, and in fact $\mathrm{Set}_s$ usually has nonstandard integers; in the material incarnation, this leads to non-wellfounded models. So one uses generic filters to make $\mathrm{Set}=\mathrm{Set}_s$.
From the structural point of view, wellfoundedness is not visible, but of course it may still be disconcerting that the extension has nonstandard integers.