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?
Best Answer
Yes, this is a model of ETCSR. Unfortunately, I don't know of a proof of this in the literature, which is in general sadly lacking as regards replacement/collection axioms in topos theory. But here's a sketch.
As Zhen says, the filterquotient construction preserves finitary properties such as Booleanness and the axiom of choice. Moreover, a maximal filterquotient will be two-valued. But as you point out, a nondegenerate two-valued topos satisfying the (external) axiom of choice is necessarily well-pointed; I wrote out an abstract proof at https://ncatlab.org/nlab/show/well-pointed+topos#boolean_properties. Thus, $\varinjlim_{U\ni s} \mathrm{Sh}_{\neg\neg}(S)$ is a model of ETCS.
As for replacement, the proof that I know (which is not written out in the literature) goes by way of the notions of "stack semantics" and "autology" in my preprint Stack semantics and the comparison of material and structural set theories (the other half, not the part that became the paper of mine cited in the question).
Briefly, stack semantics is an extension of the internal logic of a topos to a logic containing unbounded quantifiers of the form "for all objects" or "there exists an object". (My current perspective, sketched in these slides, is that this is a fragment of the internal dependent type theory of a 2-topos of stacks -- hence the name!) This language allows us to ask whether a topos is "internally" a model of structural set theories such as ETCS or ETCSR.
It turns out that every topos is "internally (constructively) well-pointed", and moreover satisfies the internal collection axiom schema. But the internal separation axiom schema is a strong condition on the topos, which I called being "autological". If a topos is autological and also Boolean, then the logic of its stack semantics is classical; thus it is internally a model of ETCSR. Since Grothendieck toposes are autological, your $\mathrm{Sh}_{\neg\neg}(S)$ is internally a model of ETCSR.
Now we can also prove that if $\mathcal{E}$ is Boolean and autological, so is any filterquotient of it. The idea is to prove a categorical version of Łoś's theorem for the stack semantics. (I don't know whether this is true without Booleanness, which annoys me to no end, but you probably don't care. (-: ) Therefore, $\varinjlim_{U\ni s} \mathrm{Sh}_{\neg\neg}(S)$ is also autological.
Finally, another fact about autology is that a well-pointed topos is autological if and only if it satisfies the ordinary structural-set-theory axiom schemas of separation and collection. Therefore, $\varinjlim_{U\ni s} \mathrm{Sh}_{\neg\neg}(S)$ satisfies these schemas, hence is a model of ETCSR.
However, I doubt that this particular filterquotient is related to forcing at all. The point is the same one that Jacob made in a comment: when set theorists force over a countable base model to make an "actual" new model, they find an actual generic ultrafilter outside that model. A generic ultrafilter in the base model would be a point of the topos $\mathrm{Sh}_{\neg\neg}(S)$, which as Andreas pointed out in a comment, does not exist. Your "points" $x$ are not points of the topos $\mathrm{Sh}_{\neg\neg}(S)$, so it's unclear to me whether filterquotients at them have anything to do with forcing.
Let me reiterate my argument that the real content of forcing is the internal logic of the topos $\mathrm{Sh}_{\neg\neg}(S)$. In particular, if you build a model of material set theory in this internal logic, what you get is essentially the Boolean-valued model that set theorists talk about. Edit: I think the rest of this answer is off-base; see the discussion in the comments.
I'm pretty sure this is the best kind of "model" you can get if you don't want to start talking about countable models of ZFC sitting inside larger ambient models.At the moment, my best guess for a topos-theoretic gloss on the countable-transitive-model version of forcing is something like the following. Suppose that $E$ is a countable model of ETCSR, containing an internal poset $P$, which we can equip with its double-negation topology. Then treating $E$ as the base topos, we can build $\mathrm{Sh}(P,E)$, a bounded $E$-indexed elementary topos (i.e. "$E$ thinks it is a Grothendieck topos"), which contains the Boolean-valued model associated to $P$ as described above. It is the classifying topos of $P$-generic filters, hence has in general no $E$-points.
But we also have the larger topos $\rm Set$ in which $E$ is countable, and we can consider the externalization $|P|$ which is a poset in $\rm Set$, namely $|P| = E(1,P)$. Then we can build the topos $\mathrm{Sh}(|P|,\rm Set)$ which "really is" a Grothendieck topos and classifies $|P|$-generic filters. The "Rasiowa–Sikorski lemma" implies that, since $E$ is countable, in this case such a filter does actually exist in $\rm Set$, so there is a point $p:\mathrm{Set} \to \mathrm{Sh}(|P|,\rm Set)$.
Now we should also have some kind of "externalization functor" $|-| : \mathrm{Sh}(P,E) \to \mathrm{Sh}(|P|,\rm Set)$. My guess is that the set-theorists' forcing model is the "image" (whatever that means) of the Boolean-valued model in $\mathrm{Sh}(P,E)$ under the composite of this externalization functor with the inverse image functor $p^* : \mathrm{Sh}(|P|,\rm Set) \to Set$. However, I have not managed to make this precise.