Subobject classifier for sheaves on large sites with WISC

Question

Let $\mathsf{C}$ be a possibly large category with a Grothendieck topology satisfying the Weakly Initial Set of Covers condition: there is for each $X$ a set (not a proper class) of covering families of $X$ such that if $\{U_i \to X\}$ is an arbitrary covering family, it is refined by one in the set. In the case where $\mathsf{C}$ is small, Mac Lane and Moerdijk construct a subobject classifier $\Omega$ in the category of sheaves on the site $\mathsf{C}$. I want to know: if $\mathsf{C}$ is allowed to be large (but satisfying the weakly initial set of covers condition), can we still construct $\Omega$?

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For example, $\mathsf{Top}$ is a large category, and we may take the covering families on $X$ to be sets of local homeomorphisms $f_i \colon V_i \to X$ such that the union of the images $\bigcup_i f_i(V_i)$ is all of $X$. Every such covering family is, one can check, refined by an open cover of $X$, and there are clearly only a set of open covers of $X$.

If one has a weakly initial set of covers for each object of $\mathsf{C}$, then the plus construction shows that every presheaf on $\mathsf{C}$ may be sheafified, great.

If $\mathsf{C}$ is small (so that there are only a set of arrows in the entire category), then there is for each $X \in \mathsf{C}$ a set of sieves on $X$, where I remind you that a sieve is a collection $S$ of arrows $f\colon U \to X$ that forms a "right ideal" in the sense that if $h\colon V \to U$ is an arbitrary arrow and $f$ belongs to $S$, then $fh$ belongs to $S$. Say a sieve covers $X$ if it contains a covering family. Mac Lane and Moerdijk define what it means for a sieve to be closed, show that every covering sieve has a closure (which is again covering) and define

$$\Omega(X) = \{ \bar S : S \text{ covers } X \}.$$

This is clearly a set when $\mathsf{C}$ is small, and the assignment $X \mapsto \Omega(X)$ is functorial in this case as well, since covering sieves pull back along any arrow $f\colon Y \to X$ to covering sieves on $Y$, and the closure of a pullback sieve is the pullback of the closure. Anyway, this presheaf turns out to be a sheaf, and it is the subobject classifier above.

If $\mathsf{C}$ is large, one only gets a set $\Omega(X)$ if one restricts to those covering sieves on $X$ generated by, for example, a weakly initial set of covers for $X$. The problem, then, is that given such a sieve $S$ and an arrow $f\colon Y \to X$, we know that the pullback sieve $f^* S$ covers $Y$ so is refined by a sieve generated by an element of the weakly initial set of covers for $Y$, but may not be equal to such a sieve, so $\overline{f^*S}$ need not belong to $\Omega(Y)$, and—to me, at least—there doesn't appear to be a natural choice of an element of $\Omega(Y)$ to associate to $S$.

Answer 1

To answer your question directly, WISC does not imply the existence of subobject classifiers. Notice that when there are only trivial covers, WISC is trivially satisfied, so it suffices to find a category with a proper class of sieves. Consider the ordered class of ordinals with a terminal object adjoined. Then each ordinal generates a distinct sieve, so the category of presheaves cannot have a subobject classifier.

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The crux of the matter is that closed sieves are orthogonal to dense (= covering) sieves. As in the example above, not having "too many" dense sieves implies nothing about not having "too many" closed sieves. To avoid having to make pedantic distinctions between classes and conglomerates and so on, let me just talk about large and small sets. Consider the following:

Axiom G. There is a small set $\mathcal{C}'$ of objects in $\mathcal{C}$ such that every dense sieve on every object in $\mathcal{C}$ contains a dense sieve generated by morphisms with domains in $\mathcal{C}'$.

It is not hard to see that axiom G implies WISC. In fact, axiom G implies the category of sheaves is a Grothendieck topos. (I think the converse is also true, at least under some additional assumptions about the category of sheaves.) So perhaps axiom G is a bit too strong. We could weaken it as follows:

Axiom LG. For every object $X$, the slice category $\mathcal{C}_{/ X}$ satisfies axiom G.

The point is that to ensure that there are not "too many" closed sieves (resp. dense sieves) on $X$, we only need to know axiom G is satisfied for $\mathcal{C}_{/ X}$. Thus axiom LG is enough to guarantee WISC and also the existence of a subobject classifier. On the other hand, axiom LG does not imply even local-smallness of the category of sheaves!

For example, consider the category $\textbf{LH}$ of all topological spaces and local homeomorphisms between them. Say a sieve on an object in $\textbf{LH}$ is a cover if it is jointly surjective. This site satisfies axiom LG but not axiom G. Thus, the category of sheaves on $\textbf{LH}$ is a pretopos and has a subobject classifier. I claim the category of sheaves is neither well-powered nor locally small.

Observe that $\textbf{LH}$ has the unpleasant property that it has a full subcategory that is totally disconnected (i.e. all morphisms are endomorphisms) but not small. We can use this to find a non-small set of distinct closed sieves of $\textbf{LH}$ (= subsheaves of the terminal sheaf). This demonstrates the failure of the category of sheaves to be well-powered. But in a category with a subobject classifier, local-smallness implies well-poweredness, so we also deduce that the category of sheaves is not locally small.