Why is $1$ not a dense sub-site in a group with the trivial Grothendieck topology?

Question

A friend of mine had the following question while reading the section "C2.2 The topos of sheaves" in "Sketches of an Elephant".

Let $G$ be a group (considered as a category with one object) with trivial topology (the only sieve is all $G$) we get a site, we denote it by $G$. A discrete subcategory $1 < G$ is dense by Definition 2.2.1. Indeed, (i) obviously holds and (ii) holds because for a given morphism $f$ any morphism $h$ factors through $g := f^{-1}$. Then by Theorem 2.2.3 the categories of sheaves on them are equivalent, but the categories $\mathrm{Set}$ and $G\text{-}\mathrm{Set}$ are not equivalent in general.

Where is the mistake? It is probably simple (for example, we misunderstand the definitions).

Answer 1

Yes, your counterexample seems to be correct — there’s an error in Def C2.2.1 as printed. This issue is mentioned in the n-lab’s article on dense sub-site (current revision permalink), which notes a very similar counterexample, and refers to a longer nForum thread on the topic (mostly by Dexter Chua, Thomas Holder, and Mike Shulman).

An easy and clean fix is to restrict to full subcategories, as Zhen Lin suggests in comments. But Johnstone clearly intends to cover the non-full case; straight after Def C2.2.1 he remarks:

In practice, the Comparison Lemma is most often used for full subcategories, and many texts only define denseness in this case; however, the extra generality afforded by the definition we have given is occasionally useful — we shall see an instance of its use in 5.2.5 below.

The n-lab article offers another fix, modifying condition (ii) to say “for each $f : U \to V$ with $V \in \mathcal{D}$, there is some covering sieve on $U$ whose elements $g : W \to U$ all satisfy $fg \in \mathcal{D}$”. But this condition seems much too strong: it will almost never hold (as noted by Dexter Chua in the nForum thread), even in the full case. In particular, if I’m not mistaken, it won’t generally hold in the application in Johnston’s Lemma C5.2.5, showing that any étendue (aka locally localic topos, i.e. a topos for which some well-supported slice is localic) has a site of definition with all maps monic.

So the printed version is in error; but neither the restriction to full subcategories, nor the n-lab’s fix, seems to correspond to Johnstone’s intention, or suffice for his application in Lemma C5.2.5. It would be very nice to find a better fix, strong enough to imply the Comparison Lemma C2.2.3 but general enough for the application in Lemma C5.2.5! Edit: Simon Henry’s answer presents what looks like the right fix, from the original Kock–Moerdijk paper that Johnson’s C5.2.5 is based on.

Answer 2

To complement the answer by Peter, I think the "correct" statement of the comparison lemma for non-full subcategory (and in fact even non-faithful functor) can be found in a paper by (A.)Kock and Moerdijk "Presentation of étendues" where it is stated — unfortunately without proof — at the end of section 2.

I haven't rechecked it in light of the discussion above — but I remembered proving it a few years ago and I was convinced it was correct. Though maybe with all these problems regarding this lemma, it might be time to actually write a proof. In any case, this version of the lemma doesn't have a problem with the example mentioned in the Question.

The comparison lemma in the paper is stated in terms of a functor between two site, so it looks a little different than what is in the elephant, but I think the key modification is that condition (2) is altered in the following way: Both $U$ and $V$ are assumed to be $D$, and the generators of the covering sieve of $U$ (such that the composite with $U \to V$ are in $D$) need to also be themselve in $D$.

The comparison lemma by Kock and Moerdijk has a final assumption of "co-continuity" that might seem a little strange — but if I'm not mistaken it can be deduced (assuming all the other conditions) from the more natural assumption that a Sieve in the domain is a cover if and only if its image by the functor is a cover — that is the functor $u$ is not just "cover preserving" as required by condition (1) but also "cover detecting".

Finally, as the title of the paper suggests, the main motivation of this paper is exactly to prove that "Lemma C.5.2.5" about étendus that Peter is referring to in his answer, which as far as I know is the only use of the comparison lemma to a non-full subcategory that can be found in the Elephant.