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I’m a little confused by the remarks about induced coverages in Section C2.2 of the Elephant. Consider the poset $\{ a, b, c, d \}$ with $b \lt a$, $c \lt a$, $d \lt a$, $d \lt c$, and let $J$ be the Grothendieck topology given by:
$J(a) = \{ \{ b \}, \{ b, d \}, \{ b, c, d \}, \{ a, b, c, d \} \}$ $J(b) = \{ \{ b \} \}$ $J(c) = \{ \emptyset, \{ d \}, \{ c, d \} \}$ $J(d) = \{ \emptyset, \{ d \} \}$Now, consider the non-full subcategory with the same objects but only the relations $b \lt a$, $c \lt a$, $d \lt a$. As I understand it, the induced coverage $K$ takes the form below:
$K(a) = \{ \{ b \}, \{ b, d \}, \{ b, c, d \}, \{ a, b, c, d \} \}$ $K(b) = \{ \{ b \} \}$ $K(c) = \{ \emptyset, \{ c \} \}$ $K(d) = \{ \emptyset, \{ d \} \}$But this is not a Grothendieck topology, since K(a) fails to be a filter. Have I misunderstood something?
It seems that some care is required when constructing the induced topology when the subcategory is non-full: in this case, what happens is that every sieve on $a$ in the parent category either contains both $c$ and $d$ or not at all, but there are sieves on $a$ on the subcategory which only contain one of $c$ or $d$ but not both.
That’s a good point! I think that must be a slip. In fact, the same issue is pointed out later on in the proof of C2.3.19.
Actually, I just found an example of a saturated coverage and a subcategory such that the induced “coverage” is not even a coverage. Let $\mathcal{C}$ be the poset $\{ a, b, c, d \}$ with relations $b \lt a$, $c \lt a$, $d \lt a$, $d \lt b$, $d \lt c$. (This is isomorphic to the powerset on the 2-element set.) Let $\mathcal{D}$ be the subcategory with the same objects but with the relation $d \lt c$ removed. Let $T$ be the saturated coverage on $\mathcal{C}$ defined as follows:
$T(a) = \{ \mathfrak{U} sink on a \mid a \in \mathfrak{U} or c \in \mathfrak{U} \}$ $T(b) = \{ \mathfrak{U} sink on b \mid b \in \mathfrak{U} or d \in \mathfrak{U} \}$ $T(c) = \{ \mathfrak{U} sink on c \mid c \in \mathfrak{U} \}$ $T(d) = \{ \mathfrak{U} sink on d \mid d \in \mathfrak{U} \}$Then, the induced “coverage” on $\mathcal{D}$ fails to be a coverage: there is no covering of $b$ whose pushforward to $a$ is subordinate to the covering $\{ c \}$.
Also, here is another counterexample for the induced coverage being a Grothendieck topology, this time using a full subcategory. Let $\mathcal{C}$ be the poset $\{ a, b, c, d \}$ with relations $b \lt a$, $c \lt a$, $d \lt a$, $d \lt c$, and let $\mathcal{D}$ be the subposet $\{ a, b, d \}$. Let $J$ be the Grothendieck topology on $\mathcal{C}$ defined by
$J(a) = \{ \{ b, c, d \}, \{ a, b, c, d \} \}$ $J(b) = \{ \{ b \} \}$ $J(c) = \{ \{ c, d \} \}$ $J(d) = \{ \emptyset, \{ d \} \}$Then, the induced coverage on $\mathcal{D}$ is given by
$K(a) = \{ \{ b, d \}, \{ b, c, d \} \}$ $K(b) = \{ \{ b \} \}$ $K(d) = \{ \emptyset, \{ d \} \}$This is a coverage, but fails to satisfy condition (L) because $\{ b \}$ is not $K$-covering.
Here are some conditions under which the induced “coverage” is nice. Let $\mathcal{C}$ be a category, let $T$ be a coverage on $\mathcal{C}$, let $\mathcal{D}$ be a subcategory, and let $S$ be the induced “coverage” on $\mathcal{D}$.
If $\mathcal{D}$ is a full subcategory, then $S$ is actually a coverage. In fact, it is enough to assume that $f : U \to X$ factors through $g : Y \to X$ in $\mathcal{D}$ if and only if it factors in $\mathcal{C}$.
If $T$ is a sifted coverage, then $S$ is also a sifted coverage. (No assumptions on $\mathcal{D}$.)
If $\mathcal{D}$ is a full subcategory and $T$ is a saturated coverage, then $S$ is also a saturated coverage.
Ultimately though, it’s not clear to me what the motivation behind the induced coverage is. I wonder whether it should be defined via a universal property similar to the quotient topology?
I believe C2.3.19 gives it a universal property: it is the minimal coverage such that the inclusion functor is cover-reflecting.
Mmm… that is what I expected, but those pathologies made me sceptical about whether inclusions could have any good properties.
One precoverage which makes the inclusion functor manifestly cover-reflecting (and cover-preserving!) is the one that has all the covering families of $\mathcal{C}$ which are contained in $\mathcal{D}$, but this fails to be a coverage in general. For example, the precoverage obtained by this construction on the span $\bullet \rightarrow \bullet \leftarrow \bullet$ considered as a subcategory of the commutative square equipped with the atomic coverage (every non-empty sink is covering) fails to be a coverage; however, the smallest coverage containing it is precisely the induced coverage in this case, and that is indeed cover-reflecting. On the other hand, because there are empty sinks which are covering in the induced coverage, the inclusion functor cannot preserve covers.
Are there situations where the induced coverage makes the inclusion functor both cover-reflecting and cover-preserving? This would probably be a good way of getting essential geometric morphisms and/or local geometric morphisms. For example, if $\mathcal{C}$ is a category with all pullbacks and $T$ is a Grothendieck pretopology, and $\mathcal{D}$ is a left exact full subcategory closed under pullbacks and covering families, then the induced coverage $S$ on $\mathcal{D}$ would be exactly the same as the restriction (!!) of the Grothendieck pretopology to $\mathcal{D}$. Saturated coverages are too big to make this construction useful, but here I’m quite sure that the coverage induced by the saturation of $T$ turns out to be the same as the saturation of $S$.
Well, if the subcategory is dense, which is the context I expect Johnstone had in mind in C2.2, then I expect the inclusion is both cover-preserving and cover-reflecting. But that will of course only induce an equivalence, not something like a local or essential geometric morphism. Have you read about fibrations of sites in the elephant?
Also, it’s not obvious to me why you say “Saturated coverages are too big to make this construction useful”; can you explain?
By saturated coverage I mean a (non-sifted) coverage $T$ satisfying the following conditions:
This is basically the non-sifted version of a Grothendieck topology. Clearly, covering families can get very large, so it is difficult to control the size of a subcategory closed under covering families.
I have not yet had the chance to read about fibred categories in the context of topos theory. What I was trying to do was to reconcile Nick Duncan’s definitions of gros and petit toposes with the usual construction of petit toposes from a gros topos (e.g. the Zariski topos; see my other thread). One of the requirements is that there should be a local geometric morphism from the slice of gros topos to the petit topos – so I’m trying to find conditions that a subcategory of the underlying site of a gros topos ought to satisfy if it is to be the site of a petit topos.
That’s what I assumed you meant by a saturated coverage. But I guess I misunderstood what you meant by “useful” – you just meant that it’s hard to make examples maximally explicit with saturated coverages? The theory works just fine (better, in fact).
Fibrations of sites are very different from fibered categories. I mean, they are technically fibered categories, but they serve a very different purpose. You might want to have a look at them.
Have you read this paper?
No no – I can see the theory works better. By “closed under covering families” what I mean is that if $X \in ob \mathcal{D}$, and $f : U \to X$ is contained in some covering family of $X$, then $f$ (and $U$) are also in $\mathcal{D}$. Clearly, if $\mathcal{D}$ contains the terminal object and $\mathcal{C}$ is equipped with a saturated coverage, then $\mathcal{D}$ is closed under covering families if and only if $ob \mathcal{D} = ob \mathcal{C}$. The intention was to extract a basis of the topology on $Spec \mathbb{Z}$ by looking at the full subcategory of ${Aff}^{fp}_{\mathbb{Z}}$ closed under covering families and containing $Spec \mathbb{Z}$… but this is evidently an evil approach as it depends on the choice of coverage.
Thanks for the reference to the paper – it looks very interesting! I’ll have to read it properly at some point.
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