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At coverage, I just made the following change: Where the sheaf condition previously read
$X(U) \to \prod_{i\in I} X(U_i) \rightrightarrows \prod_{i,j\in I} X(U_i\times_U U_j),$it now uses the variable names “$j$” and “$k$” instead of “$i$” and “$j$”:
$X(U) \to \prod_{i\in I} X(U_i) \rightrightarrows \prod_{j,k\in I} X(U_j\times_U U_k).$I’m announcing this almost trivial change because I’d like to invite objections, in which case I’d rollback that change and also would not go on to copy this change to related entries such as sheaf. There are two tiny reasons why I prefer the new variable names:
I think that’s a good idea. Thanks.
Makes sense to me.
Thanks for the encouragement. Updated sheaf accordingly.
The last line of the definition used to be
The logic here is: $\forall f, \forall g, \exists h, \forall j, \exists k, \exists i, =$.
I have fixed this (I hope) to
The logic here is: $\forall f, \forall g, \exists h, \forall j, \exists i, \exists k, =$.
(The object $U_i$ needs to be chosen before choosing the morphism $k$ makes sense.)
Yes, I think that’s right. (At first my reaction was “existential quantifiers commute with each other”, but of course not in dependently typed logic where the type of $k$ depends on $i$, as here. If the logic isn’t dependently typed, then $\exists k,\exists i$ technically makes sense (and is equivalent), but the other way is still clearer.)
Let $C$ be a category. If $X$ is an object of $C$, then write $\underline{X}: C^{op}\to\Sets$ for the corresponding functor.
Suppose that a “cover” of $X$ is a subfunctor $i: \mathcal{U}\to \underline{X}$ such that, for every object $Y$ of $C$, the function
$i^\ast: Hom(\underline{X}, \underline{Y})\to\Hom(\mathcal{U}, \underline{Y})$is a bijection. Is this a Grothendieck topology? Is it the canonical topology? To me this seems like the most natural definition of a cover, but I have never seen this exact definition written or discussed anywhere. I am guessing this is at least close to the canonical topology because this seems to be the minimum requirement for every representable functor to be a sheaf.
Hi! I think that strictly speaking this does not quite parse: a coverage has to consists of arrows in $\mathcal{C}$ itself. A Grothendieck topology which is close to being a minimal condition for representable functors to be sheaves is the ’jointly surjective’ one, and this seems a little related to your idea here. You might be able to rework your idea into something which does parse. There is the notion of Lawvere-Tierney topology which can be used to define a Grothendieck topology on $\mathcal{C}$ by working in the category of presheaves over it; maybe you can formulate things in those terms? Or maybe you can show in some other way that working in the presheaf category in the way you suggest induces a Grothendieck topology in the ordinary sense.
I have not fully understood the intuition behind your idea yet; the best I can say so far is the remark about it seeming not so far from the ’jointly surjective’ topology :-). Maybe you can elaborate?
Thanks for the comments. I’m thinking of a Grothendieck topology in terms of covering sieves, which are subfunctors of representable functors. Thinking in terms of arrows in $C$ itself, then a collection of arrows $\{X_i\to X\}_i$ will generate a subfunctor $\mathcal{U} \to \underline{X}$ which is then a cadidate to be a cover as defined above.
To put this in perspective: Let $\text{Op}(X)$ be the category of open subsets of a topological space $X$. Let $U$ be an open subset of $X$, i.e., an object of $\text{Op}(X)$. Then, using the definition of cover above, a cover of $U$ is a subfunctor $\mathcal{U}\to \underline{U}$ such that $\{V\in\text{Op}(X)\, |\, \mathcal{U}(V)\neq \emptyset\}$ is an open covering of $U$ in the usual sense. So a cover in this situation is precisely a covering sieve for the usual Grothendieck topology on $\text{Op}(X)$.
More questions: What is a cover in other categories? How about the (opposite) category of rings? Does it produce any of the usual topologies, i.e., Zariski, etale, fppf? Even if this notion of cover does not always define a Grothendieck topology, is the resulting category of sheaves a topos? Here I am defining a sheaf on a category $C$ as a functor $F: C^{\text{op}}\to \text{Sets}$ such that, for any cover $\mathcal{U}\to \underline{X}$, the canonical map
$\text{Hom}(\underline{X}, F)\to \text{Hom}(\mathcal{U}, F)$is a bijection.
Just for terminological clarification:
Let $\mathcal{C}$ be a small category. Then a sieve $S$ on $X\in\mathcal{C}$ is a subfunctor $\mathcal{U}\to\underline{X}$. A Grothendieck topology $J$ on $\mathcal{C}$ consists of a collection of sieves $J(X)$ for each $X\in\mathcal{C}$ satisfying certain stability conditions G1-G3. A presheaf $F:\mathcal{C}^{op}\to Set$ is a sheaf (resp. a separated presheaf) for $J$ if for any $X\in\mathcal{C}$ and any sieve $S\in J(X)$ the canonical map
$\text{Hom}(\underline{X}, F)\to \text{Hom}(\mathcal{U}, F)$is a bijection (resp. an injection). A topology $J$ is called subcanonical if all representable functors $\underline{X}$ are sheaves. Sofar, the definitions and terminology are entirely standard (and, in particular, this is the definition in SGA4 p.223 or used in Horst Schubert’s book on category theory).
Now, one can make sense of your question in two different ways:
You came up with this sheaf condition yourself and you ask whether it is known. The answer to this is yes but offhand I have no other references than the above mentioned though there should be plenty of others.
You want to tinker with G1-G3 and you ask how important they are for getting a topos of sheaves. Then I guess you should be more specific what your conditions on the collections of sieves are (cf. prop.2.2 in SGA4, p. 224). Note that the condition in #7 implies that you can only hope to capture subcanonical Grothendieck topologies in this way. Note also that there float around several definitions of “generators” for J in the literature e.g. the definition of a “coverage” in the elephant (p.73) is fairly parsimonious on stability conditions.
I read the question differently: if we take $K(X)$ to be the collection of sieves which induce natural isomorphisms $Hom(\underline{X}, \underline{Y}) \to \Hom(\mathcal{U}, \underline{Y})$, how far is $K$ from being a Grothendieck topolgy?
@Hurkyl this is exactly what I am asking. And, for categories where $K$ is not a Grothendieck topology, could it be that the resulting category of sheaves is still a topos?
Apologies for mis-parsing your question a little, I was reading too quickly! I will have a think about it when I get a chance. It still strikes me as reminiscent of the ’jointly epimorphic’ topology :-). Thanks for the elaboration!
@12: I think in that case you “only” have to check whether $K$ satisfies the stability under base change axiom:
For all $S\in K(X)$ and maps $h:Y\to X$, is the sieve $h^*(S)=\{g\;|\;cod(g)=Y\; and\; h g\in S\}$ in $K(Y)$ ? In other words, whether $Hom(\underline{Y},\underline{Z})\overset{\simeq}{\to}Hom(h^*(S),\underline{Z})$ for all representables $\underline{Z}$.
If it does (and I would think it does), $K$ coincides with the canonical topology $J_c$ and the category $Sh(\mathcal{C},K)$ is a topos, otherwise it isn’t. To see that much, let us denote the smallest topology containing $K$ as $J_K$. One has: $J_c\subseteq K\subseteq J_K$. Then $Sh(\mathcal{C},K)$ is a topos iff it coincides with $Sh(\mathcal{C},J_K)$. Moreover, in that case $J_K$ is subcanonical by definition of $K$ but then $J_K\subseteq J_c$ since $J_c$ is the largest subcanonical topology whence $J_c=K$. To sum up: $Sh(\mathcal{C},K)$ is a topos iff $J_c=K$.
(Added: Hhmm, I seem to tacitly assume here that from $Sh(\mathcal{C},K)=Sh(\mathcal{C},J)$ for some topology $J$ it follows that $K\subseteq J$ which I can’t quite see why that would be, hence let’s make a question mark behind everything in the preceding paragraph relying on that! Sorry for the confusion!)
That $J_c=K$ is equivalent to stability under base change follows either from checking the other two axioms for a Grothendieck topology or the aforementioned prop.2.2 in SGA4 (which incidentally suggests that mutatis mutandis pretty much the same story might hold for other families of presheaves not just the representables).
(Added: To spell the last point out. Let $F=\{F_i\}_{i\in I}$ be a (small?) family of presheaves, $J_F$ the largest topology such that all $F_i$ are sheaves and $K_F$ the collection of all sieves $S$ such all $F_i$ satisfy the sheaf condition on $S$. Then from prop.2.2 we apparently get that $J_F\subseteq K_F$ and $K_F\subseteq J_F$ precisely iff $K_F$ satisfies the stability-under-base-change condition.)
Hopefully, a person more expert or more energetic than me takes over!
This collection of covers doesn’t necessarily satisfy the stability axiom.
This condition on a sieve is known as being effective-epimorphic, and is equivalent to saying that $X$ is the colimit of the diagram consisting of all the elements of $\mathcal{U}$ and all the morphisms between them. If colimits are stable under pullback in $C$, then the effective-epimorphic sieves constitute the canonical topology; but otherwise, one has to restrict further to those that are stable under pullback, called universally effective-epimorphic. To find a counterexample, just look at any category that contains a non-pullback-stable colimit.
Thanks to all for the comments and responses. In particular, the distinguishing between effective-epimorphic and universally effective-epimorphic clears up my questions.
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