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  1. At coverage, I just made the following change: Where the sheaf condition previously read

    X(U) iIX(U i) i,jIX(U i× UU j), 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 “jj” and “kk” instead of “ii” and “jj”:

    X(U) iIX(U i) j,kIX(U j× UU k). 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:

    • It’s more symmetric. The previous notation unjustly favored “ii”.
    • It’s slightly easier to infer the definition of the two maps. (I had a student who was briefly confused by the original notation.)
    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeFeb 1st 2018

    I think that’s a good idea. Thanks.

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 1st 2018

    Makes sense to me.

  2. Thanks for the encouragement. Updated sheaf accordingly.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJun 4th 2018

    The last line of the definition used to be

    The logic here is: f,g,h,j,k,i,=\forall f, \forall g, \exists h, \forall j, \exists k, \exists i, =.

    I have fixed this (I hope) to

    The logic here is: f,g,h,j,i,k,=\forall f, \forall g, \exists h, \forall j, \exists i, \exists k, =.

    (The object U iU_i needs to be chosen before choosing the morphism kk makes sense.)

    diff, v36, current

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeJun 4th 2018

    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 kk depends on ii, as here. If the logic isn’t dependently typed, then k,i\exists k,\exists i technically makes sense (and is equivalent), but the other way is still clearer.)

    • CommentRowNumber7.
    • CommentAuthorn.mertes
    • CommentTimeFeb 17th 2021

    Let CC be a category. If XX is an object of CC, then write X̲:C opSets\underline{X}: C^{op}\to\Sets for the corresponding functor.

    Suppose that a “cover” of XX is a subfunctor i:𝒰X̲i: \mathcal{U}\to \underline{X} such that, for every object YY of CC, the function

    i *:Hom(X̲,Y̲)Hom(𝒰,Y̲) 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.

    • CommentRowNumber8.
    • CommentAuthorRichard Williamson
    • CommentTimeFeb 19th 2021
    • (edited Feb 19th 2021)

    Hi! I think that strictly speaking this does not quite parse: a coverage has to consists of arrows in 𝒞\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 𝒞\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?

    • CommentRowNumber9.
    • CommentAuthorn.mertes
    • CommentTimeFeb 19th 2021

    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 CC itself, then a collection of arrows {X iX} i\{X_i\to X\}_i will generate a subfunctor 𝒰X̲\mathcal{U} \to \underline{X} which is then a cadidate to be a cover as defined above.

    To put this in perspective: Let Op(X)\text{Op}(X) be the category of open subsets of a topological space XX. Let UU be an open subset of XX, i.e., an object of Op(X)\text{Op}(X). Then, using the definition of cover above, a cover of UU is a subfunctor 𝒰U̲\mathcal{U}\to \underline{U} such that {VOp(X)|𝒰(V)}\{V\in\text{Op}(X)\, |\, \mathcal{U}(V)\neq \emptyset\} is an open covering of UU in the usual sense. So a cover in this situation is precisely a covering sieve for the usual Grothendieck topology on Op(X)\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 CC as a functor F:C opSetsF: C^{\text{op}}\to \text{Sets} such that, for any cover 𝒰X̲\mathcal{U}\to \underline{X}, the canonical map

    Hom(X̲,F)Hom(𝒰,F) \text{Hom}(\underline{X}, F)\to \text{Hom}(\mathcal{U}, F)

    is a bijection.

    • CommentRowNumber10.
    • CommentAuthorThomas Holder
    • CommentTimeFeb 21st 2021
    • (edited Feb 21st 2021)

    Just for terminological clarification:

    Let 𝒞\mathcal{C} be a small category. Then a sieve SS on X𝒞X\in\mathcal{C} is a subfunctor 𝒰X̲\mathcal{U}\to\underline{X}. A Grothendieck topology JJ on 𝒞\mathcal{C} consists of a collection of sieves J(X)J(X) for each X𝒞X\in\mathcal{C} satisfying certain stability conditions G1-G3. A presheaf F:𝒞 opSetF:\mathcal{C}^{op}\to Set is a sheaf (resp. a separated presheaf) for JJ if for any X𝒞X\in\mathcal{C} and any sieve SJ(X)S\in J(X) the canonical map

    Hom(X̲,F)Hom(𝒰,F) \text{Hom}(\underline{X}, F)\to \text{Hom}(\mathcal{U}, F)

    is a bijection (resp. an injection). A topology JJ is called subcanonical if all representable functors X̲\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.

    • CommentRowNumber11.
    • CommentAuthorHurkyl
    • CommentTimeFeb 21st 2021

    I read the question differently: if we take K(X)K(X) to be the collection of sieves which induce natural isomorphisms Hom(X̲,Y̲)Hom(𝒰,Y̲)Hom(\underline{X}, \underline{Y}) \to \Hom(\mathcal{U}, \underline{Y}), how far is KK from being a Grothendieck topolgy?

    • CommentRowNumber12.
    • CommentAuthorn.mertes
    • CommentTimeFeb 21st 2021

    @Hurkyl this is exactly what I am asking. And, for categories where KK is not a Grothendieck topology, could it be that the resulting category of sheaves is still a topos?

    • CommentRowNumber13.
    • CommentAuthorRichard Williamson
    • CommentTimeFeb 21st 2021
    • (edited Feb 21st 2021)

    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!

    • CommentRowNumber14.
    • CommentAuthorThomas Holder
    • CommentTimeFeb 22nd 2021
    • (edited 7 days ago)

    @12: I think in that case you “only” have to check whether KK satisfies the stability under base change axiom:

    For all SK(X)S\in K(X) and maps h:YXh:Y\to X, is the sieve h *(S)={g|cod(g)=YandhgS}h^*(S)=\{g\;|\;cod(g)=Y\; and\; h g\in S\} in K(Y)K(Y) ? In other words, whether Hom(Y̲,Z̲)Hom(h *(S),Z̲)Hom(\underline{Y},\underline{Z})\overset{\simeq}{\to}Hom(h^*(S),\underline{Z}) for all representables Z̲\underline{Z}.

    If it does (and I would think it does), KK coincides with the canonical topology J cJ_c and the category Sh(𝒞,K)Sh(\mathcal{C},K) is a topos, otherwise it isn’t. To see that much, let us denote the smallest topology containing KK as J KJ_K. One has: J cKJ KJ_c\subseteq K\subseteq J_K. Then Sh(𝒞,K)Sh(\mathcal{C},K) is a topos iff it coincides with Sh(𝒞,J K)Sh(\mathcal{C},J_K). Moreover, in that case J KJ_K is subcanonical by definition of KK but then J KJ cJ_K\subseteq J_c since J cJ_c is the largest subcanonical topology whence J c=KJ_c=K. To sum up: Sh(𝒞,K)Sh(\mathcal{C},K) is a topos iff J c=KJ_c=K.

    (Added: Hhmm, I seem to tacitly assume here that from Sh(𝒞,K)=Sh(𝒞,J)Sh(\mathcal{C},K)=Sh(\mathcal{C},J) for some topology JJ it follows that KJK\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=KJ_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} iIF=\{F_i\}_{i\in I} be a (small?) family of presheaves, J FJ_F the largest topology such that all F iF_i are sheaves and K FK_F the collection of all sieves SS such all F iF_i satisfy the sheaf condition on SS. Then from prop.2.2 we apparently get that J FK FJ_F\subseteq K_F and K FJ FK_F\subseteq J_F precisely iff K FK_F satisfies the stability-under-base-change condition.)

    Hopefully, a person more expert or more energetic than me takes over!

    • CommentRowNumber15.
    • CommentAuthorMike Shulman
    • CommentTime7 days ago

    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 XX is the colimit of the diagram consisting of all the elements of 𝒰\mathcal{U} and all the morphisms between them. If colimits are stable under pullback in CC, 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.

    • CommentRowNumber16.
    • CommentAuthorn.mertes
    • CommentTime7 days ago

    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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