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In SGAIV.1, Exp. III, sec. 5, sites $(C,J)$ are localised with respect to a presheaf $X$ on $C$ (not nec. just a representable one).

Question1: There it is asserted that if $F \to X$ is a morphism of presheaves on C, i.e., an object of $PreSh(C)/X$, then its image under the equivalence $PreSh(C)/X \to PreSh(hC/X)$ is a sheaf iff F is isomorphic to the cartesian product ${X}x_{ia(X)}ia(F)$ considered over $X$ in $PreSh(C)/X$. Has this been worked out somewhere? While working out the details one encounters some points more difficult than the material in Exp.'s II and III preceding it.

Question2: If G is a sheaf on $(hC/X)$ and $i_X$ denotes the inclusion of $Sh(hC/X)$ into $PreSh(hC/X)$, is ${j_X}_!(i_X(G))$ always a sheaf on $C$? (The topology $J$ is not necessarily subcanonical.)

Some of the details of this section can be extended to a general pair of adjoint functors and their categorical localisation to slice categories, which leads one to pose the following:

Question3: Has the discussion there been extended to arbitrary pairs of adjoint functors (L,R), $L:C \to C'$, $R:C' \to C$, (L',R'), $L':D \to D'$, $R':D' \to D$ and an equivalence of categories (A,B), $A:C \to D, B:D \to C$ (perhaps with R, R' fully faithful)?

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3 Answers 3

I will try to answer your questions 2, and what I have to say should also help with questions 1 and 3. Because I do not currently have access to a copy of SGA IV I will first restate the questions in a slightly different way in order to fix notation and make sure that I have correctly understood the question. If $(C,J)$ is a site and $X$ is a presheaf on $C$, then there is an equivalence of categories (obtained by Kan extension) between $\text{PreSh}(C)/X$ and the category of presheaves on the category of elements $\text{el}(X)$ of $X$ (this is what you have denoted by $hC/X$). The Grothendieck topology $J$ on $C$ then induces a topology on the category of elements and your questions are concerned with the connection between the resulting categories of sheaves for these two topologies. One of the nice things about sheaves is that they can be described in a number of equivalent ways and here I think it is convenient to look at the original question from a slightly different perspective. (I think the category of elements here is a bit of a red herring and that's why I address the questions using Lawvere-Tierney topologies instead of Grothendieck topologies.)

Given any (elementary) topos $\mathcal{E}$ with a Lawvere-Tierney topology $j$ on $\mathcal{E}$ and an object $X$ of $\mathcal{E}$, there is an induced Lawvere-Tierney topology on the slice topos $\mathcal{E}/X$ and we can ask about the connection between the resulting categories of sheaves. (I think, but haven't checked, that you can easily find a proof of the claim from SGA IV that you ask about in your first question when working in this setting.)

Question 2: Your original question is equivalent (if I've understood it correctly) to the question whether the image of a sheaf $F:A\to X$ with respect to the topology on $\mathcal{E}/X$ under the (inclusion of sheaves into $\mathcal{E}/X$ followed by the) left-most adjoint of the essential geometric morphism $\mathcal{E}/X\to\mathcal{E}$ is always a sheaf with respect to the topology $j$ on $\mathcal{E}$. This is false. For example, the terminal object $1_{X}:X\to X$ in $\mathcal{E}/X$ is a sheaf with respect to the induced topology on $\mathcal{E}/X$, but its image under the left-most adjoint is just $X$ itself and therefore need not be a sheaf.

Question 1 and 3: I believe the best place to consult about these matters is probably the chapter on geometric morphisms in "Sheaves in Geometry and Logic" by Mac Lane and Moerdijk. There is an extensive discussion of the connection between maps of sites and the resulting categories of sheaves there (you might also check Johnstone's "Elephant"). (I'm not sure though that your precise questions are addressed there, so this is only a partial answer.)

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Some points pertaining to question 1:

1.1/ SGA can always be found at http://wstein.org/sga/ while outside of institutional access, of course.

1.2/ The diagram of proposition 5.5 of SGAIV.1, III.5, esp. the two rightmost squares, is asserted to be commutative up to isomorphic 2-cell(s). This reduces to the equivalence of (i) and (ii) in the proof of prop. 5.4, where (ii)=>(i) is easy (along the lines of what happens to adjointness under "slicing" type of abstract nonsense). (i)=>(ii) is easy by reversing somewhat (ii)=>(i) via adjointness, PROVIDED ${j_X}_{!}(i_{X}(G))$ is a sheaf on $C$ as mentioned in question 2.

2/ The essential point of question 1 can then be (re)expressed as follows: denote ${\beta}_{X}: PreSh(C)/X \to PreSh(hC/X)$, then it remains to prove $i_{X} a_{X}({\beta}_{X}(F \to X ))$ is isom. to the sheaf ${\beta}_{X}( {X{x}_{ia(X)}{ia(F)}}\to X )$.

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Concerning (i)=>(ii):

Since this question did not generate adequate interest here, I then asked P. Deligne, who immediately replied with a correct answer via the "hypercech picture for 'associated sheaf'" and argues directly using coverings along with heavily using that the image of $F \to X$ in $PreSh(hC/X)$ belongs to $Sh(hC/X)$ (which can be re-expressed in $X$-relative terms in $PreSh(C)/X$.

This confirms that the "direct" approach works, instead of trying to argue using adjunctions.

We have now written out an even more direct argument couched in terms of SGAIV.1 Exp. II (cribles, LF, esp. Lemme 3.1) (which does not use the hypercech picture, only the properties of LF). Of course, since $ia=LL$, one must "unwind"/"lift" twice and to prove "surjectivity" glue suitably.

We also generalise pieces of Lemme 3.1 (loc. cit.).

Perhaps the topos theorists have a more general argument?

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