# Equivalence of the definitions of a sheaf in SGA4 and in “Categories and Sheaves”

I asked this question on Mathematics Stack Exchange, but got no answer.

I don't understand why the definition of a sheaf (Definition 17.3.1 (ii)) given in the book

[KS] Categories and Sheaves by Kashiwara and Schapira

is equivalent to the definition of a sheaf (Definition 2.1) given in

[V] Verdier, Exposé II, SGA4, http://www.normalesup.org/~forgogozo/SGA4/02/02.pdf

To simplify, let me consider only set-valued presheaves.

Here is, in the terminology of [KS], how I understand the two definitions. (Warning: my understanding might be incorrect!)

Let $\mathcal U$ be a universe, let $X$ be a small site and let $F$ be a $\mathcal U$-set-valued presheaf over $X$. Then:

$\bullet\ F$ is sheaf in the sense of [V] if $F(f)$ is an isomorphism for any $A$ in $(\mathcal C_X)^\wedge$, any $U$ in $\mathcal C_X$, and any local isomorphism $f:A\to U$ which is a monomorphism,

$\bullet\ F$ is sheaf in the sense of [KS] if $F(f)$ is an isomorphism for any $A$ in $(\mathcal C_X)^\wedge$, any $U$ in $\mathcal C_X$, and any local isomorphism $f:A\to U$.

The difference is that the local isomorphism $f$ is supposed to be a monomorphism in Verdier's definition.

A "KS-sheaf" is of course a "V-sheaf", but I'm unable to prove the converse.

• I think these might be equivalent by the argument that proves "Ken Brown's lemma" in model category theory. – Dylan Wilson Oct 11 '17 at 12:10
• @DylanWilson - Thanks! Unfortunately I know nothing about model categories. You're most welcome to give further details in a new comment or in an answer. (I may accept an answer I don't fully understand.) – Pierre-Yves Gaillard Oct 11 '17 at 12:35
• I've just noticed that the equivalence of the definitions follows from Proposition 5.3, implication "(i) $\implies$ (ii bis)", in Verdier’s Exposé normalesup.org/~forgogozo/SGA4/02/02.pdf – Pierre-Yves Gaillard Oct 18 '17 at 20:33

Let $L$ denote the usual sheafification functor, a la Grothendieck and Verdier etc. Then I claim it's enough to show $L$ sends local epimorphisms to epimorphisms and local monomorphisms to monomorphisms. Indeed, if this is the case then $L$ takes local isomorphisms to epi-monomorphisms, and such things are isomorphisms in toposes.
Since $L$ preserves finite limits, we need only check that $L$ takes local epis to epis (since $A \to B$ is a local mono iff $A \to A\times_BA$ is a local epi). But that's not so bad: If $A \to B$ is a local epimorphism then the map $im(A \to B) \to B$ is a local epimorphism and a monomorphism, and hence a monic, local isomorphism. But $L$ preserves images (since they're computed as colimits and L preserves those) and takes monic, local isomorphisms to isomorphisms. Thus $LA \to LB$ has the property that $im(LA \to LB) \to LB$ is an isomorphism (NB: that image is computed in the category of sheaves), and so the map is epi in the category of sheaves, which is what we wanted.
• I suddenly realize that I didn't understand your argument. In the penultimate sentence you write that $L$ "takes monic, local isomorphisms to isomorphisms". Why is this true? – Pierre-Yves Gaillard Oct 12 '17 at 18:05
• Isn't that the defining feature of $L$? A 'monic, local isomorphism' with target a representable is the same as a covering sieve I think. But every monic, local isomorphism is built out of these from colimits (express the target as a colimit of representables, and pull-back the source to each of these, getting covering sieves throughout by the axioms of a topology) – Dylan Wilson Oct 12 '17 at 18:39
• Thanks! One of the things you're saying is that, given an object $U$ of $\mathcal C$ and $S\hookrightarrow U$ a covering sieve viewed as a subobject of $U$ in $\mathcal C^\wedge$, the morphism $LS\to LU$ is an isomorphism. Right? If so, could you explain this as simply as possible? – Pierre-Yves Gaillard Oct 12 '17 at 19:26
• It's basically the definition of $L$ that it inverts those, but if you have a different definition in mind that's fine: The map $LS \to LU$ is an isomorphism iff for all sheaves $F$, $\mathrm{Hom}(LU, F) \to \mathrm{Hom}(LS, F)$ is an isomorphism. Now use that $L$ is left adjoint to the inclusion and the definition of $F$ being a sheaf. – Dylan Wilson Oct 12 '17 at 20:08