I'm hoping someone can elucidate a step in the proof of V.1.2 in Mac Lane & Moerdijk's Sheaves in Geometry and Logic.
Let $\mathcal{C}$ be a small category and $J$ a Grothendieck topology. $J$ induces an endomorphism $j$ on the subobject classifier $\Omega$ in $\mathbf{Set}^{\mathcal{C}^{op}}$ by setting, for a sieve $S\in\Omega(c)$, $$j_c(S)=\{f:\mathrm{dom}(f)\to c\:|\:f^*S\in J(\mathrm{dom}(f))\}.$$ The above mentioned proof in Sheaves in Geometry and Logic establishes that this map is a natural transformation and, moreover, a Lawvere-Tierney topology. I find most of the proof fairly straightforward, except when it comes to idempotence.
Mac Lane & Moerdijk's proof explains why $j_c(S)\subset j_cj_c(S)$, which is clear enough, and then for the inclusion $j_cj_c(S)\subset j_c(S)$ they say,
Conversely, if $g\in j_cj_c(S)$, then $j_c(S)$ covers $g$. But for each $h\in j_c(S)$ one has that $S$ covers $h$. Hence, by the transitivity property of a Grothendieck topology, $S$ covers $g$, i.e., $g\in j_c(S)$.
(Here, "$S$ covers $f$" is a synonym for "$f^*S\in J(\mathrm{dom}(f))$".)
My intuition says it's obvious that transitivity has to be invoked for this, but I can't quite see how it's doing the work here; I'm missing whatever dot connects the "hence" with the bits before it. Can someone clarify how the transitivity property gives the conclusion here?
Best Answer
Let me reword this more explicitly.
Recall the transitivity axiom for a Grothendieck topology in arrow form:
Also recall that by definition, $j_C(S) = \{g|S\text{ covers }g\}$.
Now to show $j_Cj_C(S)\subset j_C(S)$ we need to show that for $g$ covered by $j_C(S)$ (ie. $g\in j_Cj_C(S)$), $S$ also covers $g$. Now by definition, $S$ covers all arrows of $j_C(S)$. Since $j_C(S)$ covers $g$, it follows by transitivity that $S$ covers $g$.