6 ANDREJ SCEDROV
Shj((C) is then the full subcategory of S whose objects are sheaves
for the Grothendieck topology J . By a Grothendieck topos we mean the
category of sheaves on a site (this includes presheaf categories, the topo-
logy being trivial). Every Grothendieck topos is an elementary topos with
the natural number object (outline is in[Jo1,§1.1] ),and it moreover has
all (small) limits and colimits. The best, but very detailed reference
to Grothendieck topoi is [SGA4], cf. also [MR], [Jo1J.
The canonical topology on ( C is the largest topology for which all
the representable functors are sheaves. We say that on topology J is
subcanonical if it is smaller than the canonical topology, i.e. if all the
representable functors are J-sheaves.
The inclusion ShT(CC)c-^ S has a left adjoint, for any site ((C,J) .
It is called the associated sheaf functor, which also preserves finite
limits. The natural number object in a Grothendieck topos is the associ-
ated sheaf of the constant presheaf U .
The following lemma is useful in the applications (cf. [SGA4],Exp.II,
Lemma. Let K be a family of sieves on a small category C C , and assume
that K is closed under pullbacks. Let J be the Grothendieck topology
on E generated by K . Then a presheaf F on E is a J-sheaf iff
for each object X of ( E , and each R e K(X) :
R c_ X
In a Grothendieck topos, the lattice of subobjects of any object is
a complete Heyting algebra (i.e. locale). Given F -+ G in S , and
a Grothendieck topology J , we say that F '-*• G in S is the J-
closure of F *-* • G iff for each object X of C C :