6 ANDREJ SCEDROV

R C

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.

(Cop

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,

Cor.2.3):

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

X'' j3!

F

In a Grothendieck topos, the lattice of subobjects of any object is

(Eop

a complete Heyting algebra (i.e. locale). Given F -+ G in S , and

CCop

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 :