Definition 7.1. Suppose C and D are two framed sites. A functor F: C ® D is continuous if all the objects and morphisms in C are geometric (therefore strongly geometric) over D (i.e., C = Geo(C/F)). Remark 7.2. (a) Any bicontinuous functor
of framed sites is continuous.
Proposition 7.3. Suppose F: C ® D is a continuous functor. Suppose X is an object of C and W is an open sieve on F(X). Then F_{X}^{1}(W) Í F_{X}^{*}(W). If F is strict and W is open effective then F_{X}^{*}(W) = F_{X}^{1}(W). Proof. Suppose f: Y ® X is a morphism in F_{X}^{1}(W). Then F(f): F(Y) ® F(X) is in W, so 1_{D/F(Y)} Í F(f)^{1}(W). Thus 1_{C/Y} = F_{Y}^{*}(1_{D/F(Y)}) Í F_{Y}^{*}(F(f)^{1}(W)) = f^{1}(F_{X}^{*}(W)), which implies that f is in F_{X}^{*}(W). Thus F_{X}^{1}(W) Í F_{X}^{*}(W). If F is strict and W is open effective then F_{X}^{*}(W) Í F_{X}^{1}(W) by (6.2.d), hence F_{X}^{*}(W) = F_{X}^{1}(W). n Theorem 7.4. Suppose F: C ® D is a strict functor. Then F is continuous if and only if for any object X of C and any open effective cover {W_{i}} of F(X), {F_{X}^{1}(W)} is an open cover of X. Proof. One direction comes from (7.3). For the other direction suppose the condition holds for F. Since any open effective sieve W on F(X) can be included in an open effective cover of F(X), F_{X}^{1}(W) is open, and so F_{X}^{*}(W) = F_{X}^{1}(W). It follows that X is geometric over D, and each morphism f: Y ® X is geometric because f^{1}(F_{X}^{1}(W)) = F_{Y}^{1}(F(f)^{1}(W)) holds (unconditionally for any sieve W). n An object X of a framed site C is called local if the joint M(X) of all the open sieves U ¹ 1_{C/X} of X is not 1_{C/X}. A morphism f: Y ® X of local objects of C is called a local morphism if f^{1}(M(X)) ¹ 1_{C/Y}. Remark 7.5. The following conditions for
an object X of C are equivalent:
Remark 7.6. Suppose X, Y
and Z are local objects of C and f: Y ®
X, g: Z ® Y
are morphisms in C.
Proof. (a) First we show that if U is an open sieve on
F(X) and F_{X}^{*}(U) = 1_{X},
then U = 1_{F}_{(X)}. The assumption
F_{X}^{*}(U) = 1_{X} implies
that there is an open cover {V_{i}} of X such that
each restriction F(V_{j}) ®
F(X) is in U. Since X is local there is some
i such that V_{i} = 1_{X} by (7.5.c).
Thus F(X) ® F(X)
is in U, hence U = 1_{F}(X).
