9. Canonical Sites
We say that a framed site (C, G) is
canonical if G
= v_{C}, i.e., G(X)
is the set of subobjects of X Î C.
Remark 9.1. (a) Any canonical framed
site is effective by (2.4.d).
(b) Any frame viewed as a category is a neat canonical framed site.
Remark 9.2. (a) The category Loc
has colimts. Denote by J: Loc ®
Set^{op} the functor sending each locale to its underlying
set. Then J lifts colimits (cf. [Borceux
1994, Vol III, Section 1.4]).
(b) Suppose G: C ®
Loc is a framed topology on a category C such that
the composition JG: C^{op}
® Set is representable. Since any
representable functor preserving limits, JG
preserves limits in C^{op}. It follows that JG:
C ® Set^{op} preserves
colimits. Combining with (a) we conclude that JG:
C ® Loc preserve colimits.
Theorem 9.3. (a) Any elementary topos
such that the poset v_{C}(X)
is complete for any X Î C
is a canonical framed site and G: C ®
Loc preserves colimits.
(b) Any elementary topos with colimits is a complete framed site.
(c) Any Grothendieck topos is a complete canonical framed site.
(d) Any elementary topos such that v_{C}(X)
is finite for any X Î C
is a strict canonical framed site.
Proof. (a) The poset v_{C}(X)
of subobjects of each object X of an elementary topos is a Heyting
algebra, and the pullback along any morphism f: Y ®
X induces a morphism f^{1}: v_{C}(X)
® v_{C}(Y)
which has both a left adjoint and a right adjoint [Borceux
1994, Vol III, Prop. (6.2.3)]. Thus (C, v_{C})
is an effective framed site if each v_{C}(X)
is complete. The second assertion follows from (9.2.b).
(b) Suppose {U_{i}} is an open cover of an object X.
Let Y be the coproduct of {U_{i}}and p: Y
® X the canonical morphism. Suppose
p = uv where v is an epimorphism and u a monomorphism.
Since each monomorphism U_{i} ®
X factors through u, and the joint of {U_{i}}
is 1_{X}, we see that u is an isomorphism, so p
= v is an epimorphism. The epimorphism p: Y ®
X is the coequalizer of its kernel pair by [Mac
Lane and Moerdijk 1993, p. 197]. This implies that X is a colimit
of {U_{i} Ç U_{j}}
via restrictions, so C is strict.
Next we prove that C is complete. Suppose ({X_{i}},
{U_{ij}}, {u_{ij}}) is a glueing diagram
of C. Suppose (X, {v_{i}}) is the colimit
of ({X_{i}}, {U_{ij}}, {u_{ij}})
in C, where each v_{i}: X_{i} ®
X is a morphism in C. The joint of the images of all the
v_{i} is X by (a) because X is a colimit of
X_{i}. Thus it suffices to prove that each v_{i}:
X_{i} ® X is a monomorphism.
We imitate the proof of [Mac
Lane and Moerdijk 1993, p.211, Corollary 4]. For a fixed v_{i}
let PX_{i} be the power object of X_{i} with
the monomorphism t: X_{i} ®
PX_{i}. Since PX_{i} is injective and e_{j}:
U_{ij} = U_{ji} ®
X_{j} is a
monomorphism for each j, the morphism te_{j}:
U_{ij} ® X_{i}
® PX_{i} extends to X_{j},
to give a morphism g_{j}: X_{j} ®
PX_{i}. Since X is a colimit, there is a unique h:
X ® PX_{i} such that
t = hv_{i }and g_{j} = hv_{j}.
Since t = hv_{i} is a monomorphism, so is v_{i}.
This proves the assertion.
(c) is a special case of (b).
(d) The proof that C is strict is similar to (b) because any
finite Heyting algebra is complete, and any elementary topos has finite
colimits. n
Example 9.3.1. (a) The category FSet of finite sets is
a canonical framed site by (9.3.c).
(b) Set is a complete canonical framed site and is a completion
of FSet.
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