Continuous Truth I Non-constructive Objects

168 M.P. FOURMAN
1.8 Surjections. Dually, we view injective inverse image maps as giving rise
to surjections of spaces.
1.9 Right adjoints.
given by
The map q
p A
where p + r = v!q I p A q 2 r}.
morphisms are different).
Each frame map f* has a right adjoint f,, direct image,
=
V{q 1 f*q 5 pl .
f*p
q has a right adjoint r + p -f r defined by
p ~ q s r iff qsp+r
Thus frames are complete Heyting algebras (but the
1.10 Definition. A map of spaces f: X -f Y is o en if the inverse image map
f*: O(y) + O(X) has a left adjoint 3,: U(X) + O(V7 commuting with A:
3f(f*(Y) A x) = Y A jf(X)
or, equivalently, if f* preserves +.
1.11 Proposition. The category of locales is complete and cocomplete. Open
surjections are stable (under pull-back).
The theory of locales is developed extensively by Joyal and Tierney C19821.
Johnstone 119821 uses locales systematically and has a comprehensive bibliography.
52 Sites and Sheaves
2.1 Definitions. Let 0 be a small categor . A cribte K of A E !El is a suboil
functor of the representable functor A E S' : that is, for each B E 181 a set
K(B) 5 IB,AI, stable under composition; for each f E K(B) and g: C -f B in C, the
composite f 0 g E K(C).
2.2 Lemma. The cribles of A form a frame, P(A).
If f: B + A in B we have an inverse image map f*: P(A) + P(B) given by
f*K = Ig f 0 g E K} for K E P(A). By abuse we write f: B -f A for the corresponding
continuous map. This map is open.
2.3 Definition. (Lawvere-Tierney) A Grothendieck topology j on is a
family of nuclei jA: P(A) -f P(A), natural in A: that is f*o jA= jBof*for f: B + A.
2.4 Lemma. If j is a Grothendieck topology on 0, the quotient frames n(A) have
induced inverse image maps f*: n(A) + n(B) and the corresponding map of locales,
which we write f: Bj +A', is open.
2.5 Definitions.
which is
A pretopotogy J on 0 i s a family J(A) c P(A) for each A E
reflerrive A E J(A)
multiplicative
K E J(A) L E J(A)
K n L E J(A)
B
stable
K E J(A) f: B + A
f*K E J(B)
(For example, let K E J(A) iff j K = T).
A crible K E P(A) is inductively ctosed for J iff