Continuous Truth I Non-constructive Objects

Continuous Truth I 169
f: B + A f*K E J(B) .
f c K
As A is closed and an intersection of closed cribles is closed, each crible
K E P(A) has a cZosure jAK. This gives a topology j on C. We say K inductiveZy
covers A iff j K = A, and write this K E J(A).
___R_
2.6 Exam les. (1) Let U(X) be a frame viewed as a poset viewed as a category.
Let K E J A iff VK = A. (Identifying B + A with its domain.) Then n(X) 1 O(X).
(2) Let be a small category of locales closed under finite limits and open
inclusions. Let K E J(A) iff K contains some family Ifi: Bi + Ali E 11 of open
maps such that V 3fi(Bi) = A.
is closed for this topology. This assignment gives an A;V map r*: U(A) + n(A),
spatially a surjection r: AJ + A. Each closed crible contains a largest open
inclusion. This assignment gives an A V map i*: n(A) + U(A), spatially we have
an inclusion i: A c+JJ. Furthermore, r o i = idA and r*i* s id,(A) so A is an
adjoint retract of AJ (Fourman C19821).
2.7 Definition. A presheaf on C is a functor X: Cop + Sets. If f: B + A E C and
a E X(A) we use the notation alf, "a restricted along f", for X(f)(a) E X(B).
Note that alflg = alfo g and alid = a. The appropriate morpkisms between presheaves
F: Y + X are natural transformations, maps FA: Y(A) + X(A) which commute
with restrictions FB(a,lf) = (FAa)lf.
2.8 Exam les. (1) The representable functor [-,A], (or by abuse, justified by
Yone a s emma, A) is a presheaf. Restrictions are by composition glf = gof.
Yoneda's lemma tells us that for any other presheaf, X, we have X(A) [A,X]. In
-7++
The crible generated by each open inclusion U 4 A
particular the embedding C + Stop is full and faithful. Each crible K E P(A) is a
subpresheaf K H A.
(2) P and n are presheaves with restrictions given by inverse images Klf = f*(K).
2.9 Definition. A presheaf X on C is a sheaf for the (pre) topology J if whenever
K E J(A), each natural transformation x: K + X has a unique extension along
K >+ A. Equivalent1 y, if K E J(A) and we have a family xf E X(B) for f: B -f A E K
such that xgf = xflg for each g: C + 8, there is a unique x E
xf = xlf for each f E K.
Grothendieck topos.
X(A) such that
The cateaorv of sheaves and natural transformations is a
There is, as yet, no staisfactory introductory text on topos theory.
references are SGA4, Wraith C19751, Johnstone C19771, Freyd '19721.
The basic
53 Forcing over a site
Here we describe Joyal's presentation of interpretations in topoi in terms of a
notion of forcing. Let C and a (pre) topology J be fixed. The basic structures
we consider are diagrams
-
of preskeaves on C. Each presheaf A interprets a type
or sort of variable. A morphism f: A1 x ... x An + B interprets an n-ary operation.
A subobject R A1 x ... x An interprets an n-ary relation.
3.1 Definitions. Let L be a first-order language (possibly many-sorted) with
equality. An interpretation of L is given by assigning to each sort A of L a
presheaf A, to each operation F from A1,. . .,An to B a natural transformation
F: A1 x ... x An + B, and to each relation R on A1 x ... x An a subfunctor
R * A1 x .. . x An.
Given such an interpretation, for U E JCCJ we let LU be the