Continuous Truth I Non-constructive Objects

Continuous Truth I 177
For those who prefer global descriptions, we associate to A E If[ (pseudo)
functors
6/U + E/U
natural in U (i.e. comnuting with g* for g: V + U) as follows:
where
31
U
For Y
'
+
U X
nf ,
E/U
a, B .
X we have nh P npE whence nhS* * nf (as E.* 4 ng)
and nhAy * nPx (as E*Ax Ay) .
This gives the required arrow nhayA + nfAXA.
functor
OP
C+EC *
We shall show that this preserves first order logic.
What we obtain is an (internal)
liere we work concretely for the sake of computations. A simple but more abstract
treatment will appear in Fourman and Kelly C19831. We now consider a first-order
language L with sorts for the objects of E and operations symbols for its
morphisms. In fact to avoid size problems, we consider an arbitrary small fragment
of such a language. We may consider L also as a language in K as a constant
object (via A).
Working in E we consider the interpretation of L given by interpreting the sort A
by A and each operation f: A + B by the corresponding morphism 4 + &.
5.1 Lemma. For f: X + U and g: X + V
~lk x/fk 9 iff vlt- X/gl!- +
Proof. By induction, it suffices to show that if Ulk X/flk *g
v~F X/glk 0 for all g: X + v
then IF is closed under the rules of . As no rule decreases the complexity of
9 we say assume that the result holds for subformulae of 9.
Only (+)+ and (W)' present any difficulties.
result for @ and $.
is defined to mean
Me consider (-+)+, and suppose the
Suppose that for all E: W -+ U and all h: Z + g*X, if W
then WlkZ/(E*f h)Ipd(f*E-h). Then if n: W' + V and h': Z + rr*X are such
that W' Z'/(n*g 0 h') IF @l(g*n 0 h') then by induction hypothesis
Z/(E*f h) Ip*01(f*E 0 h)