Continuous Truth I Non-constructive Objects

176 M.P. FOURMAN
Proof. If Uik f: pt )# + pt W then f is represented by 5: X x U + Y in (c. For
VE()(Y) a basic open of W, w: W + U and x: W + X we have
Wlk (t;lw)(x) E
iff [S 0 l-l(v) = w
-1 -1
iff 5 (V) = w
w It- x E
iff
c5-1(v)lwl
1
regarding 5- V E O(Xx U) as an open of )# defined at U.
1
Thus Ulk 5- (V) is open. 0
v
55 Iteration
We return for a while to consideration of a general Grothendieck topos B = Sh(O,J).
We consider the internal category (I in E given by
(E(U)
with restrictions given by pulling back.
[For those who worry about coherence (one should worry), we remark that a concrete
category in E with an equivalent category of sections over U is given by
considering V/f to be represented as the element S of (PV)(U) determined by
W / ~ V E S iff ~ ~ f o v = g .
So & is an internal small full subcategory of E whose objects are subfunctors of
representables.]
We give C_ a topology by letting
xi
\/
-x
cover
Now for A E IEI we define A, E ShE(C,J) by
with restrictions for g: V
(c/u
X/f in & if Xi + X cover X in c.
UkA_(X/f)
A(X)
+ U given by restriction along f*g
and for 5: Y/h + X/g in a/U, by restriction along 5
Y A X
Any morphism A + B in E induces
U
+,B in ShE($,J).