Continuous Truth I Non-constructive Objects

172 M.P. FOURMAN
P(A) with a membership reZation E . An interpretation is standard if all this
structure is interpreted by the corresponding structure on Sh((C).
3.6 Proposition. In any standard interpretation the following schemata, which
combine comprehension and extensionality, are valid.
A
x E A 3!y E B.@(x,y) - 3 !f E B WX E A.$(x,f(x))
3!z E P(A) W X E A (X E z ++ @(x)). 0
Thus power-types and function spaces behave as they should. The categorical
characterisation of this higher-order structure in terms of adjoints is very
simple, products are categorical products,
We shall not describe this structure in general here. We shall be dealing primarily
with sorts interpreted by representables. These are particularly simple to
deal with because they have generic elements. A well-known consequence of this is
the Yoneda Lemma:
OP
F(U) [U,FI for F E ISc I and U E /cC1 .
We use this to calculate some examples of the higher-order structure. For this
exercise, we suppose that ci: has finite products and that each representable
functor is a sheaf.
3.7 Lemma.
If F is a sheaf and U,V
are representable
(1) F'(U) F(U x C)
with evaluation for u: V + U and E E F(U x V) given by ~ ( u ) = sI.
(2) (PU)(V) = n(U x V) with Ulka E R iff Rl = V.
U
Proof. F (V)
U
[V,F 1 [UxV,FI 2 F(UxV)
(PU)(V) CV,PUI Sub(UxV) = 6?(UxV). 0
A logical counterpart to Yoneda's lemma is the following.
3.8 Lemma. Generic elements for representables.
If U is representable then VIkWx E U.$ iff V x Ulk $IT~[T~/XI.
Proof. In one direction this is immediate from In the other, suppose
-+ V with a: W + U E U(W) then cf,a>: W + V x U and, by persistence, if
V x U(k $fn1C.rr2/xl then WIk $lf[a/xl.
We give an example of the use of generic elements in the simple case of a category
of presheaves.
3.9 Proposition. Choice holds for representables in categories of presheaves.
Proof.
then
Let U be a representable and suppose
Vlk WX E u.3~ E F.@(x,y)
UX VIk 3~
So by (W)' we have V[k Wx.$ . 0
F.$IT~(T~,Y)
UxVlk $ 1 ~ ~ ( . r r ~ , C for ) some 6 E
F(UxV)