6
MICHAEL MAKKAI
The forgetful functors on Boole as well as on S are taken to be the usual
underlying-set functors. Condition ZO is obviously satisfied.
There is a specific object in S, 2 ; its underlying set 121=2, and its operations are
IT ' = U, one for each ultrafilter (J, U) . If Se S, then horn(5, 2) = horn (5, 2) is a
I CI I CI
subset of 12 I , the underlying set of the power 2 of the 2-element Boolean algebra
I CI
with exponent 151 .In fact, horn (5, 2) is closed under the operations of 2 , as is
immediately realized upon considering the connection between the structures 2 and 2 . Thus,
I CI
we may consider the subalgebra of the Boolean algebra 2 with underlying set
horn (5, 2) ; let us call it horn (5, 2) . For f: S— S' in S, the Boolean homomorphism
f I C ' I I CI
f =2 :2 2 restricts to a Boolean homomorphism
hom{f, 2) :hom(S', 2) -^hom{S, 2) .
We have defined a contravariant functor
F= hom(S,2) :S Boole
o p
.
We define
G = hom(-,2) : Boole
o p
S
\A\
similarly. For A a Boolean algebra, horn (A, 2) is the subalgebra of the power-algebra 2
in S with underlying set horn (A, 2) .
It is easy to see that F is a left adjoint to G ; the counit and unit maps are evaluations:
eA : A hom{hom{A, 2), 2)
a I [ul u{a)] (uehom(A, 2) ) ,
r\s : S hom{hom{S, 2), 2)
s I [xi x(s)] ( XG hom(S, 2) ) .
In fact, horn (A, 2) , with any Boolean algebra A , is a disguised form of the Stone space of
A ; the infinitary operations U , for S = horn {A, 2) , are ultrafilter-limits familiar in the
context of compact Hausdorff spaces. Condition Zl is just the Stone representation theorem, and
Z2 can be obtained by an argument similar to one used to prove that the topological version of
e is surjective. The proofs are no harder than the ones in the familiar topological
formulation.The proof of Z2 is also very similar to the proof of (*) in Section 8 below.
The point of the present formulation, in particular, the choice of the category S, is that
Z3 becomes obvious; in fact, the regular factorization of all arrows in S is the
surjective/injective factorization. This is clear on the basis of the "algebraic" character of S.
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