Our primary interest in the first few chapters is the study of
locales in an arbitrary elementary topos S. As we mentioned in the
Introduction, S should be thought of as the basic universe of generalized
sets, and its objects will, in fact, be called "sets". Of course, we allow
ourselves the freedom of changing the universe at will, which possibility,
as will be seen in the sequel, is an important aspect of the theory.
Recall that a locale is a partially ordered set A admitting
arbitrary suprema, hence finite infima, for which the distributive law
x A
= V
(xA xi1)
holds. Regarding the supremum as a kind of addition, and the infimum as a
multiplication, a locale is clearly a kind of commutative ring. Before
studying these, we are thus led to study the simpler structure analogous
to that of abelian group, namely sup-lattice.
1. Definitions and duality
A partially ordered set M, which admits suprema of arbitrary subsets
is called a sup-lattice. A morphism f: M N of sup-lattices is a
supremum-preserving map. We denote the category of sup-lattices of 5
by s£.
The supremum of S C_M will be written VS, or sup S. The suprema
of a family (x.) of elements of M will be written \/x..
iel iel
There is a partial order on the set Hom(M,N) of sup-lattice
morphisms from M to N: f _ g iff Vx e M, f(x) £ g(x). In fact, this
partial order is itself a sup-lattice:
= VM*)-
iel x iel x
As is well-known, a sup-lattice M also admits arbitrary infima:
the infimum of a subset S C_M is calculated as the supremum of the set
of lower bounds of S. Thus, if we denote the opposite partial order on
M by M , then is also a sup-lattice.
If f: M + N is a morphism of sup-lattices, then f has a (unique)
right adjoint f*: N - M satisfying
f(x) y
x f*(y) *
Received by the editors June 20, 1983.
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