CHAPTER I - SUP-LATTICES 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 (Vxi) = V (xA x i) iel x iel 1 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.. 1 iel iel x 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: (Vyw = 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 M° 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. 1

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