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Chapter 5
Sublocales, that is, generalized subspaces
In Corollary 4.5 we have seen that strong monomorphisms in Loc are precisely the one-one localic maps. Thus, the most natural notion of a subobject of a locale L in our context is a locale M that is embedded into L by a one-one localic map.
5.1. One-one localic maps. Let h : M → L be a one-one localic map. By 4.6 we have a decomposition
L e=(a→h(a)) // h[L] ⊆ // M.
with e an epimorphism. By Corollaries 4.5 and 4.4, e is an isomorphism. Thus,
up to isomorphisms, one-one localic maps are embeddings of sub- posets that are, in the induced order, locales themselves.
This, of course, does not mean that they be subframes, that is, that the em- bedding should preserve the lattice structure (the joins generally differ). In this chapter we will start to discuss the structure of such subposets; they will model the subspaces of locales viewed as generalized spaces.
5.2. Sublocales. A subset S ⊆ L is said to be a sublocale if (S1) it is closed under all meets, and
(S2) foreverys∈Sandeveryx∈L,x→s∈S.
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