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Chapter 9
Products. Completeness of the
category Loc
Products of locales are more naturally seen as coproducts in Frm. Hence, this chapter will be more on the contravariant side, presenting the standard con- struction of coproducts of frames.
It should be noted that products of spatial locales (i.e. of the type Lc(X)) do not generally agree with the products of the corresponding spaces. Even for finite products one often has Lc(X × Y ) not isomorphic to Lc(X) × Lc(Y ). Of course, as the functor Pt is a right adjoint, we have
Pt(L × M) ≃ Pt(L) × Pt(M). 9.1. Notation. Let Li, i ∈ J be a system of locales. Set
′ Li = (xi)i ∈ Li | uptofinitelymanyi,xi =1 .
i∈J i∈J Letα∈Jbefixed.Forx∈Lα andu∈′i∈JLi set
x∗αu=(vi)i wherevi =x fori=α ui fori̸=α.
Finally define 1 ∈ ′i∈J Li by
1i =1 foralli.
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