User:Leigh.Samphier/Topology/Category of Locales with Localic Mappings is Isomorphic to Category of Locales/Lemma 3
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Theorem
Let $\mathbf{Loc}$ denote the category of locales.
Let $\mathbf{Loc_*}$ denote the category of locales with localic mappings.
Let $F : \mathbf{Loc} \to \mathbf{Loc_*}$ be defined by:
- for each locale $L$ of $\mathbf{Loc_*} : \map F L = L$
- for each continuous map $f : L_1 \to L_2$ of $\mathbf{Loc} : \map G f = f_*$
where:
- $f_* : L_1 \to L_2$ denotes the upper adjoint of the frame homomorphism $f^* : L_2 \to L_1$
- $f = \paren{f^*}^{\operatorname{op}}$
Let $G : \mathbf{Loc_*} \to \mathbf{Loc}$ be defined by:
- for each locale $L$ of $\mathbf{Loc_*} : \map G L = L$
- for each localic mapping $g : L_1 \to L_2$ of $\mathbf{Loc_*} : \map G g = \paren{g^*}^{\operatorname{op}}$
where:
- $g^* : L_2 \to L_1$ denotes the frame homomorphism that is the lower adjoint of the localic mapping $g$
- $\paren{g^*}^{\operatorname{op}}$ denotes the opposite morphism of the frame homomorphism
Then:
- $GF = \operatorname{id}_{\mathbf {Loc}}$
Proof
$\blacksquare$