Spaces, frames, locales

Here are some of the key concepts from Mac Lane and Moerdijk's SGL related to the above.

$\mathbf{(Spaces)}$ (arrows go in geometric direction), 

$\mathbf{(Frames)}$ (arrows go in algebraic direction), 

$\mathbf{(Locales)} = {\mathbf{(Frames)}}\op$ (arrows go in geometric direction),

Quoting Mac Lane-Moerdijk, Section IX.1,

For a map $f : S \to T$ of spaces, the locale-map $\Loc(f) : \Loc(S) \to \Loc(T)$ is given by the frame morphism $f\inv : \calO(T) \to \calO(S)$, 

i.e.

$\Big(f :S \to T\Big) \in \mathbf{(Spaces)} \xrightarrow{\textstyle \Loc} \mathbf{(Locales)} \ni \Big( \Loc(f) : \Loc(S) \to \Loc(T) \Big) = \Big( { \big( f\inv: \mathcal{O}(T) \to \calO(S) \big) }\op \Big) \in { \mathbf{(Frames)} }\op$.