User:Leigh.Samphier/Topology/Definition:Continuous Map (Locale)/Localic Mapping

Definition

Let $L_1 = \struct{S_1, \preceq_1}, L_2 = \struct{S_2, \preceq_2}$ be locales.

Let $f : L_1 \to L_2$ be a continuous map.

That is, $f$ is the dual morphism of a frame homomorphism $\loweradjoint f : L_2 \to L_1$.

The frame homomorphism $\loweradjoint f : L_2 \to L_1$ corresponding to $f$ is a lower adjoint of a Galois connection $\struct{\upperadjoint f, \loweradjoint f}$.

The upper adjoint $\upperadjoint f : L_1 \to L_2$ of $\struct{\upperadjoint f, \loweradjoint f}$ is called a localic mapping.

Also see

• User:Leigh.Samphier/Topology/Frame Homomorphism is Lower Adjoint of Unique Galois Connection

Sources

• 2012: Jorge Picado and Aleš Pultr: Frames and Locales: Chapter II: Frames and Locales. Spectra, $\S 2.2$