User:Leigh.Samphier/Topology/Definition:Continuous Map (Locale)
Definition
Let $\mathbf{Loc}$ denote the category of locales.
A morphism of $\mathbf{Loc}$ is called a continuous map.
That is, for locales $L_1 = \struct{S_1, \preceq_1}$ and $L_2 = \struct{S_2, \preceq_2}$:
- $f: L_1 \to L_2$ is a continuous map:
- there exists a frame homomorphism $\phi: L_2 \to L_1$ such that $f = \phi^{\operatorname{op}}$
The frame homomorphism corresponding to a continuous map $f: L_1 \to L_2$ is denoted by $\loweradjoint f: L_2 \to L_1$.
Hence $f = \paren{\loweradjoint f}^{\operatorname{op}}$.
Localic Mapping
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.
Continuous Maps as Localic Maps
From User:Leigh.Samphier/Topology/Frame Homomorphism is Lower Adjoint of Unique Galois Connection, every frame homomorphism is the lower adjoint of a unique Galois connection.
From All Infima Preserving Mapping is Upper Adjoint of Galois Connection, every continuous map is uniqely determined by the localic mapping that is the upper adjoint of the frame homomorphism that corresponds to the continuous map.
This, together with the unintuitive notion of a continuous map as the dual morphism of a frame homomorphism $\phi : L_2 \to L_1$, leads some sources to define the continuous map $f : L_1 \to L_2$ to be the localic mapping $f : L_1 \to L_2$. This definition of a continuous map results in the category of locales with localic mappings.
From User:Leigh.Samphier/Topology/Category of Locales with Localic Mappings is Isomorphic to Category of Locales we have that the category of locales with localic mappings and the category of locales are isomorphic categories but they are not identical.
Also see
Sources
- 1982: Peter T. Johnstone: Stone Spaces: Chapter II: Introduction to Locales, $\S 1.1$