User:Leigh.Samphier/Topology/Definition:Category of Locales with Localic Mappings

Definition

The category of locales with localic mappings, denoted $\mathbf{Loc_*}$, is the category with:

Objects:		locales, that is, Complete Lattices $\struct{L, \preceq}$ satisfying infinite join distributive law
Morphisms:		Localic Mappings, that is, mappings that are the upper adjoint to a frame homomorphism
Composition:		Standard composition of mappings
Identity morphisms:		$\operatorname{id}_{\struct {L, \preceq} } := \operatorname{id}_L$, the identity mapping on $L$

Also see

• User:Leigh.Samphier/Topology/Category of Locales with Localic Mappings is Category

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

• User:Leigh.Samphier/Topology/Category of Locales with Localic Mappings is Isomorphic to Category of Locales

Sources

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