Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf sheaves simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeSep 6th 2024
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexCreated: ## Definition The __dissolution locale__ $\mathfrak{C}L$ of a [[locale]] $L$ is defined as the poset of its [[sublocales]] (equivalently: [[nuclei]] on $L$) equipped with the relation of reverse inclusion. There is a canonical morphism of locales $$\iota\colon\mathfrak{C}L \to L$$ such that the map $\iota^*$ sends an open $a\in L$ to the open in $\mathfrak{C}L$ given by the open sublocale of $a$. ## Interpretation The map $\mathfrak{C}L\to L$ can be considered an analogue of the canonical map $T_d \to T$ for a [[topological space]] $T$, where $T_d$ is the underlying set of $T$ equipped with the [[discrete topology]]. In particular, *discontinuous maps* $L\to M$ could be defined as morphisms of locales $\mathfrak{C}L\to M$, see Picado–Pultr, XIV.7.3. ## References Original reference: * [[John R. Isbell]], On dissolute spaces, Topology and its Applications 40:1 (1991), 63–70. [doi](https://doi.org/10.1016/0166-8641(91)90058-T). Expository account: * [[Frames and Locales]], see Sections III.3, VI.4-6, and others. The dissolution frame is denoted there by $\mathcal{Sl}(L)^{op}$ (III.3.2) or by $\mathfrak{C}(L)$ (III.5.2) and the dissolution locale is denoted by $\mathfrak{S}(L)$ (XIV.7.2). <a href="https://ncatlab.org/nlab/revision/dissolution+locale/1">v1</a>, <a href="https://ncatlab.org/nlab/show/dissolution+locale">current</a>

   Created:

   Definition

   The dissolution locale $\mathfrak{C}L$ of a locale $L$ is defined as the poset of its sublocales (equivalently: nuclei on $L$) equipped with the relation of reverse inclusion.

   There is a canonical morphism of locales

   $$\iota\colon\mathfrak{C}L \to L$$

   such that the map $\iota^*$ sends an open $a\in L$ to the open in $\mathfrak{C}L$ given by the open sublocale of $a$.

   Interpretation

   The map $\mathfrak{C}L\to L$ can be considered an analogue of the canonical map $T_d \to T$ for a topological space $T$, where $T_d$ is the underlying set of $T$ equipped with the discrete topology.

   In particular, discontinuous maps $L\to M$ could be defined as morphisms of locales $\mathfrak{C}L\to M$, see Picado–Pultr, XIV.7.3.

   References

   Original reference:

   • John R. Isbell, On dissolute spaces, Topology and its Applications 40:1 (1991), 63–70. doi.

   Expository account:

   • Frames and Locales, see Sections III.3, VI.4-6, and others. The dissolution frame is denoted there by $\mathcal{Sl}(L)^{op}$ (III.3.2) or by $\mathfrak{C}(L)$ (III.5.2) and the dissolution locale is denoted by $\mathfrak{S}(L)$ (XIV.7.2).

   v1, current