nLab dissolution locale

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. doi90058-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).