Essential sublocales are a generalization of locally connected sublocales.
A sublocale $X_j$ of a locale $X$, given by a nucleus $j : \mathcal{O}(X) \to \mathcal{O}(X)$, is called essential (sometimes also principal) if and only if the following equivalent conditions are satisfied:
This also shows the equivalence of (1) and (3).
The left adjoint $b$ automatically satisfies $b(u) \preceq u$, $b(b(u)) = u$, $j(b(u)) = j(u)$, and $b(j(u)) = b(u)$ for all $u \in \mathcal{O}(X)$. These relations follow from playing around with the adjunction.
Any open sublocale is essential: The nucleus for an open sublocale is of the form $j = (u \rightarrow -)$. Its left adjoint is $b = (- \wedge u)$.
More generally, any locally connected sublocale is essential.
For a general sublocale $i : X_j \hookrightarrow X$ and a sheaf $\mathcal{F}$ on $X$, the pullback $i^{-1} \mathcal{F}$ is the sheafification of the presheaf $u \mapsto \colim_{u \preceq j(v)} \mathcal{F}(v)$. In the case that $X_j$ is an essential sublocale, this presheaf is simply given by the formula $u \mapsto \mathcal{F}(b(u))$ and is already a sheaf, so sheafification is not necessary.
A sublocale of the one-point locale is essential if and only if it is open. This is because the extra Frobenius reciprocity condition demanded by openness is automatically satisfied (in classical mathematics and in impredicative constructive mathematics).
The lattice of essential sublocales of a given locale is complete. Suprema are calculated as in the lattice of all sublocales; infima, however, are not. There are counterexamples in (Kelly and Lawvere (1989)), however in the slightly different context of essential localizations of categories.
Let $X_j \hookrightarrow X$ be a sublocale. From the internal point of view of the topos $Sh(X)$, this sublocale looks like a sublocale of the one-point locale and itβs interesting to compare the properties of $X_j$ with this internal locale.
The following statements are equivalent:
Note that none of these statements are equivalent to $X_j \hookrightarrow X$ being an essential sublocale.
G. M. Kelly, F. W. Lawvere, On the complete lattice of essential localizations, Bull. Soc. Math. de Belgique XLI (1989) pp. 289β319 [pdf]
Guilherme Frederico Lima, From essential inclusions to local geometric morphisms, talk at Topos Γ lβIHΓS in September 2015.
Last revised on March 27, 2023 at 07:19:37. See the history of this page for a list of all contributions to it.