nLab essential sublocale

Idea

Essential sublocales are a generalization of locally connected sublocales.

Definition

Definition

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:

1. There is a monotone map $b : \mathcal{O}(X) \to \mathcal{O}(X)$ which is left adjoint to $j$.
2. The nucleus $j$ preserves arbitrary (not only finite) meets.
3. For any $u \in \mathcal{O}(X)$, there is a smallest $v \in \mathcal{O}(X)$ such that $u \preceq j(v)$.
4. The pullback functor $\mathrm{Sh}(X) \to \mathrm{Sh}(X_j)$ possesses a left adjoint (it always has a right adjoint).
5. The geometric embedding $\mathrm{Sh}(X_j) \hookrightarrow \mathrm{Sh}(X)$ is an essential geometric morphism.

Proof of equivalence

• The equivalence of (1) and (2) is by the adjoint functor theorem for complete lattices, which furthermore gives an explicit formula for the left adjoint $b$, namely
  $$b(u) \coloneqq inf \{ v \in \mathcal{O}(X) \,|\, u \preceq j(v) \}.$$

  This also shows the equivalence of (1) and (3).

• The equivalence of (4) and (5) is by definition.
• Recall that the pullback of the representable sheaf $Hom_{\mathcal{O}(X)}(\cdot,u)$ is $Hom_{\mathcal{O}(X)}(\cdot,j(u))$. Therefore continuity of the pullback functor translates to continuity of $j$. This shows that (4) implies (2).
• Conversely, assume (3). Then the explicit description of the pullback functor given below (which is valid under this assumption) shows that the pullback functor preserves arbitrary limits. By the adjoint functor theorem for Grothendieck toposes, statement (4) follows.

Remark

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.

Examples

• 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.

Properties

• 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.

In the internal language

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 it’s interesting to compare the properties of $X_j$ with this internal locale.

Proposition

The following statements are equivalent:

1. $Sh(X) \models X_j \hookrightarrow 1 \,\text{is an essential sublocale}$.
2. $Sh(X) \models X_j \hookrightarrow 1 \,\text{is an open sublocale}$.
3. $X_j \hookrightarrow X$ is an open sublocale.

Note that none of these statements are equivalent to $X_j \hookrightarrow X$ being an essential sublocale.

References

• G. M. Kelly, F. W. Lawvere, On the complete lattice of essential localizations, Bull. Soc. Math. de Belgique XLI (1989) pp. 289–319 [pdf]

• Guilherme Frederico Lima, From essential inclusions to local geometric morphisms, talk at Topos à l’IHÉS in September 2015.

Related concepts

• essential subtopos

• essential geometric morphism