nLab localic geometric morphism

Definition

A geometric morphism $f\colon E\to F$ between topoi is localic if every object of $E$ is a subquotient of an object in the inverse image of $f$: of the form $f^*(x)$.

Examples

• Any geometric morphism between localic topoi is localic.

• Any geometric embedding is localic.

• Any étale geometric morphism is localic. From the point of view of a base topos $F$, an étale geometric morphism $F/A \to F$ looks like the unique geometric morphism $\Sh(A) \to Set$ attached to the topos of sheaves over the discrete locale $A$.

• If $g:C\to D$ is a faithful functor between small categories, then the induced geometric morphism $Set^C \to Set^D$ is localic.

Properties

Proposition

A Grothendieck topos is a localic topos if and only if its unique global section geometric morphism to Set is a localic geometric morphism.

Thus, in general we regard a localic geometric morphism $E\to S$ as exhibiting $E$ as a “localic $S$-topos”.

This is supported by the following fact.

Proposition

For any base $S$, the 2-category of localic $S$-toposes (i.e. the full sub-2-category

$$(Topos/S)_{loc} \subset Topos/S$$

of the over-category Topos over $S$ spanned by the localic morphisms into $S$) is equivalent to the 2-category of internal locales in $S$

$$Loc(S) \simeq (Topos/S)_{loc}$$

Concretely, the internal locale in $\mathcal{E}$ defined by a localic geometric morphism $(f^* \dashv f_*) : \mathcal{F} \to \mathcal{E}$ is the formal dual to the direct image $f_*(\Omega_{\mathcal{F}})$ of the subobject classifier of $\mathcal{F}$, regarded as an internal poset (as described there) and $\mathcal{F}$ is equivalent to the internal category of sheaves over $f_*(\Omega_{\mathcal{F}})$.

The last bit is lemma 1.2 in (Johnstone).

Proposition

Localic geometric morphisms are the right class of a 2-categorical orthogonal factorization system on the 2-category Topos of topoi. The corresponding left class is the class of hyperconnected geometric morphisms.

((hyperconnected,localic) factorization system)

This is the main statement in (Johnstone).

References

Localic geometric morphisms are defined in def. 4.6.1 of

• Peter Johnstone, Sketches of an Elephant

The discussion there is based on

• Peter Johnstone, Factorization theorems for geometric morphisms Cahiers, 22, no1 (1981) (numdam)