# nLab localic geometric morphism

Localic geometric morphisms

topos theory

# Localic geometric morphisms

## 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.

This is the main statement in (Johnstone).

## References

Localic geometric morphisms are defined in def. 4.6.1 of

The discussion there is based on

Last revised on October 3, 2018 at 07:38:39. See the history of this page for a list of all contributions to it.