topos theory

# Localic geometric morphisms

## Definition

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

## Examples

• Any geometric morphism between localic topoi is localic.

• Any geometric embedding is localic.

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

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

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

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

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

Revised on November 22, 2011 14:00:37 by Urs Schreiber (131.174.41.189)