localic geometric morphism


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Localic geometric morphisms


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




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 ESE\to S as exhibiting EE as a “localic SS-topos”.

This is supported by the following fact.


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

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

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

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

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

The last bit is lemma 1.2 in (Johnstone).


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).


Localic geometric morphisms are defined in def. 4.6.1 of

The discussion there is based on

Revised on September 16, 2014 08:39:35 by Thomas Holder (