nLab
localic geometric morphism

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Topos Theory

topos theory

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Toposes

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Cohomology and homotopy

In higher category theory

Theorems

Localic geometric morphisms

Definition

A geometric morphism f:E→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

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→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⊂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

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

Concretely, the internal locale in ℰ defined by a localic geoemtric morphism (f *⊣f *):ℱ→ℰ is the formal dual to the direct image f *(Ω ℱ) 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 *(Ω 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

The discussion there is based on

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