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indexed topos

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Topos Theory

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Definition

Let 𝒮\mathcal{S} be a topos, regarded as a base topos.

Definition

An 𝒮\mathcal{S}-indexed topos 𝔼\mathbb{E} is an 𝒮\mathcal{S}-indexed category such that

  • for each object I𝒮I \in \mathcal{S} the fiber 𝔼 I\mathbb{E}^I is a topos;

  • for each morphism x:IJx : I \to J in 𝒮\mathcal{S} the corresponding transition functor x *:𝔼 J𝔼 Ix^* : \mathbb{E}^J \to \mathbb{E}^I is a logical morphism.

An 𝒮\mathcal{S}-indexed geometric morphism is an 𝒮\mathcal{S}-indexed adjunction (f *f *)(f^* \dashv f_*) between 𝒮\mathcal{S}-indexed toposes, such that f *f^* is left exact.

This yields a 2-category Topos 𝒮Topos_{\mathcal{S}} of 𝒮\mathcal{S}-indexed toposes.

This appears at (Johnstone, p. 369).

Examples

  • For p:𝒮p : \mathcal{E} \to \mathcal{S} a geometric morphism, the induced morphism 𝔼𝕊\mathbb{E} \to \mathbb{S} (discussed at base topos) is an 𝒮\mathcal{S}-indexed topos.

Properties

Proposition

Write Topos/𝒮/\mathcal{S} for the slice 2-category of toposes over 𝒮\mathcal{S}. This is a full sub-2-category 𝒮\mathcal{S}-indexed toposes

Topos/𝒮Topos 𝒮. Topos/{\mathcal{S}} \hookrightarrow Topos_{\mathcal{S}} \,.

This appears as (Johnstone, prop. 3.1.3).

References

Section B3.1 of

Revised on August 30, 2013 11:29:33 by David Corfield (87.114.166.212)