indexed topos (changes) in nLab
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Context
Topos Theory
Background
Toposes
Internal Logic
Topos morphisms
Cohomology and homotopy
In higher category theory
Theorems
Contents
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
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for each object I∈𝒮I \in \mathcal{S} the fiber 𝔼 I\mathbb{E}^I is a topos;
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for each morphism x:I→Jx : I \to J in 𝒮\mathcal{S} the corresponding transition functor x *:𝔼 J→𝔼 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:ℰ→𝒮p : \mathcal{E} \to \mathcal{S} a geometric morphism, the induced morphism 𝔼→𝕊\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 of the 2-category of𝒮\mathcal{S} -indexed toposes toposes:
Topos/𝒮↪Topos 𝒮. Topos/{\mathcal{S}} \hookrightarrow Topos_{\mathcal{S}} \,.
This appears as (Johnstone, prop. 3.1.3).
References
Section B3.1 of
Last revised on June 12, 2024 at 08:13:50. See the history of this page for a list of all contributions to it.