nLab
indexed topos

Contents

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

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

  • 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

Last revised on August 30, 2013 at 11:29:33. See the history of this page for a list of all contributions to it.