nLab
indexed topos

Definition

Let $𝒮$ be a topos, regarded as a base topos.

Definition

An $𝒮$-indexed topos $𝔼$ is an $𝒮$-indexed category such that

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

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

An $𝒮$-indexed geometric morphism is a $𝒮$-nindexed adjunction $(f^{*}⊣f_{*})$ between $𝒮$-indexed toposes, such that $f^{*}$ is left exact.

This yields a 2-category ${\mathrm{Topos}}_{𝒮}$ of $𝒮$-indexed toposes.

This appears at (Johnstone, p. 369).

Examples

• For $p:ℰ\to 𝒮$ a geometric morphism, the induced morphism $𝔼\to 𝕊$ (discussed at base topos) is an $𝒮$-indexed topos.

Properties

Proposition

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

$$\mathrm{Topos}/𝒮\hookrightarrow {\mathrm{Topos}}_{𝒮}.$$Topos/{\mathcal{S}} \hookrightarrow Topos_{\mathcal{S}} \,.

This appears as (Johnstone, prop. 3.1.3).

Related concepts

• topos, base topos

• indexed category

• hyperdoctrine

References

Section B3.1 of

• Peter Johnstone, Sketches of an Elephant