nLab indexed topos

Definition

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

Definition

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

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

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

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

This yields a 2-category $Topos_{\mathcal{S}}$ of $\mathcal{S}$-indexed toposes.

This appears at (Johnstone, p. 369).

Examples

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

Properties

Proposition

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

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

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