base topos

Every sheaf topos $\mathcal{E}$ of sheaves with values in Set is canonically and essentially uniquely equipped with its global section geometric morphism $\Gamma : \mathcal{E} \to Set$. So in particular for $\mathcal{E} \to \mathcal{F}$ any other geometric morphism, we have necessarily a diagram

$\array{
\mathcal{E} &&\to&& \mathcal{F}
\\
& \searrow &\swArrow_{\simeq}& \swarrow
\\
&& Set
}$

in the 2-category Topos.

Accordingly, if $\mathcal{E}$ and $\mathcal{F}$ are both equipped with geometric morphism to some other topos $\mathcal{S}$, it makes sense to restrict attention to those geometric morphisms between them that do form commuting triangles as before

$\array{
\mathcal{E} &&\to&& \mathcal{F}
\\
& \searrow &\swArrow_{\simeq}& \swarrow
\\
&& \mathcal{S}
}$

but now over the new *base topos* $\mathcal{S}$. This is a morphism in the slice 2-category Topos$/\mathcal{S}$.

One can develop essentially all of topos theory in $Topos/\mathcal{S}$ instead of in $Topos$ itself.

To some extent it is also possible to speak of a base topos entirely internally to a given topos. See for instance (AwodeyKishida).

To $\mathcal{S}$ itself we associate the $\mathcal{S}$-indexed category (the canonical self-indexing) $\mathbb{S}$ given by

$\mathbb{S}^I = \mathcal{S}/I
\,.$

To $p : \mathcal{E} \to \mathcal{S}$ a topos over a base $\mathcal{S}$, we associate the $\mathcal{S}$-indexed category

$\mathbb{E} : \mathcal{S}^{op} \to Cat$

which sends an object $I \in \mathcal{S}$ to the over-topos of $\mathcal{E}$ over the inverse image of $I$ under the geometric morphism $p$

$\mathcal{E}^I := \mathcal{E}/p^*(I)
\,.$

The geometric morphism $p : \mathcal{E} \to \mathcal{S}$ induces an $\mathcal{S}$-indexed geometric morphism (hence a geometric morphism internal to the slice 2-category Topos$/\mathcal{S}$)

$\mathbb{p} : \mathbb{E} \to \mathbb{S}
\,.$

By the discussion at indexed category.

The general notion of base toposes is the topic of section B3 of

An internal description of base toposes in the context of modal logic appears in

- Steve Awodey, Kohei Kishida,
*Topology and modality: the topological interpretation of first-order modal logic*(pdf)