nLab base topos

Idea

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).

Constructions

Definition

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 $p$

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

Proposition

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.

Related concepts

• base (∞,1)-topos
• internal site
• internal sheaves
• bounded geometric morphism
• unbounded topos

References

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

• Peter Johnstone, Sketches of an Elephant

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)