topos theory

# Contents

## Idea

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

$\begin{array}{ccccc}ℰ& & \to & & ℱ\\ & ↘& {⇙}_{\simeq }& ↙\\ & & \mathrm{Set}\end{array}$\array{ \mathcal{E} &&\to&& \mathcal{F} \\ & \searrow &\swArrow_{\simeq}& \swarrow \\ && Set }

in the 2-category Topos.

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

$\begin{array}{ccccc}ℰ& & \to & & ℱ\\ & ↘& {⇙}_{\simeq }& ↙\\ & & 𝒮\end{array}$\array{ \mathcal{E} &&\to&& \mathcal{F} \\ & \searrow &\swArrow_{\simeq}& \swarrow \\ && \mathcal{S} }

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

One can develop essentially all of topos theory in $\mathrm{Topos}/𝒮$ instead of in $\mathrm{Topos}$ 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 $𝒮$ itself we associate the $𝒮$-indexed category (the canonical self-indexing) $𝕊$ given by

${𝕊}^{I}=𝒮/I\phantom{\rule{thinmathspace}{0ex}}.$\mathbb{S}^I = \mathcal{S}/I \,.

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

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

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

${ℰ}^{I}:=ℰ/{p}^{*}\left(I\right)\phantom{\rule{thinmathspace}{0ex}}.$\mathcal{E}^I := \mathcal{E}/p^*(I) \,.
###### Proposition

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

$𝕡:𝔼\to 𝕊\phantom{\rule{thinmathspace}{0ex}}.$\mathbb{p} : \mathbb{E} \to \mathbb{S} \,.

By the discussion at indexed category.

## References

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)

Revised on May 8, 2013 10:13:39 by David Roberts (192.43.227.18)