topos theory

bundles

# Contents

## Idea

For $\mathcal{T}$ a topos and $X \in \mathcal{T}$ any object the over category $\mathcal{T}/X$ – the slice topos or over-topos – is itself a topos: the “big little topos” incarnation of $X$. This fact is sometimes called the “Fundamental Theorem of Topos Theory”.

## Definition / Existence

###### Proposition

For $\mathcal{T}$ a topos and $X \in \mathcal{T}$ any object, the slice category $\mathcal{T}_{/X}$ is itself again a topos.

A proof is spelled out for instance in MacLane-Moerdijk, IV.7 theorem 1. In particular we have

###### Proposition

If $\Omega \in \mathcal{T}$ is the subobject classifier in $\mathcal{T}$, then the projection $\Omega \times X \to X$ regarded as an object in the slice over $X$ is the subobject classifier of $\mathcal{T}_{/X}$.

## Properties

### Étale geometric morphism

###### Proposition

For $\mathcal{T}$ a Grothendieck topos and $X \in \mathcal{T}$ any object, the canonical projection functor $X_! : \mathcal{T}/X \to \mathcal{T}$ is part of an essential geometric morphism

$(X_! \dashv X^* \dashv X_*) : \mathcal{T}/X \stackrel{\overset{X_!}{\to}}{\stackrel{\overset{X^*}{\leftarrow}}{\underset{X_*}{\to}}} \mathcal{T} \,.$
###### Proof

The functor $X^*$ is given by taking the product with $X$:

$X^* : K \mapsto (p_2 : K \times X \to X) \,,$

since commuting diagrams

$\array{ A &&\to&& K \times X \\ & \searrow && \swarrow_{\mathrlap{p_2}} \\ && X }$

are evidently uniquely specified by their components $A \to K$.

Moreover, since in the Grothendieck topos $\mathcal{T}$ we have universal colimits, it follows that $(-) \times X$ preserves all colimits. Therefore by the adjoint functor theorem a further right adjoint $X_*$ exists.

###### Remark

One also says that $X_!$ is the dependent sum operation and $X_*$ the dependent product operation. As discussed there, this can be seen to compute spaces of sections of bundles over $X$.

Moreover, in terms of the internal logic of $\mathcal{T}$ the functor $X_!$ is the existential quantifier $\exists$ and $X_*$ is the universal quantifier $\forall$.

###### Definition

A geometric morphism $\mathcal{E} \to \mathcal{T}$ equivalent to one of the form $(X_! \dashv X^* \dashv X_*)$ is called an etale geometric morphism.

More generally:

###### Proposition

For $\mathcal{E}$ a Grothendieck topos and $f : X \to Y$ a morphism in $\mathcal{E}$, there is an induced essential geometric morphism

$(f_! \dashv f^* \dashv f_*) : \mathcal{E}/X \stackrel{\overset{f_!}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}} \mathcal{E}/Y \,,$

where $f_!$ is given by postcomposition with $f$ and $f^*$ by pullback along $X$.

###### Proof

By universal colimits in $\mathcal{E}$ the pullback functor $f^*$ preserves both limits and colimits. By the adjoint functor theorem and using that th over-toposes are locally presentable categories, this already implies that it has a left adjoint and a right adjoint. That the left adjoint is given by postcomposition with $f$ follows from the universality of the pullback: for $(a : A \to X)$ in $\mathcal{E}/X$ and $(b : B \to Y)$ in $\mathcal{E}/Y$ we have unique factorizations

$\array{ A &\to& X \times_X B &\to& B \\ &{}_{\mathllap{a}}\searrow& \downarrow^{\mathrlap{f^*(b)}} && \downarrow^{\mathrlap{b}} \\ && X &\stackrel{f}{\to}& Y }$

in $\mathcal{E}$, hence an isomorphism

$\mathcal{E}/Y(f_*(A \to X), (B \to Y)) \simeq \mathcal{E}/X((A \to X), f^*(B \to Y)) \,.$

### Presheaf over-topos

We discuss special properties of over-presheaf toposes.

Let $C$ be a small category, $c$ an object of $C$ and let $C/c$ be the over category of $C$ over $c$.

Write

• $PSh(C/c) = [(C/c)^{op}, Set]$ for the category of presheaves on $C/c$

• and write $PSh(C)/Y(y)$ for the over category of presheaves on $C$ over the presheaf $Y(c)$, where $Y : C \to PSh(c)$ is the Yoneda embedding.

###### Proposition

There is an equivalence of categories

$e : PSh(C/c) \stackrel{\simeq}{\to} PSh(C)/Y(c) \,.$
###### Proof

The functor $e$ takes $F \in PSh(C/c)$ to the presheaf $F' : d \mapsto \sqcup_{f \in C(d,c)} F(f)$ which is equipped with the natural transformation $\eta : F' \to Y(c)$ with component map

$\eta_d : \sqcup_{f \in C(d,c)} F(f) \to C(d,c) : ((f \in C(d,c), \theta \in F(f)) \mapsto f \,.$

A weak inverse of $e$ is given by the functor

$\bar e : PSh(C)/Y(c) \to PSh(C/c)$

which sends $\eta : F' \to Y(c)$ to $F \in PSh(C/c)$ given by

$F : (f : d \to c) \mapsto F'(d)|_c \,,$

where $F'(d)|_c$ is the pullback

$\array{ F'(d)|_c &\to& F'(d) \\ \downarrow && \downarrow^{\eta_d} \\ pt &\stackrel{f}{\to}& C(d,c) } \,.$
###### Example

Suppose the presheaf $F \in PSh(C/c)$ does not actually depend on the morphsims to $c$, i.e. suppose that it factors through the forgetful functor from the over category to $C$:

$F : (C/c)^{op} \to C^{op} \to Set \,.$

Then $F'(d) = \sqcup_{f \in C(d,c)} F(f) = \sqcup_{f \in C(d,c)} F(d) \simeq C(d,c) \times F(d)$ and hence $F ' = Y(c) \times F$ with respect to the closed monoidal structure on presheaves.

### Geometric morphisms by slicing

###### Proposition

For $(f^* \dashv f_*) : \mathcal{T} \to \mathcal{E}$ a geometric morphism of toposes and $X \in \mathcal{E}$ any object, there is an induced geometric morphism between the slice-toposes

$(f^*/X \dashv f_*) : \mathcal{T}/f^*X \to \mathcal{E}/X \,,$

where the inverse image $f^*/X$ is the evident application of $f^*$ to diagrams in $\mathcal{E}$.

###### Proof

The slice adjunction $(f^*/X \dashv f_*/X)$ is discussed here: the left adjoint $f^*/X$ is the evident induced functor. Since limits in an over-category $\mathcal{E}/X$ are computed as limits in $\mathcal{E}$ of diagrams with a single bottom element $X$ adjoined, $f^*/X$ preserves finite limits, since $f^*$ does, so that $(f^*/X \dashv f_*/X)$ is indeed a geometric morphism.

### Topos points

We discuss topos points of over-toposes.

###### Observation

Let $\mathcal{E}$ be a topos, $X \in \mathcal{E}$ an object and

$(e^* \dashv e_*) : Set \to \mathcal{E}$

a point of $\mathcal{E}$. Then for every element $x \in e^*(X)$ there is a point of the slice topos $\mathcal{E}/X$ given by the composite

$(e,x) : Set \stackrel{\overset{x^*}{\leftarrow}}{\underset{x_*}{\to}} Set/e^*(X) \stackrel{\overset{e^*/X}{\leftarrow}}{\underset{e_*/X}{\to}} \mathcal{E}/X \,.$

Here $(e^*/X \dashv e_*/X)$ is the slice geometric morphism of $e$ over $X$ discussed above and $(x^* \dashv x_*)$ is the étale geometric morphism discussed above induced from the morphism $* \stackrel{x}{\to} e^*(X)$.

Hence the inverse image of $(e,x)$ sends $A \to X$ to the fiber of $e^*(A) \to e^*(X)$ over $x$.

###### Corollary

If $\mathcal{E}$ has enough points then so does the slice topos $\mathcal{E}/X$ for every $X \in \mathcal{E}$.

###### Proof

That $\mathcal{E}$ has enough points means that a morphism $f : A \to B$ in $\mathcal{E}$ is an isomorphism precisely if for every point $e : Set \to \mathcal{E}$ the function $e^*(f) : e^*(A) \to e^*(B)$ is an isomorphism.

A morphism in the slice topos, given by a diagram

$\array{ A &&\stackrel{f}{\to}&& B \\ & \searrow && \swarrow \\ && X }$

in $\mathcal{E}$ is an isomorphism precisely if $f$ is. By the above observation we have that under the inverse images of the slice topos points $(e,x \in e^*(X))$ this maps to the fibers of

$\array{ e^*(A) &&\stackrel{e^*(f)}{\to}&& e^*(B) \\ & \searrow && \swarrow \\ && e^*(X) }$

over all points $* \stackrel{x}{\to} e^*(X)$. Since in Set every object $S$ is a coproduct of the point indexed over $S$, $S \simeq \coprod_S *$ and using universal colimits in $S$, we have that if $x^* e^*(f)$ is an isomorphism for all $x \in e^*(X)$ then $e^*(f)$ was already an isomorphism.

The claim the follows with the assumption that $\mathcal{E}$ has enough points.

## References

Section IV.7 of

Revised on September 22, 2012 14:24:52 by Urs Schreiber (89.204.130.57)