nLab
over-topos

Idea

For $𝒯$ a topos and $X\in 𝒯$ any object the over category $𝒯/X$ – 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 $𝒯$ a topos and $X\in 𝒯$ any object, the slice category ${𝒯}_{/X}$ 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 $\Omega \in 𝒯$ is the subobject classifier in $𝒯$, then the projection $\Omega \times X\to X$ regarded as an object in the slice over $X$ is the subobject classifier of ${𝒯}_{/X}$.

Properties

Étale geometric morphism

Proposition

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

$$(X_{!}⊣X^{*}⊣X_{*}):𝒯/X\stackrel{\stackrel{X_{!}}{\to }}{\stackrel{\stackrel{X^{*}}{\leftarrow }}{\underset{X_{*}}{\to }}}𝒯.$$(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 $X$:

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

since commuting diagrams

$$\begin{array}{ccccc}A& & \to & & K\times X\\ & \searrow & & {\swarrow }_{p_2}\\ & & X\end{array}$$\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 $𝒯$ 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 $𝒯$ the functor $X_{!}$ is the existential quantifier $\exists$ and $X_{*}$ is the universal quantifier $\forall$.

Definition

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

More generally:

Proposition

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

$$(f_{!}⊣f^{*}⊣f_{*}):ℰ/X\stackrel{\stackrel{f_{!}}{\to }}{\stackrel{\stackrel{f^{*}}{\leftarrow }}{\underset{f_{*}}{\to }}}ℰ/Y,$$(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 $ℰ$ 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 $ℰ/X$ and $(b:B\to Y)$ in $ℰ/Y$ we have unique factorizations

$$\begin{array}{ccccc}A& \to & X{\times }_{X}B& \to & B\\ & {}_a\searrow & {\downarrow }^{f^{*}(b)}& & {\downarrow }^b\\ & & X& \stackrel{f}{\to }& Y\end{array}$$\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 $ℰ$, hence an isomorphism

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

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

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

Proposition

There is an equivalence of categories

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

Proof

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

$${\eta }_d:{\bigsqcup }_{f\in C(d,c)}F(f)\to C(d,c):((f\in C(d,c),\theta \in F(f))\mapsto f.$$\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

$$\overline{e}:\mathrm{PSh}(C)/Y(c)\to \mathrm{PSh}(C/c)$$\bar e : PSh(C)/Y(c) \to PSh(C/c)

which sends $\eta :F\prime \to Y(c)$ to $F\in \mathrm{PSh}(C/c)$ given by

$$F:(f:d\to c)\mapsto F\prime (d){\mid }_c,$$F : (f : d \to c) \mapsto F'(d)|_c \,,

where $F\prime (d){\mid }_c$ is the pullback

$$\begin{array}{ccc}F\prime (d){\mid }_c& \to & F\prime (d)\\ \downarrow & & {\downarrow }^{{\eta }_d}\\ \mathrm{pt}& \stackrel{f}{\to }& C(d,c)\end{array}.$$\array{ F'(d)|_c &\to& F'(d) \\ \downarrow && \downarrow^{\eta_d} \\ pt &\stackrel{f}{\to}& C(d,c) } \,.

Example

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

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

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

Geometric morphisms by slicing

Proposition

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

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

Proof

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

Topos points

We discuss topos points of over-toposes.

Observation

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

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

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

$$(e,x):\mathrm{Set}\stackrel{\stackrel{x^{*}}{\leftarrow }}{\underset{x_{*}}{\to }}\mathrm{Set}/e^{*}(X)\stackrel{\stackrel{e^{*}/X}{\leftarrow }}{\underset{e_{*}/X}{\to }}ℰ/X.$$(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 $(e^{*}/X⊣e_{*}/X)$ is the slice geometric morphism of $e$ over $X$ discussed above and $(x^{*}⊣x_{*})$ 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 $ℰ$ has enough points then so does the slice topos $ℰ/X$ for every $X\in ℰ$.

Proof

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

A morphism in the slice topos, given by a diagram

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

in $ℰ$ 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

$$\begin{array}{ccccc}{e}^{*}(A)& & \stackrel{e^{*}(f)}{\to }& & e^{*}(B)\\ & \searrow & & \swarrow \\ & & e^{*}(X)\end{array}$$\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 $ℰ$ has enough points.

Related concepts

• over-category

  • slice category

  • under category

  • over-topos

• over (∞,1)-category,

  • model structure on an over category

  • over-(∞,1)-topos

References

Section IV.7 of

• Saunders MacLane, Ieke Moerdijk, Sheaves in Geometry and Logic