nLab over-topos

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 has been called the fundamental theorem of topos theory (McLarty 1992, Thm. 17.4).

More generally, given a functor $u : E \to F$ between toposes that preserves pullbacks, the comma category $(id_F/u)$ is again a topos, called the Artin gluing.

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 $\mathcal{T}{/X}$.

Proposition

The power object of a map $f: A \to X$ is given by the equalizer of the maps $p, t$:

$$\array{ P_X f && \dashrightarrow && P A \times X && \underoverset{t}{p}{\rightrightarrows} && P A \\ &&&&\downarrow^{\mathrlap{1 \times \{\cdot\}_X}} &&&& \uparrow^{\mathrlap{\wedge}} \\ &&&& P A \times P X && \underset{1 \times P f}{\rightarrow} && P A \times P A },$$

where $p$ is the projection map and $t$ is the composition $\wedge \circ (1 \times P f) \circ (1 \times \{\cdot\}_X)$. In the internal language, this says

$$P_X f = \{(B, x) \in P A \times X: (\forall b \in B) f(b) = x\}.$$

The map to $X$ is given by projection onto the second factor.

Remark

The fact that the slice $\mathcal{T}/X$ is a topos, and particularly the construction of power objects above, can be deduced from a more general result: that the category of coalgebras of a pullback-preserving comonad $G: \mathcal{T} \to \mathcal{T}$ is a topos. See at topos of coalgebras over a comonad. In the case of a slice topos, the comonad would be $X \times -: \mathcal{T} \to \mathcal{T}$ (with comultiplication induced by the diagonal $X \to X \times X$, and counit induced by the projection $!: X \to 1$). This result also subsumes the weaker result where $G$ is assumed to preserve finite limits. See the Elephant, Section A, Remark 4.2.3. A proof of a still more general result may be found here.

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 the 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

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

In terms of sheaves on a slice site

Generally, for $C$ a site, $c \in C$ an object, and $L(y(c)) \in Sh(C)$ the sheafification of its image under the Yoneda embedding, there is an equivalence of categories

$$Sh\big( C /c \big) \;\simeq\; Sh(C)/(L(y(c)))$$

between the category of sheaves on the slice category $C/c$ with its evident induced structure of a site, and the slice topos of the category of sheaves on $C$, sliced over $L(y(c))$.

This is for instance in Verdier’s exposé III.5 prop.5.4 (SGA4, p.295).

We now discuss this in more detail for the special case of over-presheaf toposes.

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

Write

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

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

Proposition

(slice of presheaves is presheaves on slice)

There is an equivalence of categories

$$e \;\colon\; PSh(C/c) \xrightarrow{\;\;\simeq\;\;} PSh(C)/Y(c) \,.$$

(SGA4 I, Ex. 1 Prop. 5.11, p. 27; Kashiwara-Schapira 2006, Lemma 1.4.12, p. 26)

Proof

The functor $e$ takes $F \in PSh(C/c)$ to the presheaf

$$F' \,\colon\, d \;\mapsto\; \underset{ f \in C(d,c) }{\sqcup} F(f)$$

which is equipped with the natural transformation $\eta : F' \to Y(c)$ with component map

$$\array{ \eta_d & \colon & \underset {f \in C(d,c)} {\sqcup } F(f) & \longrightarrow & C(d,c) \\ && \theta \in F(f) &\mapsto& f \,. }$$

One readily checks (for more details see here) that a weak inverse of $e$ is given by the functor

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

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

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

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 morphisms 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.

Remark

Consider $\int_C Y(c)$ , the category of elements of $Y(c):C^{op}\to Set$. This has objects $(d_1,p_1)$ with $p_1\in Y(c)(d_1)$, hence $p_1$ is just an arrow $d_1\to c$ in $C$. A map from $(d_1, p_1)$ to $(d_2, p_2)$ is just a map $u:d_1\to d_2$ such that $p_2\circ u =p_1$ but this is just a morphism from $p_1$ to $p_2$ in $C/c$.

Hence, the above Prop. can be rephrased as $PSh(\int_C Y(c))\simeq PSh(C)/Y(c)$ which is an instance of the following formula:

Proposition

Let $P:C^{op}\to Set$ be a presheaf. Then there is an equivalence of categories

$$PSh(\int_C P) \simeq PSh(C)/P \,.$$

On objects this takes $F : (\int_C P)^{op} \to Set$ to

$$i(F)(A \in C) = \{ (p,a) | p \in P(A), a \in F(A,p) \} = \Sigma_{p \in P(A)} F(A,p)$$

with obvious projection to $P$. The inverse takes $f : Q \to P$ to

$$i^{-1}(f)(A, p \in P(A)) = f_A^{-1}(p)\;.$$

Kashiwara-Schapira (2006, Lemma 1.4.12, p. 26). For a more general statement involving slices of Grothendieck toposes see Mac Lane-Moerdijk (1994, p.157).

In particular, this equivalence shows that slices of presheaf toposes are presheaf toposes.

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

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

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.

Lemma

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

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

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

It turns out that all points of $\mathcal{E}/X$ correspond to pairs $(e,x)$ as above, with $e$ a point of $\mathcal{E}$ and $x \in e^*(X)$ an element. More precisely:

Proposition

Let $\mathcal{E}$ be a topos and $X$ an object in $\mathcal{E}$. Then the category of points of the over-topos $\mathcal{E}/X$ is equivalent to the category with: as objects the pairs $(e,x)$ with $e$ a point of $\mathcal{E}$ and $x \in e^*(X)$ an element; and as morphisms $(e,x) \to (e',x')$ the natural transformations $\eta : e^* \to (e')^*$ such that $\eta_X(x) = x'$.

This is SGA4 (1972, Exposé IV, Proposition 5.12, p. 382), in the special case where $E' = Set$. In the statement of the proposition, we used the (now standard) convention that a morphism of points (or geometric transformation) $e \to e'$ is a natural transformation $e^* \to (e')^*$. Note however that SGA4 uses the opposite convention, see SGA4 (1972, Exposé IV, 3.2, p. 328).

The point corresponding the pair $(e,x)$ is the one constructed in Observation .

Related concepts

• over category, under category

• Artin gluing

• fundamental theorem of topos theory

• over (∞,1)-category,

  • model structure on an over category

  • over-(∞,1)-topos

References

• Michael Artin, Alexander Grothendieck, Jean-Louis Verdier, Théorie des Topos et Cohomologie Etale des Schémas (SGA4) Tome 1: Théorie des Topos Springer LNM 269 (1972) (doi:10.1007/BFb0081551, pdf)

  (In particular, exposé III.5 and exposé IV.5 on the “induced topos” - topos induit = slice topos)

• Colin McLarty, Elementary Categories, Elementary Toposes, Oxford University Press 1992 (ISBN:9780198514732)

• Masaki Kashiwara, Pierre Schapira, Categories and Sheaves, Springer 2006 ( doi:10.1007/3-540-27950-4, pdf)

• Saunders Mac Lane, Ieke Moerdijk, Sheaves in Geometry and Logic , Springer Heidelberg 1994. (Especially section IV.7)