over-topos in nLab

Context

Topos Theory

Bundles

• Idea
• Definition / Existence
• Properties
• Étale geometric morphism
• In terms of sheaves on a slice site
• Geometric morphisms by slicing
• Topos points
• Related concepts
• References

Proposition

If Ω∈𝒯\Omega \in \mathcal{T} is the subobject classifier in 𝒯\mathcal{T}, then the projection Ω×X→X\Omega \times X \to X regarded as an object in the slice over XX is the subobject classifier of 𝒯/X\mathcal{T}{/X}.

Proposition

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

P Xf ⤏ PA×X ⇉tp PA ↓ 1×{⋅} X ↑ ∧ PA×PX →1×Pf PA×PA, \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 pp is the projection map and tt is the composition ∧∘(1×Pf)∘(1×{⋅} X)\wedge \circ (1 \times P f) \circ (1 \times \{\cdot\}_X). In the internal language, this says

P Xf={(B,x)∈PA×X:(∀b∈B)f(b)=x}. P_X f = \{(B, x) \in P A \times X: (\forall b \in B) f(b) = x\}.

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

Proposition

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

(X !⊣X *⊣X *):𝒯/X→X *←X *→X !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 *X^* is given by taking the product with XX:

X *:K↦(p 2:K×X→X), X^* : K \mapsto (p_2 : K \times X \to X) \,,

since commuting diagrams

A → K×X ↘ ↙ p 2 X \array{ A &&\to&& K \times X \\ & \searrow && \swarrow_{\mathrlap{p_2}} \\ && X }

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

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

Proposition

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

(f !⊣f *⊣f *):ℰ/X→f *←f *→f !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 !f_! is given by postcomposition with ff and f *f^* by pullback along XX.

Proof

By universal colimits in ℰ\mathcal{E} the pullback functor f *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 ff follows from the universality of the pullback: for (a:A→X)(a : A \to X) in ℰ/X\mathcal{E}/X and (b:B→Y)(b : B \to Y) in ℰ/Y\mathcal{E}/Y we have unique factorizations

A → X× XB → B a↘ ↓ f *(b) ↓ b X →f Y \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\mathcal{E}, hence an isomorphism

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

Proof

The functor ee takes F∈PSh(C/c)F \in PSh(C/c) to the presheaf

F′:d↦⊔f∈C(d,c)F(f) F' \,\colon\, d \;\mapsto\; \underset{ f \in C(d,c) }{\sqcup} F(f)

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

η d : ⊔f∈C(d,c)F(f) ⟶ C(d,c) θ∈F(f) ↦ f. \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 ee is given by the functor

e¯:PSh(C)/Y(c)→PSh(C/c) \bar e \;\colon\; PSh(C)/Y(c) \to PSh(C/c)

which sends η:F′→Y(c)\eta \,\colon\, F' \to Y(c) to F∈PSh(C/c)F \in PSh(C/c) given by

F:(f:d→c)↦F′(d)| c, F \;\colon\; (f \,\colon\, d \to c) \mapsto F'(d)|_c \,,

where F′(d)| cF'(d)|_c is the pullback

F′(d)| c → F′(d) ↓ ↓ η d pt →f C(d,c). \array{ F'(d)|_c &\to& F'(d) \\ \downarrow && \downarrow^{\eta_d} \\ pt &\stackrel{f}{\to}& C(d,c) } \,.

Example

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

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

Then F′(d)=⊔ f∈C(d,c)F(f)=⊔ f∈C(d,c)F(d)≃C(d,c)×F(d) 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)×FF ' = Y(c) \times F with respect to the closed monoidal structure on presheaves.

Consider ∫ CY(c)\int_C Y(c) , the category of elements of Y(c):C op→SetY(c):C^{op}\to Set. This has objects (d 1,p 1)(d_1,p_1) with p 1∈Y(c)(d 1)p_1\in Y(c)(d_1), hence p 1p_1 is just an arrow d 1→cd_1\to c in CC. A map from (d 1,p 1)(d_1, p_1) to (d 2,p 2)(d_2, p_2) is just a map u:d 1→d 2u:d_1\to d_2 such that p 2∘u=p 1p_2\circ u =p_1 but this is just a morphism from p 1p_1 to p 2p_2 in C/cC/c.

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

Proposition

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

PSh(∫ CP)≃PSh(C)/P. PSh(\int_C P) \simeq PSh(C)/P \,.

Proposition

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

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

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

Proof

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

Lemma

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

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

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

(e,x):Set→x *←x *Set/e *(X)→e */X←e */Xℰ/X. (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 \,.

Corollary

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

Proof

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

A morphism in the slice topos, given by a diagram

A →f B ↘ ↙ X \array{ A &&\stackrel{f}{\to}&& B \\ & \searrow && \swarrow \\ && X }

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

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

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

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

Proposition

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