Topos Theory

topos theory



Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory





For 𝒯\mathcal{T} a topos and X𝒯X \in \mathcal{T} any object the over category 𝒯/X\mathcal{T}/X – the slice topos or over-topos – is itself a topos: the “big little topos” incarnation of XX. This fact is has been called the fundamental theorem of topos theory (McLarty 1992, Thm. 17.4).

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

Definition / Existence


For 𝒯\mathcal{T} a topos and X𝒯X \in \mathcal{T} any object, the slice category 𝒯/X\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


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


The power object of a map f:AXf: 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:(bB)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.


The fact that the slice 𝒯/X\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:𝒯𝒯G: \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×:𝒯𝒯X \times -: \mathcal{T} \to \mathcal{T} (with comultiplication induced by the diagonal XX×XX \to X \times X, and counit induced by the projection !:X1!: X \to 1). This result also subsumes the weaker result where GG 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.


Étale geometric morphism


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

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

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

X *:K(p 2:K×XX), 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 AKA \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.


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

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


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

More generally:


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

(f !f *f *):/Xf *f *f !/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.


By universal colimits in \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:AX)(a : A \to X) in /X\mathcal{E}/X and (b:BY)(b : B \to Y) in /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 \mathcal{E}, hence an isomorphism

/Y(f !(AX),(BY))/X((AX),f *(BY)). \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 CC a site, cCc \in C an object, and L(y(c))Sh(C)L(y(c)) \in Sh(C) the sheafification of its image under the Yoneda embedding, there is an equivalence of categories

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

between the category of sheaves on the slice category C/cC/c with its evident induced structure of a site, and the slice topos of the category of sheaves on CC, sliced over L(y(c))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 CC be a small category, cc an object of CC and let C/cC/c be the slice category of CC over cc.



(slice of presheaves is presheaves on slice)

There is an equivalence of categories

e:PSh(C/c)PSh(C)/Y(c). 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)


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

F:dfC(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:FY(c)\eta : F' \to Y(c) with component map

η d : fC(d,c)F(f) C(d,c) θF(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:FY(c)\eta \,\colon\, F' \to Y(c) to FPSh(C/c)F \in PSh(C/c) given by

F:(f:dc)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) } \,.

Suppose the presheaf FPSh(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) opC opSet. F : (C/c)^{op} \to C^{op} \to Set \,.

Then F(d)= fC(d,c)F(f)= fC(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 opSetY(c):C^{op}\to Set. This has objects (d 1,p 1)(d_1,p_1) with p 1Y(c)(d 1)p_1\in Y(c)(d_1), hence p 1p_1 is just an arrow d 1cd_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 1d 2u:d_1\to d_2 such that p 2u=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:


Let P:C opSetP: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 \,.

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

i(F)(AC)={(p,a)|pP(A),aF(A,p)}=Σ pP(A)F(A,p)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 PP. The inverse takes f:QPf : Q \to P to

i 1(f)(A,pP(A))=f A 1(p).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


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

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


The slice adjunction (f */Xf */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 /X\mathcal{E}/X are computed as limits in \mathcal{E} of diagrams with a single bottom element XX adjoined, f */Xf^*/X preserves finite limits, since f *f^* does, so that (f */Xf */X)(f^*/X \dashv f_*/X) is indeed a geometric morphism.

Topos points

We discuss topos points of over-toposes.


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

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

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

(e,x):Setx *x *Set/e *(X)e */Xe */X/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 \,.

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

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


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


That \mathcal{E} has enough points means that a morphism f:ABf : A \to B in \mathcal{E} is an isomorphism precisely if for every point e:Sete : 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 \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,xe *(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 xe *(X)x \in e^*(X) then e *(f)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 /X\mathcal{E}/X correspond to pairs (e,x)(e,x) as above, with ee a point of \mathcal{E} and xe *(X)x \in e^*(X) an element. More precisely:


Let \mathcal{E} be a topos and XX an object in \mathcal{E}. Then the category of points of the over-topos /X\mathcal{E}/X is equivalent to the category with: as objects the pairs (e,x)(e,x) with ee a point of \mathcal{E} and xe *(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'.

This is SGA4 (1972, Exposé IV, Proposition 5.12, p. 382), in the special case where E=SetE' = Set. In the statement of the proposition, we used the (now standard) convention that a morphism of points (or geometric transformation) eee \to e' is a natural transformation e *(e) *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)(e,x) is the one constructed in Observation .


Last revised on October 12, 2021 at 04:47:50. See the history of this page for a list of all contributions to it.