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 sometimes called the “Fundamental Theorem of Topos Theory”.

More generally, given a fibred product-preserving functor u:EFu : E \to F between toposes, 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)) \,.

Presheaf over-topos

We discuss special properties of over-presheaf toposes.

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



There is an equivalence of categories

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

The functor ee takes FPSh(C/c)F \in PSh(C/c) to the presheaf F:d fC(d,c)F(f)F' : d \mapsto \sqcup_{f \in C(d,c)} 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):((fC(d,c),θF(f))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 ee is given by the functor

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

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

F:(f:dc)F(d)| c, F : (f : 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 proposition 6 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)

For a proof see Kashiwara-Schapira (2006, 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) : 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.


Revised on June 19, 2017 18:23:27 by Max S. New? (