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point of a topos

Context

Topos Theory

Could not include topos theory - contents

The points of a topos

Definition

Definition

A point xx of a topos EE is a geometric morphism

x:Setx *x *E x : Set \stackrel{\overset{x^*}{\leftarrow}}{\underset{x_*}{\to}} E

from the base topos Set to \mathcal{E}.

For AA \in \mathcal{E} an object, its inverse image x *ASetx^* A \in Set under such a point is called the stalk of AA at xx.

If xx is given by an essential geometric morphism we say that it is an essential point of EE.

Remark

Since Set is the terminal object in the category GrothendieckTopos of Grothendieck toposes, for \mathcal{E} a sheaf topos this is a global element of the topos.

Since Set=Sh(*)Set = Sh(*) is the category of sheaves on the one-point locale, the notion of point of a topos is indeed the direct analog of a point of a locale (under localic reflection).

Definition

A topos is said to have enough points if isomorphy can be tested stalkwise, i.e. if the inverse image functors from all of its points are jointly conservative.

More precisely, EE has enough points if for any morphism f:ABf : A \to B, we have that if for every point pp of EE, the morphism of stalks p *f:p *Ap *Bp^* f : p^* A \to p^* B is an isomorphism, then ff itself is an isomorphism.

Properties

In presheaf toposes

Proposition

For CC a small category, the points of the presheaf topos [C op,sSet][C^{op}, sSet] are the flat functors CSetC \to Set:

there is an equivalence of categories

Topos(Set,[C op,Set])FlatFunc(C,Set). Topos(Set, [C^{op}, Set]) \stackrel{\overset{}{\leftarrow}}{\underset{}{\to}} FlatFunc(C,Set) \,.

This appears for instance as (MacLaneMoerdijk, theorem VII 2).

In localic sheaf toposes

For the special case that E=Sh(X)E = Sh(X) is the category of sheaves on a category of open subsets Op(X)Op(X) of a topological space XX the notion of topos points comes from the ordinary notion of points of XX.

For notice that

  • Set=Sh(*)Set = Sh(*) is simply the topos of sheaves on a one-point space.

  • geometric morphisms f:Sh(Y)Sh(X)f : Sh(Y) \to Sh(X) between sheaf topoi are in a bijection with continuous functions of topological spaces f:YXf : Y \to X (denoted by the same letter, by convenient abuse of notation).

It follows that for E=Sh(X)E = Sh(X) points of EE in the sense of points of topoi are in bijection with the ordinary points of XX.

The action of the direct image x *:SetSh(X)x_* : Set \to Sh(X) and the inverse image x *:Sh(X)Setx^* : Sh(X) \to Set of a point x:SetSh(X)x : Set \to Sh(X) of a sheaf topos have special interpretation and relevance:

  • The direct image of a set SS under the point x:*Xx : {*} \to X is, by definition of direct image the sheaf

    x *(S):(UX)S(x 1(U))={S if xU * otherwise x_*(S) : (U \subset X) \mapsto S(x^{-1}(U)) = \left\{ \array{ S & \text{ if } x \in U \\ {*} & \text{otherwise} } \right.

    This is the skyscraper sheaf skysc x(S)skysc_x(S) with value SS supported at xx. (In the first line on the right in the above we identify the set SS with the unique sheaf on the point it defines. Notice that S()=ptS(\emptyset) = pt).

  • The inverse image of a sheaf AA under the point x:*Xx : {*} \to X is by definition of inverse image (see the Kan extension formula discussed there), the set

    x *(A) =colim *x 1(V)A(V) =colim VX|xVF(V). \begin{aligned} x^*(A) & = colim_{{*} \to x^{-1}(V)} A(V) \\ &= colim_{V\subset X| x \in V} F(V) \end{aligned} \,.

    This is the stalk of AA at the point xx,

    x *()=stalk x(). x^*(-) = stalk_x(-) \,.

By definition of geometric morphisms, taking the stalk at xx is left adjoint to forming the skyscraper sheaf at xx:

for all SSetS \in Set and ASh(X)A \in Sh(X) we have

Hom Set(stalk x(A),S)Hom Sh(X)(A,skysc x(S)). Hom_{Set}(stalk_x(A), S) \simeq Hom_{Sh(X)}(A, skysc_x(S)) \,.

In sheaf toposes

The following characterization of points in sheaf toposes a special case of the general statements at morphism of sites.

Proposition

For CC a site, there is an equivalence of categories

Topos(Set,Sh(C))ConFlatFunc(C,Set). Topos(Set, Sh(C)) \simeq ConFlatFunc(C,Set) \,.

This appears for instance as (MacLaneMoerdijk, corollary VII, 4).

Proposition

If EE is a Grothendieck topos, there is a small set of points of EE which are jointly conservative, and therefore a geometric morphism Set/XESet/X \to E, for some set XX, which is surjective.

This appears as (Johnstone, lemma 2.2.11, 2.2.12).

(In general, of course, a topos can have a proper class of non-isomorphic points.)

Proposition

A Grothendieck topos has enough points precisely when it underlies a bounded ionad.

In classifying toposes

From the above it follows that if EE is the classifying topos of a geometric theory TT, then a point of EE is the same as a model of TT in Set.

Of toposes with enough points

Proposition

If a sheaf topos EE has enough points then

This is due to (Butz) and (Moerdijk).

Examples

General

Of a local topos

A local topos (ΔΓcoDisc):ESet(\Delta \dashv \Gamma \dashv coDisc) : E \to Set has a canonical point, (ΓcoDisc):SetE(\Gamma \dashv coDisc) : Set \to E. Morover, this point is an initial object in the category of all points of EE (see Equivalent characterizations at local topos.)

Over \infty-cohesive sites

  • Let Diff be a small category version of the category of smooth manifolds (for instance take it to be the category of manifolds embedded in \mathbb{R}^\infty). Then the sheaf topos Sh(Diff)Sh(Diff) has precisely one point p np_n per natural number nn \in \mathbb{N} , corresponding to the nn-ball: the stalk of a sheaf on DiffDiff at that point is the colimit over the result of evaluating the sheaf on all nn-dimensional smooth balls.

    This is discussed for instance in (Dugger, p. 36) in the context of the model structure on simplicial presheaves.

Toposes with enough points

The following classes of topos have enough points.

References

Textbook references are section 7.5 of

as well as section C2.2 of

In

  • Carsten Butz, Logical and cohomological aspects of the space of points of a topos (ps)

is a discussion of how for every topos with enough points there is a topological space whose cohomology and homotopy theory is related to the intrinsic cohomology and intrinsic homtopy theoryof the topos.

More on this is in

  • Ieke Moerdijk, Classifying toposes for toposes with enough points , Milan Journal of Mathematics Volume 66, Number 1, 377-389

See also

  • Sam Zoghaib, A few points in topos theory (pdf)

Points of the sheaf topos over the category of manifolds are discussed in

Revised on June 2, 2014 23:03:56 by Urs Schreiber (82.113.106.149)