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point of a topos in nLab

Context

Topos Theory

topos theory

Toposes

Background

category theory
- category
- functor

Toposes

Internal Logic

Topos morphisms

Cohomology and homotopy

In higher category theory

Theorems

Examples

Definition

A point xx of a topos EE is a geometric morphism

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

from the base topos Set to EE.

For A∈EA \in E an object, its inverse image x *A∈Setx^* 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.

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 explicity: EE has enough points if for any morphism f:A→Bf : A \to B, we have that if for every point pp of EE, the morphism of stalks p *f:p *A→p *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,Set][C^{op}, Set] are the flat functors C→SetC \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.5.2).

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 sober 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 morphismsf:Sh(Y)→Sh(X)f : Sh(Y) \to Sh(X) between sheaf topoi are in a bijection with continuous functions of topological spaces f:Y→Xf : Y \to X (denoted by the same letter, by convenient abuse of notation); for this to hold XX needs to be sober.

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 *:Set→Sh(X)x_* : Set \to Sh(X) and the inverse image x *:Sh(X)→Setx^* : Sh(X) \to Set of a point x:Set→Sh(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):(U⊂X)↦S(x −1(U))={S if x∈U * 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 V⊂X|x∈VA(V). \begin{aligned} x^*(A) & = colim_{{*} \to x^{-1}(V)} A(V) \\ &= colim_{V\subset X| x \in V} A(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 S∈SetS \in Set and A∈Sh(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)) \,.

Note that the observation that the points of Sh(X)Sh(X) are in bijection with the points of XX actually factors over an intermediate concept, namely that of points of a locale. Firstly, any topological space gives rise to a locale; if the space is sober, its points are in bijection with the locale-theoretic points of the induced locale. Secondly, for any locale (spatial or not), its locale-theoretic points correspond to the points of its induced sheaf topos.

In sheaf toposes

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

This appears for instance as (MacLaneMoerdijk, corollary VII.5.4).

This appears as (Johnstone, Lemma C2.2.11, C2.2.12).

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

Proposition

A Grothendieck topos has enough points (def. ) 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

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

Examples

General

For XX any topological space, the topos of sheaves on (the category of open subsets of) XX has enough points (def. ): a morphism of sheaves is a mono-/epi-/isomorphism precisely if it is so on every stalk.
Points of over-toposes are discussed at over topos – points.

Rings and algebraic theories

The category of points of the presheaf topos over Ring fp opRing_{fp}^{op}, the dual of the category of finitely presented rings, is the category of all rings (without a size or presentation restriction). In fact this holds for any algebraic theory, not only for the theory of commutative rings. See also at Gabriel-Ulmer duality, flat functors.

Of a local topos

A local topos (Δ⊣Γ⊣coDisc):E→Set(\Delta \dashv \Gamma \dashv coDisc) : E \to Set has a canonical point, (Γ⊣coDisc):Set→E(\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 ℝ ∞\mathbb{R}^\infty). Then the sheaf topos Sh(Diff)Sh(Diff) has precisely one point p np_n per natural number n∈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 (def. ).

every presheaf topos;
every coherent topos (due to the Deligne completeness theorem);
every Galois topos (see (Zoghaib)).

Toposes without points

In contrast with point-set topology where non empty spaces tautologically have points there are non trivial toposes lacking points just like in pointfree topology there are nontrivial locales without points. From a logical perspective these toposes correspond to consistent geometric theories lacking models in Set. Classical examples are provided by toposes of sheaves on complete atomless Boolean algebras (see Barr (1974) or for an example due to Deligne SGA4, p.412).

References

Textbook references are section 7.5 of

Saunders MacLane, Ieke Moerdijk, Sheaves in Geometry and Logic

as well as section C2.2 of

Peter Johnstone, Sketches of an Elephant

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

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 homotopy theory of the topos.