nLab point of a topos

Definition

Definition

A point $x$ of a topos $E$ is a geometric morphism

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

from the base topos Set to $E$.

For $A \in E$ an object, its inverse image $x^* A \in Set$ under such a point is called the stalk of $A$ at $x$.

If $x$ is given by an essential geometric morphism we say that it is an essential point of $E$.

Remark

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

Since $Set = Sh(*)$ 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 explicity: $E$ has enough points if for any morphism $f : A \to B$, we have that if for every point $p$ of $E$, the morphism of stalks $p^* f : p^* A \to p^* B$ is an isomorphism, then $f$ itself is an isomorphism.

Properties

In presheaf toposes

Proposition

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

there is an equivalence of categories

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

This appears for instance as (MacLaneMoerdijk, theorem VII.5.2).

In localic sheaf toposes

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

For notice that

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

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

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

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

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

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

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

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

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

  This is the stalk of $A$ at the point $x$,

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

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

for all $S \in Set$ and $A \in Sh(X)$ we have

$$Hom_{Set}(stalk_x(A), S) \simeq Hom_{Sh(X)}(A, skysc_x(S)) \,.$$

Note that the observation that the points of $Sh(X)$ are in bijection with the points of $X$ 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.

Proposition

For $C$ a site, there is an equivalence of categories

$$Topos(Set, Sh(C)) \simeq Site(C,Set) \,.$$

(Morphisms of sites $C\to Set$ are precisely the continuous flat functors.)

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

Proposition

If $E$ is a Grothendieck topos with enough points (def. ), there is a small set of points of $E$ which are jointly conservative, and therefore a geometric morphism $Set/X \to E$, for some set $X$, which is surjective.

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 $E$ is the classifying topos of a geometric theory $T$, then a point of $E$ is the same as a model of $T$ in Set.

Of toposes with enough points

Proposition

If a sheaf topos $E$ has enough points (def. ) then

• there exists a topological space $X$ whose cohomology and homotopy theory is the intrinsic cohomology and intrinsic homotopy theory of the topos;

• such that $E$ is the category of equivariant objects in the sheaf topos $Sh(X)$ with respect to some groupoid action on $X$.

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

Examples

General

• For $X$ any topological space, the topos of sheaves on (the category of open subsets of) $X$ 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 $Ring_{fp}^{op}$, 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 $(\Delta \dashv \Gamma \dashv coDisc) : E \to Set$ has a canonical point, $(\Gamma \dashv coDisc) : Set \to E$. Morover, this point is an initial object in the category of all points of $E$ (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 $Sh(Diff)$ has precisely one point $p_n$ per natural number $n \in \mathbb{N}$ , corresponding to the $n$-ball: the stalk of a sheaf on $Diff$ at that point is the colimit over the result of evaluating the sheaf on all $n$-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).

Related concepts

• topological locale

• Barr's theorem

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

In

• 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.

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

• Dan Dugger, Sheaves and homotopy theory (web, pdf)

Toposes without points are discused in the homonymous paper

• Michael Barr, Toposes without points, J. Pure Appl. Algebra 5 (1974) 265-280 [pdf, pdf]