topos theory

# The points 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 $\mathcal{E}$.

For $A \in \mathcal{E}$ 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 $\mathcal{E}$ 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 precisely, $E$ 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 $[C^{op}, sSet]$ 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 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} F(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

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

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

###### Proposition

If $E$ is a Grothendieck topos with enough points, 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 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 $E$ 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 then

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

## Examples

### 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.

## 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 (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 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