topos theory

# The points of a topos

## Definition

###### Definition

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

$x:\mathrm{Set}\stackrel{\stackrel{{x}^{*}}{←}}{\underset{{x}_{*}}{\to }}E$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 ${x}^{*}A\in \mathrm{Set}$ 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 $\mathrm{Set}sime\mathrm{Sh}\left(*\right)$ 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 $\left[{C}^{\mathrm{op}},\mathrm{sSet}\right]$ are the flat functors $C\to \mathrm{Set}$:

there is an equivalence of categories

$\mathrm{Topos}\left(\mathrm{Set},\left[{C}^{\mathrm{op}},\mathrm{Set}\right]\right)\stackrel{\stackrel{}{←}}{\underset{}{\to }}\mathrm{FlatFunc}\left(C,\mathrm{Set}\right)\phantom{\rule{thinmathspace}{0ex}}.$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 $E=\mathrm{Sh}\left(X\right)$ is the category of sheaves on a category of open subsets $\mathrm{Op}\left(X\right)$ of a topological space $X$ the notion of topos pointscomes from the ordinary notion of points of $X$.

For notice that

• $\mathrm{Set}=\mathrm{Sh}\left(*\right)$ is simply the topos of sheaves on a one-point space.

• geometric morphisms $f:\mathrm{Sh}\left(Y\right)\to \mathrm{Sh}\left(X\right)$ 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).

It follows that for $E=\mathrm{Sh}\left(X\right)$ 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 ${x}^{*}:\mathrm{Set}\to \mathrm{Sh}\left(X\right)$ and the inverse image ${x}_{*}:\mathrm{Sh}\left(X\right)\to \mathrm{Set}$ of a point $x:\mathrm{Set}\to \mathrm{Sh}\left(X\right)$ 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

${x}_{*}\left(S\right):\left(U\subset X\right)↦S\left({x}^{-1}\left(U\right)\right)=\left\{\begin{array}{cc}S& \mathrm{if}x\in U\\ *& \mathrm{otherwise}\end{array}$x_*(S) : (U \subset X) \mapsto S(x^{-1}(U)) = \left\{ \array{ S & if x \in U \\ {*} & otherwise } \right.

This is the skyscraper sheaf ${\mathrm{skysc}}_{x}\left(S\right)$ 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 $S\left(\varnothing \right)=\mathrm{pt}$).

• 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

$\begin{array}{rl}{x}^{*}\left(A\right)& ={\mathrm{colim}}_{*\to {x}^{-1}\left(V\right)}A\left(V\right)\\ & ={\mathrm{colim}}_{V\subset X\mid x\in V}F\left(V\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 he point $x$,

${x}^{*}\left(-\right)={\mathrm{stalk}}_{x}\left(-\right)\phantom{\rule{thinmathspace}{0ex}}.$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 $S\in \mathrm{Set}$ and $A\in \mathrm{Sh}\left(X\right)$ we have

${\mathrm{Hom}}_{\mathrm{Set}}\left({\mathrm{stalk}}_{x}\left(A\right),S\right)\simeq {\mathrm{Hom}}_{\mathrm{Sh}\left(X\right)}\left(A,{\mathrm{skysc}}_{x}\left(S\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$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 $C$ a site, there is an equivalence of categories

$\mathrm{Topos}\left(\mathrm{Set},\mathrm{Sh}\left(C\right)\right)\simeq \mathrm{ConFlatFunc}\left(C,\mathrm{Set}\right)\phantom{\rule{thinmathspace}{0ex}}.$Topos(Set, Sh(C)) \simeq ConFlatFunc(C,Set) \,.

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

###### Proposition

If $E$ is a Grothendieck topos, there is a small set of points of $E$ which are jointly conservative, and therefore a geometric morphism $\mathrm{Set}/X\to E$, 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 $\left(\Delta ⊣\Gamma dahv\mathrm{coDisc}\right):E\to \mathrm{Set}$ has a canonical point, $\left(\Gamma ⊣\mathrm{coDisc}\right):\mathrm{Set}\to E$. 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 ${ℝ}^{\infty }$). Then the sheaf topos $\mathrm{Sh}\left(\mathrm{Diff}\right)$ has precisely one point ${p}_{n}$ per natural number $n\in ℕ$ , corresponding to the $n$-ball: the stalk of a sheaf on $\mathrm{Diff}$ 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 (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

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

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

Revised on April 26, 2012 07:50:13 by Urs Schreiber (82.113.121.164)