Could not include topos theory - contents
A point $x$ of a topos $E$ is a geometric morphism
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 $E$.
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).
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.
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
This appears for instance as (MacLaneMoerdijk, theorem VII 2).
For the special case that $E = Sh(X)$ is the category of sheaves on a category of open subsets $Op(X)$ of a 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).
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
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
This is the stalk of $A$ at the point $x$,
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
The following characterization of points in sheaf toposes a special case of the general statements at morphism of sites.
For $C$ a site, there is an equivalence of categories
This appears for instance as (MacLaneMoerdijk, corollary VII, 4).
If $E$ is a Grothendieck topos, 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.)
A Grothendieck topos has enough points precisely when it underlies a bounded ionad.
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.
If a sheaf topos $E$ has enough points then
there exists a topological space $X$ whose cohomology and homotopy theory is the intrinsic cohomology and intrinsic homtopy 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).
For $X$ any topological space, the topos of sheaves on (the category of open subsets of) $X$ has enough points: 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.
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.)
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.
The following classes of topos have enough points.
every presheaf topos;
every coherent topos (due to the Deligne completeness theorem);
every Galois topos (see (Zoghaib)).
Textbook references are section 7.5 of
as well as section C2.2 of
In
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.
See also
Points of the sheaf topos over the category of manifolds are discussed in