Cohomology and homotopy
In higher category theory
The points of a topos
A point of a topos is a geometric morphism
from the base topos Set to .
For an object, its inverse image under such a point is called the stalk of at .
If is given by an essential geometric morphism we say that it is an essential point of .
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, has enough points if for any morphism , we have that if for every point of , the morphism of stalks is an isomorphism, then itself is an isomorphism.
In presheaf toposes
For a small category, the points of the presheaf topos are the flat functors :
there is an equivalence of categories
This appears for instance as (MacLaneMoerdijk, theorem VII 2).
In localic sheaf toposes
For the special case that is the category of sheaves on a category of open subsets of a sober topological space the notion of topos points comes from the ordinary notion of points of .
For notice that
is simply the topos of sheaves on a one-point space.
geometric morphisms between sheaf topoi are in a bijection with continuous functions of topological spaces (denoted by the same letter, by convenient abuse of notation); for this to hold needs to be sober.
It follows that for points of in the sense of points of topoi are in bijection with the ordinary points of .
The action of the direct image and the inverse image of a point of a sheaf topos have special interpretation and relevance:
The direct image of a set under the point is, by definition of direct image the sheaf
This is the skyscraper sheaf with value supported at . (In the first line on the right in the above we identify the set with the unique sheaf on the point it defines. Notice that ).
The inverse image of a sheaf under the point is by definition of inverse image (see the Kan extension formula discussed there), the set
This is the stalk of at the point ,
By definition of geometric morphisms, taking the stalk at is left adjoint to forming the skyscraper sheaf at :
for all and we have
Note that the observation that the points of are in bijection with the points of 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.
For a site, there is an equivalence of categories
This appears for instance as (MacLaneMoerdijk, corollary VII, 4).
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.
In classifying toposes
From the above it follows that if is the classifying topos of a geometric theory , then a point of is the same as a model of in Set.
Of toposes with enough points
This is due to (Butz) and (Moerdijk).
Of a local topos
A local topos has a canonical point, . Morover, this point is an initial object in the category of all points of (see Equivalent characterizations at local topos.)
Over -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 ). Then the sheaf topos has precisely one point per natural number , corresponding to the -ball: the stalk of a sheaf on at that point is the colimit over the result of evaluating the sheaf on all -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.
Textbook references are section 7.5 of
as well as section C2.2 of
- 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
- Sam Zoghaib, A few points in topos theory (pdf)
Points of the sheaf topos over the category of manifolds are discussed in