Cohomology and homotopy
In higher category theory
Classes of bundles
Examples and Applications
For a topos and any object the over category – the slice topos or over-topos – is itself a topos: the “big little topos” incarnation of . This fact is sometimes called the “Fundamental Theorem of Topos Theory”.
More generally, given a fibred product-preserving functor between toposes, the comma category is again a topos, called the Artin gluing.
Definition / Existence
A proof is spelled out for instance in MacLane-Moerdijk, IV.7 theorem 1. In particular we have
If is the subobject classifier in , then the projection regarded as an object in the slice over is the subobject classifier of .
The power object of a map is given by the equalizer of the maps :
where is the projection map and is the composition . In the internal language, this says
The map to is given by projection onto the second factor.
Étale geometric morphism
For a Grothendieck topos and any object, the canonical projection functor is part of an essential geometric morphism
The functor is given by taking the product with :
since commuting diagrams
are evidently uniquely specified by their components .
Moreover, since in the Grothendieck topos we have universal colimits, it follows that preserves all colimits. Therefore by the adjoint functor theorem a further right adjoint exists.
For a Grothendieck topos and a morphism in , there is an induced essential geometric morphism
where is given by postcomposition with and by pullback along .
By universal colimits in the pullback functor preserves both limits and colimits. By the adjoint functor theorem and using that the over-toposes are locally presentable categories, this already implies that it has a left adjoint and a right adjoint. That the left adjoint is given by postcomposition with follows from the universality of the pullback: for in and in we have unique factorizations
in , hence an isomorphism
We discuss special properties of over-presheaf toposes.
Let be a small category, an object of and let be the over category of over .
There is an equivalence of categories
The functor takes to the presheaf which is equipped with the natural transformation with component map
A weak inverse of is given by the functor
which sends to given by
where is the pullback
Suppose the presheaf does not actually depend on the morphisms to , i.e. suppose that it factors through the forgetful functor from the over category to :
Then and hence with respect to the closed monoidal structure on presheaves.
Consider , the category of elements of . This has objects with , hence is just an arrow in . A map from to is just a map such that but this is just a morphism from to in .
Hence, the above proposition 6 can be rephrased as which is an instance of the following formula:
Let be a presheaf. Then there is an equivalence of categories
For a proof see Kashiwara-Schapira (2006, p.26). For a more general statement involving slices of Grothendieck toposes see Mac Lane-Moerdijk (1994, p.157).
In particular, this equivalence shows that slices of presheaf toposes are presheaf toposes.
Geometric morphisms by slicing
For a geometric morphism of toposes and any object, there is an induced geometric morphism between the slice-toposes
where the inverse image is the evident application of to diagrams in .
The slice adjunction is discussed here: the left adjoint is the evident induced functor. Since limits in an over-category are computed as limits in of diagrams with a single bottom element adjoined, preserves finite limits, since does, so that is indeed a geometric morphism.
We discuss topos points of over-toposes.
Let be a topos, an object and
a point of . Then for every element there is a point of the slice topos given by the composite
Here is the slice geometric morphism of over discussed above and is the étale geometric morphism discussed above induced from the morphism .
Hence the inverse image of sends to the fiber of over .
If has enough points then so does the slice topos for every .
That has enough points means that a morphism in is an isomorphism precisely if for every point the function is an isomorphism.
A morphism in the slice topos, given by a diagram
in is an isomorphism precisely if is. By the above observation we have that under the inverse images of the slice topos points this maps to the fibers of
over all points . Since in Set every object is a coproduct of the point indexed over , and using universal colimits in , we have that if is an isomorphism for all then was already an isomorphism.
The claim then follows with the assumption that has enough points.