Cohomology and homotopy
In higher category theory
For and topological spaces, a continuous function induces (in particular) two functors
between the corresponding Grothendieck topoi of sheaves on and . These are such that:
Morever, if and are sober topological spaces every pair of functors with these properties comes uniquely from a continuous map (see the theorem below).
A geometric morphism between arbitrary topoi is the direct generalization of this situation.
Another motivation of the concept comes from the the fact that a functor such as that preserves finite limits and arbitrary colimits (since it is a left adjoint) necessarily preserves all constructions in geometric logic. See also classifying topos.
If and are toposes, a geometric morphism consists of an pair of adjoint functors
such that the left adjoint preserves finite limits.
We say that
of the geometric morphism.
If moreover the inverse image has also a left adjoint , then is an essential geometric morphism.
See also (Johnstone, p. 162/163).
We discuss some general properties of geometric morphisms. The
also serves as a motivation or justification of the notion of geometric morphism. The
is a fairly straightforward generalization of that situation, reflecting the passage from (sheaf-) (0,1)-toposes to general (1,1)-toposes.
A somewhat subtle point about geometric morphisms of toposes is that there is also another sensible notion of topos homomorphisms: logical morphisms. In
aspects of the relation between the two concepts are discussed.
The reader wishing to learn about geometric morphisms systematically might want to first read the section on Geometric morphisms between presheaf toposes below, as much of the following discussion makes use of a few basic facts discussed there.
Relation to homomorphisms of locales
The definition of geometric morphisms may be motivated as being a categorification of the definition of morphisms of locales.
Such a preservation of finite limits and arbitrary colimits is precisely what characterizes the inverse image part of a geometric morphism, and hence by the adjoint functor theorem already characterizes the full notion of geometric morphisms. Since a locale may equivalently be thought of as a (0,1)-topos, this means that geometric morphisms are direct generalization of the notion of locale homorphisms to 1-toposes.
The following says this in more precise fashion.
For a homomorphism of locales, let
be the functor between their sheaf toposes that sends a sheaf to the composite
where is the corresponding frame morphism as in def. 2.
The functor in def. 3 is the direct image part of a geometric morphism of sheaf toposes
Moreover, the corresponding inverse image functor does restrict on representables to the frame morphism that we also denoted .
In (Johnstone) this appears as lemma C1.4.1 and theorem C1.4.3.
See also at locale the section relation to toposes.
Relation to morphisms of sites
See at site the section Relation between geometric morphisms and morphisms of sites .
Relation to logical morphisms
This is a restatement of this proposition at logical morphism. See there for a proof.
But inverse images can be nontrivial logical morphisms:
Generally, a geometric morphism with logical inverse image is called an atomic geometric morphism. See there for more details.
Structure preserved by geometric morphisms
The inverse images of geometric morphisms preserves the structure of toposes in the sense of their characterization as categories with finite limits that are well-powered indexed categories with respect to the canonical indexing over themselves.
This appears in (Johnstone) as remark B2.2.7 based on example B1.3.17 and prop. B1.3.14. See at indexed category the section Well-poweredness.
Every geometric morphism factors, essentially uniquely, as a geometric surjection followed by a geometric embedding. See geometric surjection/embedding factorization for more on this.
Special classes of geometric morphisms
There are various special cases and types of classes of geometric morphisms. For instance
The following subsections describe some of these in more detail.
Between presheaf toposes
Let and be any two categories. We write and for their opposite categories and , for the corresponding presheaf toposes over and , respectively.
Every functor induces an (essential, even) geometric morphism
where is the functor given by precomposition presheaves with .
Moreover, for a natural transformation between two such functors there is an induced geometric transformation . This is compatible with composition in that it makes forming presheaf toposes a 2-functor
from the 2-category Cat to the 2-category Topos.
This appears as (Johnstone, example 4.1.4).
Since categories of presheaves have all limits and colimits, the left and right Kan extensions and along exists, and form with an adjoint triple
Hence and . Notice that left adjoints and right adjoints to a functor are, if they exist, unique up to unique isomorphism.
Next we consider extra property on , and such that induces also a second geometric morphism, going the other way round. This plays a role for the discussion of morphisms of sites. For that reason we pass now from and to their opposite categories hence consider genuine presheaves on and .
Let and by categories with finite limits and let be a finite-limit preserving functor.
Then in the adjoint triple
the left Kan extension also preserves finite limits and hence in this case is also the direct image of a geometric morphism going the other way round:
This appears as (Johnstone, example 4.1.10).
Recall that for a functor, the left Kan extension is computed over each object by the colimit
where is the comma category and
is the evident forgetful functor. This is natural in and so is the functor
By the above argument has a left adjoint (the left Kan extension along ) hence itself preserves all limits.
It then suffices to observe (see below) that by the fact that preserves finite limits we have that the categories are filtered categories. Then by the fact (see there) that filtered colimits commute with finite limits, it follows that also preserves finite limits, and hence does. Since colimits of presheaves are computed objectwise, this shows that preserves finite limits. This completes the proof.
Here is an explicit desciption of the filteredness of the comma category for any object .
We check the axioms on a filtered category:
non-emptiness : There is an object in : since by assumption preserves the terminal object, take the terminal morphism ;
connectedness : for any two objects and form the product and use that preserves this to produce the object . Then the image under of the two projections provides the required span
two parallel morphism, let be the equalizer of the underlying morphism in . Since preserves equalizers we have an object and a morphism to that equalizes the above two morphisms.
Surjections and embeddings
A geometric morphism is a surjection if is faithful. It is an embedding if is fully faithful.
Up to equivalence, every embedding of toposes is of the form
where is the topos of sheaves with respect to a Lawvere-Tierney topology on .
This means in particular that fully faithful geometric morphisms into Grothendieck topoi are an equivalent way of encoding a Grothendieck topology.
Up to equivalence, every surjection of topoi is of the form
where is the category of coalgebras for a finite-limit-preserving comonad on .
Every geometric morphism factors, uniquely up to equivalence, as a surjection followed by an embedding. There are two ways to produce this factorization: either construct where is the comonad induced by the adjunction , or construct where is the smallest Lawvere-Tierney topology on such that factors through . In fact, surjections and embeddings form a 2-categorical orthogonal factorization system on the 2-category of topoi.
Global sections and constant sheaves
For every Grothendieck topos , there is a geometric morphism
called the global sections functor. It is given by the hom-set out of the terminal object
and hence assigns to each object its set of global elements . If we think of as a sheaf, then is the set of global sections.
The left adjoint of the global section functor is the canonical Set-tensoring functor
applied to the terminal object
which sends a set to the coproduct of copies of the terminal object
This is called the constant object of on the set . Notably when is a sheaf topos this is the constant sheaf on .
The left adjointness is just the defining property of the tensoring
This left adjoint preserves products, using that colimits in a topos are stable by base change (see commutativity of limits and colimits)
and it preserves equalizers and therefore limits. So it is left exact and we do have a geometric morphism.
Point of a topos
For a topos, a geometric morphism
is called a point of a topos.
For any topos and any morphism in there is the change-of-base functor of over categories
by pullback. As described at dependent product this functor has both a left adjoint as well as a right adjoint . Therefore
is a geometric morphism. Hence is an essential geometric morphism.
A category of sheaves is a geometric embedding into a presheaf topos
Geometric morphisms of sheaf topoi
Geometric morphisms between localic topoi are equivalent to continuous maps of locales, which in turn are equivalent to continuous maps of topological spaces if you restrict to sober spaces.
Unrolling this: For a topological space, write as usual for the topos given by the category of sheaves on the category of open subsets with the standard coverage
For every continuous map of sober topological spaces with the induced functor of sites, the direct image
and the inverse image
constitute a geometric morphism
(denoted by the same symbol, by convenient abuse of notation).
This map is an bijection of sets.
That the induced pair forms a geometric morphism is (or should eventually be) discussed at inverse image.
We now show that every geometric morphism of sheaf toposes arises this way from a continuous function, at least up to isomorphism. (In fact, more is true: the category of geometric morphisms is equivalent to the poset of continuous functons with the specialization ordering.) We follow MacLane-Moerdijk, page 348.
One reconstructs the continuous map from a geometric morphism as follows.
Write for the sheaf on constant on the singleton set, the terminal object in .
Notice that since the inverse image preserves finite limits, every subobject is taken by to a subobject , obtained by applying to the pullback diagram
that characterizes the subobject in the topos.
But, as the notation already suggests, the subobjects of are just the open sets, i.e. the representable sheaves.
This yields a function from open subsets to open subsets. By assumption, this preserves finite limits and arbitrary colimits, i.e. finite intersections and arbitrary unions of open sets. In other words, it is a frame homomorphism, and thus can be regarded as a morphism of locales.
We can now use this to define a function of the sets underlying the topological spaces and by setting
This yields a well defined function for the following reasons (which for the moment we spell out in the case where is Hausdorff, although the result should hold —and furthermore, hold constructively— whenever is sober):
there is at most one satisfying this equation: if both satisfy it, there are, by assumption of being Hausdorff, neighbourhoods and such that (using that preserves limits hence intersections) , which contradicts the assumption.
there is at least one satisfying this equation: again by contradiction: if there were none then every has a neighbourhood with , so that similarly to above we conclude with again a contradiction.
Am I right that what we are really need of our space here is not necessarily that it be Hausdorff but simply that it be sober? (Then the nonconstructive aspects of the argument —which is what made me look at this— come in only because the theorem that a Hausdorff space must be sober is not constructively valid.) —Toby
Mike Shulman: Yes, that’s exactly right. All the complication defining above is just an unrolled way of saying that geometric morphisms between localic topoi are equivalent to continuous maps of locales, which are equivalent to continuous functions if you have sober spaces. I think that should be clarified.
Toby: OK, I added a paragraph at the beginning of the example to clarify this. I still need to rewrite the argument immediately above to apply to sober spaces. (Everything else seems to go through exactly the same.)
So our function is well defined and satisfies for every open set . In particular it is therefore a continuous map.
It remains to check that this map reproduces the geometric morphism that we started with. For that we compute its direct image on any sheaf as
The points of the topological space are in canonical bijection with the points of in the sense of point of a topos.
Geometric morphisms are the topic of section VII of
Embeddings and surjections are discussed in section VII.4.
Geometric morphisms are defined in section A4 of
The special classes of geometric morphisms are discussed in section C3.