Cohomology and homotopy
In higher category theory
Every geometric morphism between toposes factors into a geometric surjection followed by a geometric embedding. This exhibits an image construction in the topos-theoretic sense, and gives rise to a factorization system in a 2-category for Topos.
There is a factorization system on the 2-category Topos whose left class is the surjective geometric morphisms and whose right class is the geometric embeddings.
Moreover, the factorization of a given geometric morphism is, up to equivalence, through the canonical surjection onto the topos of coalgebras of the comonad :
This appears for instance as (MacLaneMoerdijk, VII 4., theorem 6).
We use the following lemma
This appears as (MacLaneMoerdijk, VII 4. prop. 2).
Proof of the lemma
We first show the first statement, that for to factor it is sufficient for to take values in -sheaves: in that case, set
Since by assumption the unit map is an isomorphism on the image of this indeed serves to factor :
The left adjoint to is then
where in the middle steps we used that is a -sheaf, by assumption, and that is full and faithful.
It is clear that is left exact, and so is indeed a factorizing geometric morphism.
We now show that taking values in sheaves is equivalent to mapping dense monos to isos.
Let be a -dense monomorphism and any object. Consider the induced naturality square
of the adjunction natural isomorphism. If now is a -sheaf and a dense monomorphism, then by definition the left vertical morphism is also an isomorphism and so is the right one. By the Yoneda lemma this being an iso for all is equivalent to being an iso. And conversely.
Proof of the proposition
Let be any geometric morphism.
We first discuss the existence of the factorization, then its uniqueness.
To construct the factorization, we shall give a Lawvere-Tierney topology on and factor through the geometric embedding of the corresponding sheaf topos.
Take the closure operator to be given by sending to the pullback
where the bottom morphism is the -unit. One checks that this is indeed a closure operator by the fact that preserves both pullbacks and pushouts.
Notice that this implies that for two subobjects we have
Write for the corresponding Lawvere-Tierney topology and
for the corresponding geometric embedding.
By lemma 1 we get a factorization through if sends -dense monomorphisms to isomorphisms. But if is dense so that then, by (1), and hence .
for the factorization thus established. It remains to show that here is a geometric surjection. By one of the equivalent characterizations discussed there, this is the case if induces an injective homomorphism of subobject lattices.
So suppose that for subobjects we have . Observe that then also , because
by the fact that is full and faithful. With (1) it follows that also
by the very fact that includes -sheaves in general, hence -closed subobjects in particular. Finally since if a full and faithful functor this means that
So is indeed injective on subobjects and so is a geometric surjection.
This establishes the existence of a surjection/embedding factorization. Next we discss that this is unique.
So consider two factorizations
into a geometric surjection followed by a geometric embedding.
We will now argue that factors – essentially uniquely – through in a way that makes
commute up to natural isomorphisms. By the same argument for the upside-down situation we find an essentially unique middle vertical morphism the other way round. Then essential uniqueness of these factorizations implies that and . This means that the original two factorizations are equivalent.
To find and , use again that every geometric embedding (by the discussion there) is, up to equivalence, an inclusion of -sheaves for some . Find such a the bottom morphism and then use again lemma 1 that factors through – essentially uniquely – precisely if sends dense monomorphisms to isomorphisms.
To see that it does, let be a dense mono and consider the naturality square
Since is an iso by definition, the left vertical morphism is, and thus so is the right vertical morphism. But since is a geometric surjection we have (by the discussion there) that is conservative, and hence also is an isomorphism.
Hence factors via some through and the proof is completed by the above argument.
For a continuous function between topological spaces and its ordinary image factorization through an embedding, the corresponding composite of geometric morphisms of sheaf toposes
is a geometric surjection/geometric embedding factorization.
For any topos, any morphism in , and its image factorization, the corresponding composite of base change geometric morphisms
is a geometric surjection/embedding factorization.
For any functor between categories, write for its essential image factorization. Then the induced composite geometric morphism of presheaf toposes
is a geometric surjection/embedding factorization.
See (MacLaneMoerdijk, p. 377).
As idempotent approximation
A geometric morphism induces via the adjunction a monad on . Due to a general result by S. Fakir this induces an associated idempotent monad on and this idempotent approximation coincides with the monad induced by given by the inclusion from the factorization .
For references and further details on the idempotent approximation see at idempotent monad.
A logical description
Let be a geometric theory over a signature and a geometric morphism to its classifying topos. Then by the general properties of a classifying topos, corresponds to a certain -model in .
Notice that every geometric morphism between Grothendieck toposes is of this form for some geometric theory and hence corresponds to some model ! This models permits to attach a geometric theory to as well:
The theory of M consists of all geometric sequents over such that .
Then the following holds (Caramello 2009, p.57):
The topos occurring in the middle of the surjection-embedding factorization of is precisely the classifying topos for : .