Cohomology and homotopy
In higher category theory
A geometric embedding is the right notion of embedding or inclusion of topoi , i.e. of subtoposes.
Notably the inclusion of a category of sheaves into its presheaf topos or more generally the inclusion of sheaves in a topos into itself, is a geometric embedding. Actually every geometric embedding is of this form, up to equivalence of topoi.
Another perspective is that a geometric embedding is the localizations of at the class or morphisms that the left adjoint sends to isomorphisms in .
The induced geometric morphism of a topological immersion is a geometric embedding. The converse holds if is a space. (Example A4.2.12(c) in (Johnstone))
For and two topoi, a geometric morphism
is a geometric embedding if the following equivalent conditions are satisfied
the direct image functor is full and faithful (so that is a full subcategory of );
the counit of the adjunction is an isomorphism
there is a Lawvere-Tierney topology on and an equivalence of categories such that the diagram of geometric morphisms commutes up to natural isomorphism
That the first two conditions are equivalent is standard, that the third one is equivalent to the first two is for instance corollary 7 in section VII, 4 of (MacLaneMoerdijk)
Relation to localization
There is a close relation between geometric embedding and localization.
Let be a geometric embedding and let be the class of morphisms sent by to isomorphisms in .
This fact connects for instance the description of sheafification in terms of geometric embedding as described for instance in
with that in terms of localization at local isomorphisms, as described in
Moreover, this is the basis on which sheafification is generalized to (∞,1)-sheafification in
The following gives a detailed proof of the above assertion.
Write for the unit of the adjunction.
Since is fully faithful we will identify objects and morphism of with their images in . To further trim down the notation write for the left adjoint.
Write for the class of morphism that are sent to isomorphism under ,
Follows since isomorphisms satisfy 2-out-of-3.
This follows using the fact that is left exact and hence preserves finite limits.
In more detail:
We have already seen in the previous proposition that
It remains to check the following points:
with , we have to show that there is
To get this, take this to be the pullback diagram, . Since preserves pullbacks, it follows that
is a pullback diagram in with . But by assumption is an isomorphism. Therefore is an isomorphism, therefore is in .
Finally for every
with such that the two composites coincide, we need to find
with such that the composites again coincide.
To get this, take to be the equalizer of the two morphisms. Sending everything with to we find from
that , since is an isomorphism. This implies that is the equalizer
of two equal morphism, hence an identity. So is in .
For every object
This follows from the zig-zag identities of the adjoint functors.
In components they say that
for every we have
for every we have
This implies the claim.
An object is -local object if for every in the map
obtained by precomposition is an isomorphism.
Up to isomorphism, the -local objects are precisely the objects of in
First assume that . We need to show that is -local.
Notice that the existence of the required isomorphism is equivalent to the statement that for every diagram
there is a unique extension
To see the existence of this extension, hit the original diagram with to get
By the assumption that is in the morphism here is an isomorphism. By the assumption that is already in we have since the counit is an isomorphism. Therefore this diagram clearly has a unique extension
By the hom-isomorphism (using full faithfullness of to work entirely in )
this defines a morphism . Chasing through the naturality diagram of the hom-isomorphism
shows that does extend the original diagram. Again by the Hom-isomorphism, it is the unique morphism with this property.
So is -local.
Now for the converse, assume that a given is -local.
By one of the above propositions we know that the unit is in , so by the -locality of it follows that
has an extension
By the 2-out-of-3 property of shown in one of the above propositions, (using that , being an isomorphism, is in ) it follows that is in .
Since is in and therefore -local by the above, it follows that also
has an extension
So has a left inverse which itself has a left inverse . It follows that is also a right inverse to , since
So if is -local we find that is an isomorphism, hence that is isomorphic to an object of .
By standard reasoning (e.g. KS lemma 1.3.11) there is a functor and a natural isomorphism
Since and are full and faithful, so is . Since by the above it is also essentially surjective, it establishes the equivalence .
By one of the above propositions we know that is a left multiplicative systems.
This implies that the localization is (equivalent to) the category with the same objects as , and with hom-sets given by
There is an obvious candidate for a functor
given on objects by the usual embedding by and on morphism by the map which regards a morphism trivially as a span with left leg the identity
For this to be an equivalence of categories we need to show that this is a essentially surjective and full and faithful functor.
To see essential surjectivity, let be any object in and let be the component of the unit of our adjunction on , as above. By one of the above propositons, is in . This means that the span
represents an element in , and this element is clearly an isomorphism: the inverse is represented by
Since every is in the image of our functor, this shows that it is essentially surjective.
To see fullness and faithfulness, let be any two objects. By one of the above propositions this means in particular that is a -local object. As discussed above, this means that every span
with has a unique extension
But this implies that in the colimit that defines the hom-set of all these spans are identified with spans whose left leg is the identiy. And these are clearly in bijection with the morphisms in so that indeed
for all . Hence our functor is also full and faithful and therefore define an equivalence of categories
Factorizations and images
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. The factorization of a geometric morphism can be said to construct its image in the topos-theoretic sense.
See geometric surjection/embedding factorization.
Moreover, each geometric embedding itself has a (dense,closed)-factorization.
In the more general context of (∞,1)-topos theory an -geometric embedding is an (∞,1)-geometric morphism
such that the right adjoint direct image is a full and faithful (∞,1)-functor.
See reflective sub-(∞,1)-category for more details.
Section VII, 4 of
and section A4.2 of