Discrete and concrete objects
Could not include topos theory - contents
When regarding a sheaf as a space defined by how it is probed by test spaces, a concrete sheaf is a generalized space that has (at least) an underlying set of points out of which it is built.
So a concrete sheaf models a space that is given by a set of points and a choice of which morphisms of sets from concrete test spaces into it count as “structure preserving” (e.g. count as smooth, when the sheaf models a smooth space).
More in the intrinsic language of sheaves, a concrete sheaf is a sheaf on a concrete site that, while perhaps not representable, is “quasi-representable” in that it is a subobject of a sheaf of the form
where is a set and is the set of points underlying the object in the concrete site .
We discuss two definitions: the first one is more elementary and describes concrete sheaves explicitly in terms of properties of the underlying site.
The second one is more abstract and more general, and describes them entirely topos theoretically.
On a concrete site
For any presheaf, write
for the adjunct of the restriction map
which in turn is the adjunct of the component map of the functor
A presheaf on a concrete site is a concrete presheaf if for each the map is injective.
A concrete sheaf is a presheaf that is both concrete and a sheaf.
So a concrete presheaf is a subobject of the presheaf .
Write for the full subcategory of the category of sheaves on concrete sheaves.
In a local topos
A more abstract perspective on the previous definition is obtained by noticing the following.
Taking in the second condition defining a concrete site implies that any covering family of contains a split epimorphism, or equivalently that the only covering sieve of is the maximal sieve consisting of all morphisms with target . This means that a concrete site is in particular a local site, which implies that its topos of sheaves is a local topos.
In fact, we can formulate the definition of concrete sheaf inside any local topos over any base topos :
be a local geometric morphism. Since then by definition is a subtopos the morphisms that are inverted by are the local isomorphisms with respect to which the objects of are sheaves/-local objects in .
The concrete sheaves are the objects of that are the -separated objects.
Since is concrete, in the global sections geometric morphism , the direct image is evaluation on the point: . The further right adjoint , sends a set to the functor . Moreover, this right adjoint is fully faithful and thus embeds as a subtopos of .
We observe that is the localization of at the set of counits of the adjunction on representables: because if for we have that
is an isomorphism, then clearly .
On the other hand, comparison with the previous definition shows that this is a monomorphism precisely if is a concrete sheaf. But this is also the definition of a separated object.
So the concrete sheaves on are precisely the separated objects for this Lawvere-Tierney topology on that corresponds to the subtopos .
Let be a local topos. From the definition of concrete sheaves as separated presheaves it follows immediately that
The category of concrete sheaves forms a reflective subcategory of
which is a quasitopos.
The left adjoint is concretization which sends a sheaf to the image sheaf
Slice topos over a concrete object
Let be a Grothendieck topos that is a local topos over and let be a concrete object, equivalently an object such that the -counit is a monomorphism.
We discuss properties of the over-topos .
is the canonical topos point of .
For every global element (for every ) there is a topos point of the form
This is discussed in detail at over-topos – points.
Let be concrete. Then the image under the -monad of any object is an object with being concrete.
This is the finest concrete sheaf structure on that extends to a morphism of concrete sheaves.
By definition of the slice geometric morphism we have that is the pullback in
where the bottom morphism is the -unit. Since this is a monomorphism by assumption on it follows that is a monomorphism. Since is a full and faithful functor by assumption on and is a right adjoint it follows that the adjunct is a monomorphism, as is its image under the right adjoint .
Then by the universal property of the unit we have a commuting diagram
where the bottom and the right morphisms are monomorphisms. Therefore also the diagonal morphism, the -unit on , is a monomorphism, and hence is concrete.
The notion of quasitoposes of concrete sheaves goes back to
- Eduardo Dubuc, Concrete quasitopoi , Lecture Notes in Math. 753 (1979), 239–254
and is further developed in
A review of categories of concrete sheaves, with special attention to sheaves on CartSp, i.e. to diffeological spaces is in
The characterization of concrete sheaves in terms of the extra right adjoint of a local topos originated in discussion with David Carchedi.