typical contexts
For a functor we say that it has codiscrete objects if it has a full and faithful right adjoint .
This is for instance the case for the global section geometric morphism of a local topos .
In this situation, we say that a concrete object is one for which the -unit of an adjunction is a monomorphism.
If is a sheaf topos, this is called a concrete sheaf.
If is a cohesive (∞,1)-topos then this is called a concrete (∞,1)-sheaf or the like.
The dual notion is that of a co-concrete object.
is a faithful functor on morphisms whose codomain is concrete.
For a local topos, write
for its full subcategory of concrete objects.
Then there is a sequence of reflective subcategory-inclusions that factor the -adjunction as
Here morphisms on top are left adjoint to morphisms below, hence
exhibits the concrete objects as a reflective subcategory, the reflector being “concretification”.
For the adjunction on the right, we just need to observe that for every set , the codiscrete object is concrete, which is immediate by idempotency of and the fact that every isomorphism is also a monomorphism.
For the adjunction on the left we claim that the left adjoint (concretification), is given by sending each object to the image of its adjunction unit :
hence to the object which exhibits the epi/mono-factorization of
First we need to show that , thus defined, is indeed concrete, hence that is a monomorphism. For this, consider the following naturality square of the -adjunction hom-isomorphism
By chasing the identity morphism on through this diagram, as shown by the diagram on the right, we obtain the equality displayed in the bottom right entry, where we used the general formula for adjuncts and the definition .
But observe that , and hence also , is an isomorphism, as indicated above: Since is both a left adjoint as well as a right adjoint, it preserves both epimorphisms as well as monomorphisms, hence it preserves image factorizations. This implies that is the epimorphism onto the image of . But by idempotency of , the latter is an isomorphism, and hence so is the epimorphism in its image factorization.
Therefore the equality in (2) says that
where in the second line we remembered that is, by definition, the epimorphism in the epi/mono-factorization of .
Now the defining property of epimorphisms allows to cancel this commmon factor on both sides, which yields
This shows that is indeed concrete.
It remains to show that this construction is left adjoint to the inclusion. We claim that the adjunction unit of is provided by (1).
To see this, first notice that, since the epi/mono-factorization is orthogonal and hence functorial, we have commuting diagrams of the form
Now to demonstrate the adjunction, it is sufficient, to show that is a universal morphism in the sense discussed at adjoint functors. Hence consider any morphism with . Then we need to show that there is a unique diagonal morphism as below, that makes the following top left triangle commute:
Now, from (3), we have a commuting square as shown. Here the left morphism is an epimorphism by construction, while the right morphism is a monomorphism by assumption on . With this, the epi/mono-factorization says that there is a diagonal lift which makes both triangles commute.
It remains to see that the lift is unique with just the property of making the top left triangle commute. But this is equivalently the statement that the left morphism is an epimorphism.
(cohesive maps to concrete objects glue)
Let be a cohesive -topos, and consider a concrete object, in that its sharp modality unit is (-1)-truncated/monomorphic: . Then cohesive maps to glue (satisfy the respective sheaf property):
For any and any open cover, namely any (-1)-connected/effective epi-morphism , cohesive maps whose maps of underlying -groupoids descend/extend to , then they also descend/extend to as cohesive maps, in an essentially unique way, in that all solid homotopy-commutative square as follows have essentially unique dashed lifts:
Under the given assumptions, the essentially unique existence of the lift is an instance of the (n-connected, n-truncated) factorization system for .
Mike Shulman, Discreteness, Concreteness, Fibrations, and Scones (blog post)
Urs Schreiber, Section 3.7.2 in Differential cohomology in a cohesive -topos (arXiv:1310.7930)
Last revised on October 6, 2021 at 07:42:44. See the history of this page for a list of all contributions to it.