This entry is about special properties of functors on comma categories. See also category of presheaves.
Presheaves on over-categories and over-categories of presheaves
Let be a category, an object of and let be the over category of over . Write for the category of presheaves on and write for the over category of presheaves on over the presheaf , where is the Yoneda embedding.
There is an equivalence of categories
The functor takes to the presheaf which is equipped with the natural transformation with component map
A weak inverse of is given by the functor
which sends to given by
where is the pullback
Suppose the presheaf does not actually depend on the morphsims to , i.e. suppose that it factors through the forgetful functor from the over category to :
Then and hence with respect to the closed monoidal structure on presheaves.
See over-topos for more.
Over-categories of presheaf categories and presheaves on categories of elements
Generalizing the above,
For every , there is an equivalence of categories
where is the category of elements of .
The construction is completely analogous to the above; Given , is defined pointwise as a coproduct:
where is an object of . The action on morphisms is defined analogously. This comes equipped with a natural transformation , with component
Given an object of the action of a weak inverse can be specified as , that is, the wedge of the pullback:
The action of on arrows of , functoriality, etc is derived from its definition as a pullback and the def of morphisms in .
Relationship with the over-categories statement
Putting in the above yields:
Now it is easy to see that ; we get then:
In higher category theory
For the analogous result in the context of (∞,1)-category theory see (∞,1)-Category of (∞,1)-presheaves – Interaction with overcategories