Contents
Context
Category theory
category theory
Concepts
Universal constructions
Theorems
Extensions
Applications
This entry is about special properties of functors on comma categories. See also category of presheaves.
Contents
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.
Proposition
There is an equivalence of categories
Proof
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
Example
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,
Proposition
For every , there is an equivalence of categories
where is the category of elements of .
Proof
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