The slice category or over category of a category over an object has * objects that are all arrows such that , and * morphisms from to such that .
The slice category is a special case of a comma category.
There is a forgetful functor which maps an object to its domain and a morphism (from to such that ) to the morphism .
The dual notion is an under category.
If is a poset and , then the slice category is the down set of elements with .
If is a terminal object in , then is isomorphic to .
Relation to codomain fibration
The assignment of overcategories to objects extends to a functor
Under the Grothendieck construction this functor corresponds the the codomain fibration
from the arrow category of . (Note that unless has pullbacks, this functor is not actually a fibration, though it is always an opfibration.)
Adjunctions on overcategories
be a pair of adjoint functors, where the category has all pullbacks.
Then for every object there is induced a pair of adjoint functors between the slice categories
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 also functors and comma categories.
For the analog statement in (∞,1)-category theory see (∞,1)-category of (∞,1)-presheaves – Interaction with overcategories
at (∞,1)-category of (∞,1)-presheaves.
Limits and colimits
A limit in an under category is computed as a limit in the underlying category.
Precisely: let be a category, an object, and the corresponding under category, and the obvious projection.
Let be any functor. Then, if it exists, the limit over in is the image under of the limit over :
and is uniquely characterized by .
Over a morphism in the limiting cone over (which exists by assumption) looks like
By the universal property of the limit this has a unique lift to a cone in the under category over :
It therefore remains to show that this is indeed a limiting cone over . Again, this is immediate from the universal property of the limit in . For let be another cone over in , then is another cone over in and we get in a universal morphism
A glance at the diagram above shows that the composite constitutes a morphism of cones in into the limiting cone over . Hence it must equal our morphism , by the universal property of , and hence the above diagram does commute as indicated.
This shows that the morphism which was the unique one giving a cone morphism on does lift to a cone morphism in , which is then necessarily unique, too. This demonstrates the required universal property of and thus identifies it with .
For a category, a diagram, the comma category (the over-category if is the point) and a diagram in the comma category, then the limit in coincides with the limit in .
For a proof see at (∞,1)-limit here.
Initial and terminal objects
As a special case of the above discussion of limits and colimits in a slice we obtain the following statement, which of course is also immediately checked explicitly.
If has an initial object , then has an initial object, given by .
The terminal object is .