A functor is final, if we can restrict diagrams on to diagrams on along without changing their colimit.
Dually, a functor is initial if pulling back diagrams along it does not change the limits of these diagrams.
Beware that this property is pretty much unrelated to that of a functor being an initial object or terminal object in the functor category . The terminology comes instead from the fact that an object is initial (resp. terminal) just when the corresponding functor is initial (resp. final).
A functor is final if for every object the comma category is (non-empty and) connected (the non-emptiness condition is redundant since connected categories are non-empty by convention).
A functor is initial if the opposite is final, i.e. if for every object the comma category is non-empty and connected.
Let be a functor
The following conditions are equivalent.
is final.
For all functors the natural function between colimits
is a bijection.
For all categories and all functors the natural morphism between colimits
is a isomorphism.
For all functors the natural function between limits
is a bijection.
For all categories and all functors the natural morphism
is an isomorphism.
For all
If is final then is connected precisely if is.
If and are final, then so is their composite .
If and the composite are final, then so is .
If is a full and faithful functor and the composite is final, then both functors seperately are final.
The first two statements of Proposition in fact follow from the stability properties of orthogonal factorization systems:
Final functors and discrete fibrations form an orthogonal factorization system.
The generalization of the notion of final functor from category theory to (∞,1)-higher category theory is described at
The characterization of final functors is also a special case of the characterization of exact squares.
If has a terminal object then the functor that picks that terminal object is final: for every the comma category is equivalent to . The converse is also true: if a functor is final, then its image is a terminal object.
In this case the statement about preservation of colimits states that the colimit over a category with a terminal object is the value of the diagram at that object. Which is also readily checked directly.
Every right adjoint functor is final.
Let be a pair of adjoint functors.To see that is final, we may for instance check that for all the comma category is non-empty and connected:
It is non-empty because it contains the adjunction unit . Similarly, for
two objects, they are connected by a zig-zag going through the unit, by the universal factorization property of adjunctions
The inclusion of any category into its idempotent completion is final.
See at idempotent completion in the section on Finality.
By the characterization (here) of limits in a slice category, this implies that fiber products in a slice category are computed as fiber products in the underlying category, or in other words that dependent sum to the point preserves fiber products.
final functor, cofinal diagram
Section 2.5 of
Section 2.11 of
Notice that this says “final functor” for the version under which limits are invariant.
Section IX.3 of
Last revised on January 27, 2021 at 18:14:49. See the history of this page for a list of all contributions to it.