A constant functor is a functor that maps each object of the category to a fixed object and each morphism of to the identity morphism of that fixed object.
(The notation is suggested by the fact that if names an object of , then the composite
names a functor which is the constant functor at . Here denotes a diagonal functor.)
Viewing , it is possible to see that is a base change of the functor categories and , given the functor .
More generally, we may say that a functor is essentially constant if it is naturally isomorphic to a constant functor.
Note that a constant functor can be expressed as the composite
Here is a terminal category (exactly one object and exactly one morphism, namely the identity), and denotes the unique functor from with and . It follows that is essentially constant if and only if it factors through up to isomorphism.
If has at least one object, then there is a codescent object
Therefore, in this case a functor is essentially constant if and only if we have a natural isomorphism between the two composites (i.e. isomorphisms natural in ) that satisfies a cocycle condition?. This is a categorified version of the statement that a function with inhabited domain is a constant function if and only if the images of any two elements are equal. Just as in that case, there is a distinction in the trivial case: the identity functor of the empty category satisfies this weaker condition, but is not essentially constant because there is no object for it to be constant at.
For any functor, a natural transformation
from a constant functor into is precisely a cone over . Similarly a natural transformation
is a cocone.
Last revised on July 15, 2021 at 01:47:02. See the history of this page for a list of all contributions to it.