category theory

# Contents

## Definition

A constant functor $\Delta(d):C\to D$ is a functor that maps each object of the category $C$ to a fixed object $d\in D$ and each morphism of $C$ to the identity morphism of that fixed object.

More generally, we may say that a functor $F:C\to D$ is essentially constant if it is naturally isomorphic to a constant functor.

## Properties

Note that a constant functor can be expressed as the composite

$C \stackrel{!}{\to} 1 \stackrel{[d]}{\to} D.$

Here $1$ is a terminal category (exactly one object and exactly one morphism, namely the identity), and $[d]$ denotes the unique functor from $1$ with $F(\bullet) = d$ and $F(Id_\bullet) = Id_d$. It follows that $F:C\to D$ is essentially constant if and only if it factors through $1$ up to isomorphism.

If $C$ has at least one object, then there is a codescent object

$C\times C\times C \underoverset{\to}{\to}{\to} C\times C \rightrightarrows C \to 1.$

Therefore, in this case a functor $F:C\to D$ is essentially constant if and only if we have a natural isomorphism between the two composites $C\times C \rightrightarrows C \xrightarrow{F} D$ (i.e. isomorphisms $F(c_1)\cong F(c_2)$ natural in $c_1,c_2\in C$) 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.

## Examples

• For $F$ any functor, a natural transformation

$\Delta_d \Rightarrow F$

from a constant functor into $F$ is precisely a cone over $F$. Similarly a natural transformation

$F \Rightarrow \Delta_d$

is a cocone.

Revised on June 26, 2017 12:33:49 by Mike Shulman (76.167.222.204)