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.

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 $F(Id_\bullet) = Id_d$.

## 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 are natural transformation

$F \Rightarrow \Delta_d$

is a cocone.

Revised on June 20, 2013 12:47:01 by Urs Schreiber (82.169.65.155)