nLab
constant morphism

Idea

The notion of a constant morphism in a category generalises the notion of constant function.

Definition

Definition

A constant morphism in a category $𝒞$ is a morphism $c:B\to C$ with the property that if $f,g:A\to B$ are morphisms in $𝒞$ then $c\circ f=c\circ g$. In other words, for every object $A$, at most one morphism from $A$ to $C$ factors through $f$.

Thus, $c$ is a constant morphism if the function $c_{*}:𝒞(A,B)\to 𝒞(A,C)$ given by composition with $c$ is a constant function for every object $A$.

Another definition that is sometimes used is the following.

Definition

A morphism $c:B\to C$ in a category $𝒞$ is constant if, for every object $A$, exactly one morphism from $A$ to $C$ factors through $f$. Assuming that $𝒞$ has a terminal object, $f$ is constant iff it factors through this terminal object.

This second definition implies the first, but they are not equivalent in general. In the category of sets, the first implies the second if the set $B$ is inhabited. More generally, if $c:B\to C$ is a morphism in a category $𝒞$, then the two definitions are equivalent if $𝒞(A,B)$ is inhabited for every $A$. If $𝒞$ has a terminal object $1$, then this is equivalent to the existence of a global section $b:1\to B$.

See the forum for further discussion of this.

Relation to (sub)terminal objects

The identity morphism on an object $B$ satisfies definition 1 if and only if $B$ is subterminal; it satisfies definition 2 iff $B$ is terminal.

Examples

Using the two-point set, it is simple to show that the constant morphisms in Set are precisely the constant functions.