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

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

Another definition that is sometimes used is the following.

Definition

If $\mathcal{C}$ has a terminal object, then $c:B\to C$ is constant iff it factors through this terminal object.

If $\mathcal{C}$ does not have a terminal object, then we can reformulate this by saying that the image of $c$ under the Yoneda embedding, $(c\circ -) \colon \mathcal{C}(-,B)\to \mathcal{C}(-,C)$, factors through the terminal presheaf. In elementary terms, this means that we can choose for each object $X$ a morphism $f_X:X\to C$ such that (1) $f_B=c$ and (2) the $f_X$ are natural in $X$, i.e. for any $g:Y\to X$ we have $f_X g = f_Y$.

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, or if the set $B$ is empty set and the set $C$ is inhabited. (By excluded middle, $B$ is either inhabited or empty, so it suffices to assume that $C$ is inhabited with no assumption about $B$.)

More generally, if $c\colon B \to C$ is a morphism in a category $\mathcal{C}$, then the two definitions are equivalent if $\mathcal{C}(A,B)$ is inhabited for every $A$, since we can then define $f_X = c g$ for some (hence any) $g:A\to B$. If $\mathcal{C}$ has a terminal object $1$, then this is equivalent to the existence of a global section $b\colon 1 \to B$.

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. In particular, the identity function on the empty set satisfies definition 1 but not definition 2.

Examples

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

Results

Using the first definition:

• If $f$ is constant, then also $g \circ f$ is constant (whenever this composition is defined).
• If $f$ is constant, then also $f \circ g$ is constant (whenever this composition is defined).
• Any right adjoint functor preserves constant morphisms.
• A representable functor maps constant morphisms to constant maps.
• In a category with zero morphisms, constant morphisms coincide with the zero morphisms.

Dual concept

As usual, we may dualize the concept: A morphism $c\colon A \to B$ is coconstant if for all pairs of morphisms $f,g \colon B \to C$ we have $f \circ c = g \circ c$.

The results on constant morphisms dualize to results on coconstant morphisms. For example, any morphism that factors through an initial object is coconstant.

For a concrete example, the evaluation homomorphism $\mathbb{Z}[X] \to \mathbb{Z}$, $f \mapsto f(0)$ from the polynomial ring is coconstant in the category of rings.

In a category with zero morphisms, coconstant morphisms agree with the zero morphisms (which also coincide with the constant morphisms).

A morphism that is both constant and coconstant is a zero morphism.

Related concepts

• constant function

• steady function

• global element