nLab
constant morphism

Contents

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

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

Another definition that is sometimes used is the following.

Definition

A morphism c:BCc\colon B \to C in a category 𝒞\mathcal{C} is constant if, for every object AA, exactly one morphism from AA to CC factors through ff. Assuming that 𝒞\mathcal{C} has a terminal object, ff 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 BB is inhabited. More generally, if c:BCc\colon B \to C is a morphism in a category 𝒞\mathcal{C}, then the two definitions are equivalent if 𝒞(A,B)\mathcal{C}(A,B) is inhabited for every AA. If 𝒞\mathcal{C} has a terminal object 11, then this is equivalent to the existence of a global section b:1Bb\colon 1 \to B.

See the forum for further discussion of this.

Relation to (sub)terminal objects

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

Examples

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

Revised on August 20, 2013 06:52:35 by Mike Shulman (107.194.22.192)