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

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

If 𝒞\mathcal{C} does not have a terminal object, then we can reformulate this by saying that the image of cc under the Yoneda embedding, (c)::𝒞(,B)𝒞(,C)(c\circ -) : :\mathcal{C}(-,B)\to \mathcal{C}(-,C), factors through the terminal presheaf?. In elementary terms, this means that we can choose for each object XX a morphism f X:XCf_X:X\to C such that (1) f B=cf_B=c and (2) the f Xf_X are natural in XX, i.e. for any g:YXg:Y\to X we have f Xg=f Yf_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 BB is inhabited, or if the set BB is empty set and the set CC is inhabited. (By excluded middle, BB is either inhabited or empty, so it suffices to assume that CC is inhabited with no assumption about BB.)

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, since we can then define f X=cgf_X = c g for some (hence any) g:ABg:A\to B. 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. 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.

Revised on July 8, 2016 11:50:25 by Mike Shulman (76.167.222.204)