nLab
zero morphism

Definition

Definition

In a category $C$ with zero object $0$ the zero morphism $0_{c,d}:c\to d$ between two objects $c,d\in C$ is the unique morphism that factors through $0$:

$$0_{c,d}:c\to 0\to d.$$0_{c,d} : c \to 0 \to d \,.

More generally, in any category enriched over the closed monoidal category of pointed sets (with tensor product the smash product), the zero morphism $0_{c,d}:c\to d$ is the basepoint of the hom-object $[c,d]$.

Remark

In fact, an enrichment over pointed sets consists precisely of the choice of a ‘zero’ morphism $0_{c,d}:c\to d$ for each pair of objects, with the property that $0_{c,d}\circ f=0_{b,d}$ and $f\circ 0_{a,b}=0_{a,c}$ for any morphism $f:b\to c$. Such an enrichment is unique if it exists, for if we are given a different collection of zero morphisms $0{\prime }_{c,d}$, we must have

$$0{\prime }_{c,d}=0{\prime }_{c,d}\circ 0_{c,c}=0_{c,d}$$0'_{c,d} = 0'_{c,d} \circ 0_{c,c} = 0_{c,d}

for any $c,d$. Thus, the existence of zero morphisms can be regarded as a property of a category, rather than structure on it. (To be more precise, it is an instance of property-like structure, since not every functor between categories with zero morphisms will necessarily preserve the zero morphisms, although an equivalence of categories will.)

Examples

See zero object for examples.

Related concepts

• null homotopy