📂Abstract Algebra

Zero Morphism

Definition ¹

$$ W \to X \overset{f}{\to} Y \to Z $$

Consider a morphism $f : X \to Y$.

1. A morphism $g,h : W \to X$ is called a constant morphism if $fg = fh$ implies $f$.
2. A morphism $g,h : Y \to Z$ is called a coconstant morphism if $gf = hf$ implies $f$.
3. A morphism $f$ that is both a constant morphism and a coconstant morphism is called a zero morphism.

Description

In the definition, $fg$ refers to the composition of functions, not their product. To be constant in either direction, the function must essentially have a value like $0$, and thus, the zero morphism is aptly named as it denotes $0$. That a morphism is zero indicates that no matter which object is input, it results in an object based on $\left\{ 0 \right\}$.