Zero Element - Zero Morphisms

Zero Morphisms

A zero morphism in a category is a generalised absorbing element under function composition: any morphism composed with a zero morphism gives a zero morphism. Specifically, if 0_XY : X → Y is the zero morphism among morphisms from X to Y, and f : A → X and g : Y → B are arbitrary morphisms, then g ∘ 0_XY = 0_XB and 0_XY ∘ f = 0_AY.

If a category has a zero object 0, then there are canonical morphisms X → 0 and 0 → Y, and composing them gives a zero morphism 0_XY : X → Y. In the category of groups, for example, zero morphisms are morphisms which always return group identities, thus generalising the function z(x) = 0.