In category theory, a branch of mathematics, a zero morphism is a special kind of morphism exhibiting properties like the morphisms to and from a zero object.
Suppose C is a category, and f : X â Y is a morphism in C. The morphism f is called a constant morphism (or sometimes left zero morphism) if for any object W in C and any , fg = fh. Dually, f is called a coconstant morphism (or sometimes right zero morphism) if for any object Z in C and any g, h : Y â Z, gf = hf. A zero morphism is one that is both a constant morphism and a coconstant morphism.
A category with zero morphisms is one where, for every two objects A and B in C, there is a fixed morphism 0<sub>AB</sub> : A â B, and this collection of morphisms is such that for all objects X, Y, Z in C and all morphisms f : Y â Z, g : X â Y, the following diagram commutes:
The morphisms 0<sub>XY</sub> necessarily are zero morphisms and form a compatible system of zero morphisms.
If C is a category with zero morphisms, then the collection of 0<sub>XY</sub> is unique.
This way of defining a "zero morphism" and the phrase "a category with zero morphisms" separately is unfortunate, but if each hom-set has a unique "zero morphism", then the category "has zero morphisms".
If a category has zero morphisms, then one can define the notions of kernel and cokernel for any morphism in that category.