Zero morphism

In category theory, a zero morphism is a special kind of morphism exhibiting properties like those to and from a zero object.

Suppose C is a category, and f : XY 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 g, h : WX, 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 ∈ MorC(Y, Z), gf = hf. A zero morphism is one that is both a constant morphism and a coconstant morphism.

If C has a zero object 0, given two objects X and Y in C, there are canonical morphisms f : 0X and g : Y0. Then, fg is a zero morphism in MorC(Y, X).

A category with zero morphisms is one where, for any two objects A and B in C, there is a fixed morphism 0AB : AB such that for all objects X, Y, Z in C and all morphisms f : YZ, g : XY, the following diagram commutes:

The morphisms 0XY are forced to be zero morphisms and form a compatible system of such. If C is a category with zero morphisms, then the collection of 0XY is unique. A category with a zero object is a category with zero morphisms (given by the composition 0XY : X0Y described above).

If a category has zero morphisms, then one can define the notions of kernel and cokernel for any morphism in that category.

Examples

0XY : X0Y
The family of all morphisms so constructed endows C with the structure of a category with zero morphisms.

References