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 : X → 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 g, h : W → X, 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 : 0 → X and g : Y → 0. 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 : A → B such that for all objects X, Y, Z in C and all morphisms f : Y → Z, g : X → Y, 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 : X → 0 → Y described above).
If a category has zero morphisms, then one can define the notions of kernel and cokernel for any morphism in that category.
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