Zero morphism

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 : 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 : YZ, 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 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 necessarily are zero morphisms and form a compatible system of zero morphisms.

If C is a category with zero morphisms, then the collection of 0XY is unique.[1]

This way of defining a "zero morphism" and the phrase "a category with zero morphisms" separately is unfortunate, but if each homset has a ″zero morphism", then the category "has zero morphisms".


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

Related concepts

If C has a zero object 0, given two objects X and Y in C, there are canonical morphisms f : X0 and g : 0Y. Then, gf is a zero morphism in MorC(X, Y). Thus, every category with a zero object is a category with zero morphisms given by the composition 0XY : X0Y.

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



  1. "Category with zero morphisms - Mathematics Stack Exchange". 2015-01-17. Retrieved 2016-03-30.
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