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Zero morphism

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Title: Zero morphism  
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Subject: Category of rings, Injective cogenerator, Biproduct, Kernel (category theory), Pre-abelian category
Collection: 0 (Number), Morphisms
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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.

Contents

  • Definitions 1
  • Examples 2
  • Related concepts 3
  • References 4
  • Notes 5

Definitions

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".

Examples

  • More generally, suppose C is any category with a zero object 0. Then for all objects X and Y there is a unique sequence of morphisms
0XY : X0Y
The family of all morphisms so constructed endows C with the structure of a category with zero morphisms.
  • If C is a preadditive category, then every morphism set Mor(X,Y) is an abelian group and therefore has a zero element. These zero elements form a compatible family of zero morphisms for C making it into a category with zero morphisms.
  • The category Set (sets with functions as morphisms) does not have a zero object, but it does have an initial object, the empty set ∅. The only right zero morphisms in Set are the functions ∅ → X for a set X.

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.

References

  • Section 1.7 of Pareigis, Bodo (1970), Categories and functors, Pure and applied mathematics 39,  
  • Herrlich, Horst; Strecker, George E. (2007), Category Theory, Heldermann Verlag .

Notes

  1. ^ http://math.stackexchange.com/questions/189818/category-with-zero-morphisms
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