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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.

Definitions

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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 : Y → 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 0AB : A → B, and this collection of morphisms is 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 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 hom-set has a unique "zero morphism", then the category "has zero morphisms".

Examples

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  • In the category of groups (or of modules), a zero morphism is a homomorphism f : G → H that maps all of G to the identity element of H. The zero object in the category of groups is the trivial group 1 = {1}, which is unique up to isomorphism. Every zero morphism can be factored through 1, i. e., f : G → 1 → H.
  • 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 : X → 0 → Y
    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 hom-set Hom(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 of sets 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.
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If a category has zero morphisms, then one can define the notions of kernel and cokernel for any morphism in that category.

References

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  • Section 1.7 of Pareigis, Bodo (1970), Categories and functors, Pure and applied mathematics, vol. 39, Academic Press, ISBN 978-0-12-545150-5
  • Herrlich, Horst; Strecker, George E. (2007), Category Theory, Heldermann Verlag.

Notes

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

📚 Artikel Terkait di Wikipedia

Zero element

zero morphism in a category is a generalised absorbing element under function composition: any morphism composed with a zero morphism gives a zero morphism

Initial and terminal objects

called a zero object or null object. A pointed category is one with a zero object. A strict initial object I is one for which every morphism into I is

Morphism

and existence of an identity morphism for every object), and the outcome of the composition is a morphism. Morphisms and categories recur in much of

Kernel (category theory)

algebra. Intuitively, the kernel of the morphism f : X → Y is the "most general" morphism k : K → X that yields zero when composed with (followed by) f. Kernel

Cokernel

question must have zero morphisms. The cokernel of a morphism f : X → Y is defined as the coequalizer of f and the zero morphism 0XY : X → Y. Explicitly

Preadditive category

is bilinear, the composition of a zero morphism and any other morphism (on either side) must be another zero morphism. If you think of composition as analogous

Zero object (algebra)

which means that a morphism A → {0} must exist and be unique for an arbitrary object A. This morphism maps any element of A to 0. The zero object, also by

0M

for zero manifold Several terms related to 0 (number) Zero map, see constant function Zero morphism, a kind of morphism in category theory Zero matrix