In category theory

Category theory

Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...

, a zero morphism is a special kind of morphism

Morphism

In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...

 exhibiting properties like those to and from a zero object.

Suppose C is a category

Category (mathematics)

In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...

, 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 ∈ Mor_C(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 Mor_C(Y, X).

A category with zero morphisms is one where, for any two objects A and B in C, there is a fixed morphism 0_AB : 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 0_XY 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 0_XY is unique. A category with a zero object is a category with zero morphisms (given by the composition 0_XY : X → 0 → Y described above).

If a category has zero morphisms, then one can define the notions of kernel

Kernel (category theory)

In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms, the kernels of module homomorphisms and certain other kernels from algebra...

 and cokernel

Cokernel

In mathematics, the cokernel of a linear mapping of vector spaces f : X → Y is the quotient space Y/im of the codomain of f by the image of f....

 for any morphism in that category.

Examples

• In the category of groups
  Category of groups
  In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category...
  
   (or of modules
  Module (mathematics)
  In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
  
  ), a zero morphism is a homomorphism
  Homomorphism
  In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...
  
   f : G → H that maps all of G to the identity element
  Identity element
  In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
  
   of H. The null object in the category of groups is the trivial group
  Trivial group
  In mathematics, a trivial group is a group consisting of a single element. All such groups are isomorphic so one often speaks of the trivial group. The single element of the trivial group is the identity element so it usually denoted as such, 0, 1 or e depending on the context...
  
   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

0_XY : 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
  Preadditive category
  In mathematics, specifically in category theory, a preadditive category is a category that is enriched over the monoidal category of abelian groups...
  
  , then every morphism set Mor(X,Y) is an abelian group
  Abelian group
  In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
  
   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
  Category of sets
  In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets A and B are all functions from A to B...
  
   (sets with function
  Function (mathematics)
  In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
  
  s as morphisms) does not have a zero object, but it does have an initial object
  Initial object
  In category theory, an abstract branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X...
  
  , the empty set
  Empty set
  In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
  
  ∅. The only zero morphisms in Set are the functions ∅ → X for a set X.