Zero morphism

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In category theory, a zero morphism is a special kind of "trivial" morphism. Suppose C is a category, and for any two objects X and Y in C we are given a morphism 0_XY : X → Y with the following property: for any two morphism f : R → S and g : U → V we obtain a commutative diagram:

Then the morphisms 0_XY are called a family of zero morphisms in C.

By taking f or g to be the identity morphism in the diagram above, we see that the composition of any morphism with a zero morphism results in a zero morphism. Furthermore, if a category has a family of zero morphisms, then this family is unique.

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

A morphism is zero if and only if it is constant and coconstant.

Examples

• 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 null 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

0_XY : X → 0 → Y

The family of all morphisms so constructed is a family of zero morphisms for C.

• 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 family of zero morphisms for C.
• The category Set (sets with functions as morphisms) does not have zero morphisms; nor does Top (topological spaces, with continuous functions). However, the category of sets with partial functions as morphisms does have a zero object (the empty set) and thus a family of zero morphisms (the partial functions with empty range). Similarly, one can adjoin to each object of Top a distinguished isolated point ø (the "null point"), and expand the set of morphisms to include all continuous partial functions, i. e., continuous maps such that f(ø) = ø. (Note that {ø} is an isolated open neighborhood, complementary to the remainder Y of the codomain of f. Since Y is also open in , must consist of {ø} plus zero or more isolated components of X.) The resulting category has the zero object {ø} and thus a set of zero morphisms .