A zero morphism in a category is a generalised absorbing element under function composition: any morphism composed with a zero morphism gives a zero morphism. Specifically, if 0XY : X → Y is the zero morphism among morphisms from X to Y, and f : A → X and g : Y → B are arbitrary morphisms, then g ∘ 0XY = 0XB and 0XY ∘ f = 0AY.
If a category has a zero object 0, then there are canonical morphisms X → 0 and 0 → Y, and composing them gives a zero morphism 0XY : X → Y. In the category of groups, for example, zero morphisms are morphisms which always return group identities, thus generalising the function z(x) = 0.
Read more about this topic: List Of Zero Terms
Other articles related to "morphisms, morphism":
... Unlike many categories studied in mathematics, there do not always exist morphisms between pairs of objects in Ring ... For example, there are no morphisms from the trivial ring 0 to any nontrivial ring ... A necessary condition for there to be morphisms from R to S is that the characteristic of S divide that of R ...
... A factorization system (E, M) for a category C consists of two classes of morphisms E and M of C such that E and M both contain all isomorphisms of C and are closed under composition ... Every morphism f of C can be factored as for some morphisms and ... The factorization is functorial if and are two morphisms such that for some morphisms and, then there exists a unique morphism making the following diagram commute ...