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.
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Introduction to Higher Mathematics  Lecture 18: Morphisms

Morphism of Sheaves
Transcription
Definitions
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 0_{AB} : 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 0_{XY} necessarily are zero morphisms and form a compatible system of zero morphisms.
If C is a category with zero morphisms, then the collection of 0_{XY} 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
 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
 0_{XY} : X → 0 → Y
 If C is a preadditive category, then every homset 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.
Related concepts
If C has a zero object 0, given two objects X and Y in C, there are canonical morphisms f : X → 0 and g : 0 → Y. Then, gf is a zero morphism in Mor_{C}(X, Y). Thus, every category with a zero object is a category with zero morphisms given by the composition 0_{XY} : X → 0 → Y.
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, vol. 39, Academic Press, ISBN 9780125451505
 Herrlich, Horst; Strecker, George E. (2007), Category Theory, Heldermann Verlag.
Notes
 ^ "Category with zero morphisms  Mathematics Stack Exchange". Math.stackexchange.com. 20150117. Retrieved 20160330.