Zero morphism

---

title: "Zero morphism" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["morphisms", "0-(number)"] description: "Bi-universal property in category theory" topic_path: "general/morphisms" source: "https://en.wikipedia.org/wiki/Zero_morphism" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Bi-universal property in category theory ::

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 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:

::figure[src="https://upload.wikimedia.org/wikipedia/commons/5/5b/ZeroMorphism.png"] ::

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.

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

|1= 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. |2= 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. |3= 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. |4= 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 MorC(X, Y). Thus, every category with a zero object is a category with zero morphisms given by the composition 0XY : 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 {{Citation | last=Pareigis | first=Bodo | title=Categories and functors | year=1970 | isbn=978-0-12-545150-5 | publisher=Academic Press | series=Pure and applied mathematics | volume=39
• .

Notes

References

1. (2015-01-17). "Category with zero morphisms - Mathematics Stack Exchange".

::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::