# Zero morphism

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.

## 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* 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 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*

- 0
- The family of all morphisms so constructed endows
**C**with the structure of a category with zero morphisms.

- 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 compatible family of zero morphisms for**C**making it into a category with zero morphisms. - The category
**Set**(sets with functions as morphisms) 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,**39**, Academic Press, ISBN 978-0-12-545150-5 - Herrlich, Horst; Strecker, George E. (2007),
*Category Theory*, Heldermann Verlag.

## Notes

- ↑ "Category with zero morphisms - Mathematics Stack Exchange".
*Math.stackexchange.com*. 2015-01-17. Retrieved 2016-03-30.