# Image (category theory)

In category theory, a branch of mathematics, the **image** of a morphism is a generalization of the image of a function.

## General definition[edit]

Given a category and a morphism in , the **image**^{[1]}
of is a monomorphism satisfying the following universal property:

- There exists a morphism such that .
- For any object with a morphism and a monomorphism such that , there exists a unique morphism such that .

**Remarks:**

- such a factorization does not necessarily exist.
- is unique by definition of monic.
- , therefore by monic.
- is monic.
- already implies that is unique.

The image of is often denoted by or .

**Proposition:** If has all equalizers then the in the factorization of (1) is an epimorphism.^{[2]}

**Proof**

Let be such that , one needs to show that . Since the equalizer of exists, factorizes as with monic. But then is a factorization of with monomorphism. Hence by the universal property of the image there exists a unique arrow such that and since is monic . Furthermore, one has and by the monomorphism property of one obtains .

This means that and thus that equalizes , whence .

## Second definition[edit]

In a category with all finite limits and colimits, the **image** is defined as the equalizer of the so-called **cokernel pair** , which is the cocartesian of a morphism with itself over its domain, which will result in a pair of morphisms , on which the equalizer is taken, i.e. the first of the following diagrams is cocartesian, and the second equalizing.^{[3]}

**Remarks:**

- Finite bicompleteness of the category ensures that pushouts and equalizers exist.
- can be called
**regular image**as is a regular monomorphism, i.e. the equalizer of a pair of morphisms. (Recall also that an equalizer is automatically a monomorphism). - In an abelian category, the cokernel pair property can be written and the equalizer condition . Moreover, all monomorphisms are regular.

**Theorem** — If always factorizes through regular monomorphisms, then the two definitions coincide.

**Proof**

**First definition implies the second:** Assume that *(1)* holds with regular monomorphism.

**Equalization:**one needs to show that . As the cokernel pair of and by previous proposition, since has all equalizers, the arrow in the factorization is an epimorphism, hence .**Universality:**in a category with all colimits (or at least all pushouts) itself admits a cokernel pair

- Moreover, as a regular monomorphism, is the equalizer of a pair of morphisms but we claim here that it is also the equalizer of .
- Indeed, by construction thus the "cokernel pair" diagram for yields a unique morphism such that . Now, a map which equalizes also satisfies , hence by the equalizer diagram for , there exists a unique map such that .
- Finally, use the cokernel pair diagram (of ) with : there exists a unique such that . Therefore, any map which equalizes also equalizes and thus uniquely factorizes as . This exactly means that is the equalizer of .

**Second definition implies the first:**

**Factorization:**taking in the equalizer diagram ( corresponds to ), one obtains the factorization .**Universality:**let be a factorization with regular monomorphism, i.e. the equalizer of some pair .

- Then so that by the "cokernel pair" diagram (of ), with , there exists a unique such that .
- Now, from (
*m*from the equalizer of (*i*) diagram), one obtains , hence by the universality in the (equalizer of (_{1}, i_{2}*d*) diagram, with_{1}, d_{2}*f*replaced by*m*), there exists a unique such that .

## Examples[edit]

In the category of sets the image of a morphism is the inclusion from the ordinary image to . In many concrete categories such as groups, abelian groups and (left- or right) modules, the image of a morphism is the image of the correspondent morphism in the category of sets.

In any normal category with a zero object and kernels and cokernels for every morphism, the image of a morphism can be expressed as follows:

- im
*f*= ker coker*f*

In an abelian category (which is in particular binormal), if *f* is a monomorphism then *f* = ker coker *f*, and so *f* = im *f*.

## See also[edit]

## References[edit]

**^**Mitchell, Barry (1965),*Theory of categories*, Pure and applied mathematics, vol. 17, Academic Press, ISBN 978-0-12-499250-4, MR 0202787 Section I.10 p.12**^**Mitchell, Barry (1965),*Theory of categories*, Pure and applied mathematics, vol. 17, Academic Press, ISBN 978-0-12-499250-4, MR 0202787 Proposition 10.1 p.12**^**Kashiwara, Masaki; Schapira, Pierre (2006),*"Categories and Sheaves"*, Grundlehren der Mathematischen Wissenschaften, vol. 332, Berlin Heidelberg: Springer, pp. 113–114 Definition 5.1.1