# Image (category theory) – Wikipedia

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

${displaystyle C}$and a morphism

${displaystyle fcolon Xto Y}$in

${displaystyle C}$, the **image**^{[1]}

of

is a monomorphism

${displaystyle mcolon Ito Y}$satisfying the following universal property:

- There exists a morphism
- For any object

**Remarks:**

- such a factorization does not necessarily exist.

The image of

${displaystyle f}$is often denoted by

${displaystyle {text{Im}}f}$or

${displaystyle {text{Im}}(f)}$.

**Proposition:** If

has all equalizers then the

${displaystyle e}$in the factorization

${displaystyle f=m,e}$ of (1) is an epimorphism.^{[2]}

**Proof**

Let

${displaystyle alpha ,,beta }$be such that

${displaystyle alpha ,e=beta ,e}$, one needs to show that

${displaystyle alpha =beta }$. Since the equalizer of

${displaystyle (alpha ,beta )}$exists,

${displaystyle e}$factorizes as

${displaystyle e=q,e’}$with

${displaystyle q}$monic. But then

${displaystyle f=(m,q),e’}$is a factorization of

${displaystyle f}$with

${displaystyle (m,q)}$monomorphism. Hence by the universal property of the image there exists a unique arrow

${displaystyle v:Ito Eq_{alpha ,beta }}$such that

${displaystyle m=m,q,v}$and since

${displaystyle m}$is monic

${displaystyle {text{id}}_{I}=q,v}$. Furthermore, one has

${displaystyle m,q=(mqv),q}$and by the monomorphism property of

${displaystyle mq}$one obtains

${displaystyle {text{id}}_{Eq_{alpha ,beta }}=v,q}$.

This means that

${displaystyle Iequiv Eq_{alpha ,beta }}$and thus that

${displaystyle {text{id}}_{I}=q,v}$equalizes

${displaystyle (alpha ,beta )}$, whence

${displaystyle alpha =beta }$.

## Second definition[edit]

In a category

${displaystyle C}$ with all finite limits and colimits, the **image** is defined as the equalizer

of the so-called **cokernel pair**

.^{[3]}

^{[clarification needed]}

**Remarks:**

- Finite bicompleteness of the category ensures that pushouts and equalizers exist.
- In an abelian category, the cokernel pair property can be written

**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**Universality:**in a category with all colimits (or at least all pushouts)

- Moreover, as a regular monomorphism,
- Indeed, by construction
- Finally, use the cokernel pair diagram (of

**Second definition implies the first:**

- Then
- Now, from

## Examples[edit]

In the category of sets the image of a morphism

${displaystyle fcolon Xto Y}$is the inclusion from the ordinary image

${displaystyle {f(x)~|~xin X}}$to

${displaystyle Y}$. 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

${displaystyle f}$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

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