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

C{displaystyle C}

and a morphism

f:XY{displaystyle fcolon Xto Y}


C{displaystyle C}

, the image[1]

f{displaystyle f}

is a monomorphism

m:IY{displaystyle mcolon Ito Y}

satisfying the following universal property:

  1. There exists a morphism
  2. For any object


  1. such a factorization does not necessarily exist.
Image Theorie des catégories.png

The image of

f{displaystyle f}

is often denoted by

Imf{displaystyle {text{Im}}f}


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


Proposition: If

C{displaystyle C}

has all equalizers then the

e{displaystyle e}

in the factorization

f=me{displaystyle f=m,e}

of (1) is an epimorphism.[2]



α,β{displaystyle alpha ,,beta }

be such that

αe=βe{displaystyle alpha ,e=beta ,e}

, one needs to show that

α=β{displaystyle alpha =beta }

. Since the equalizer of

(α,β){displaystyle (alpha ,beta )}


e{displaystyle e}

factorizes as

e=qe{displaystyle e=q,e’}


q{displaystyle q}

monic. But then

f=(mq)e{displaystyle f=(m,q),e’}

is a factorization of

f{displaystyle f}


(mq){displaystyle (m,q)}

monomorphism. Hence by the universal property of the image there exists a unique arrow

v:IEqα,β{displaystyle v:Ito Eq_{alpha ,beta }}

such that

m=mqv{displaystyle m=m,q,v}

and since

m{displaystyle m}

is monic

idI=qv{displaystyle {text{id}}_{I}=q,v}

. Furthermore, one has

mq=(mqv)q{displaystyle m,q=(mqv),q}

and by the monomorphism property of

mq{displaystyle mq}

one obtains

idEqα,β=vq{displaystyle {text{id}}_{Eq_{alpha ,beta }}=v,q}


E epimorphism.png

This means that

IEqα,β{displaystyle Iequiv Eq_{alpha ,beta }}

and thus that

idI=qv{displaystyle {text{id}}_{I}=q,v}


(α,β){displaystyle (alpha ,beta )}

, whence

α=β{displaystyle alpha =beta }


Second definition[edit]

In a category

C{displaystyle C}

with all finite limits and colimits, the image is defined as the equalizer

(Im,m){displaystyle (Im,m)}

of the so-called cokernel pair

(YXY,i1,i2){displaystyle (Ysqcup _{X}Y,i_{1},i_{2})}


Cokernel pair.png
Equalizer of the cokernel pair, diagram.png

[clarification needed]


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

Theorem — If

f{displaystyle f}

always factorizes through regular monomorphisms, then the two definitions coincide.


First definition implies the second: Assume that (1) holds with

m{displaystyle m}

regular monomorphism.

  • Equalization: one needs to show that
  • Universality: in a category with all colimits (or at least all pushouts)
Cokernel pair m.png
Moreover, as a regular monomorphism,
Indeed, by construction
Finally, use the cokernel pair diagram (of

Second definition implies the first:

Now, from


In the category of sets the image of a morphism

f:XY{displaystyle fcolon Xto Y}

is the inclusion from the ordinary image

{f(x) | xX}{displaystyle {f(x)~|~xin X}}


Y{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

f{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]


  1. ^ 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
  2. ^ 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
  3. ^ Kashiwara, Masaki; Schapira, Pierre (2006), “Categories and Sheaves”, Grundlehren der Mathematischen Wissenschaften, vol. 332, Berlin Heidelberg: Springer, pp. 113–114 Definition 5.1.1