The image of 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is often denoted by 
 
 
 
 
 Im
 
 f
 
 
 {\displaystyle {\text{Im}}f}
 
 or 
 
 
 
 
 Im
 
 (
 f
 )
 
 
 {\displaystyle {\text{Im}}(f)}
 
.
Proposition: If 
 
 
 
 C
 
 
 {\displaystyle C}
 
 has all <a href="/facts/Equaliser_(mathematics)/4eD7pB8y">equalizers</a> then the 
 
 
 
 e
 
 
 {\displaystyle e}
 
 in the factorization 
 
 
 
 f
 =
 m
 
 e
 
 
 {\displaystyle f=m\,e}
 
 of (1) is an <a href="/facts/Epimorphism/IrzonI01">epimorphism</a>.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>

Proof
Let 
 
 
 
 α
 ,
 
 β
 
 
 {\displaystyle \alpha ,\,\beta }
 
 be such that 
 
 
 
 α
 
 e
 =
 β
 
 e
 
 
 {\displaystyle \alpha \,e=\beta \,e}
 
, one needs to show that 
 
 
 
 α
 =
 β
 
 
 {\displaystyle \alpha =\beta }
 
. Since the equalizer of 
 
 
 
 (
 α
 ,
 β
 )
 
 
 {\displaystyle (\alpha ,\beta )}
 
 exists, 
 
 
 
 e
 
 
 {\displaystyle e}
 
 factorizes as 
 
 
 
 e
 =
 q
 
 
 e
 ′
 
 
 
 {\displaystyle e=q\,e'}
 
 with 
 
 
 
 q
 
 
 {\displaystyle q}
 
 monic. But then 
 
 
 
 f
 =
 (
 m
 
 q
 )
 
 
 e
 ′
 
 
 
 {\displaystyle f=(m\,q)\,e'}
 
 is a factorization of 
 
 
 
 f
 
 
 {\displaystyle f}
 
 with 
 
 
 
 (
 m
 
 q
 )
 
 
 {\displaystyle (m\,q)}
 
 monomorphism. Hence by the universal property of the image there exists a unique arrow 
 
 
 
 v
 :
 I
 →
 E
 
 q
 
 α
 ,
 β
 
 
 
 
 {\displaystyle v:I\to Eq_{\alpha ,\beta }}
 
 such that 
 
 
 
 m
 =
 m
 
 q
 
 v
 
 
 {\displaystyle m=m\,q\,v}
 
 and since 
 
 
 
 m
 
 
 {\displaystyle m}
 
 is monic 
 
 
 
 
 
 id
 
 
 I
 
 
 =
 q
 
 v
 
 
 {\displaystyle {\text{id}}_{I}=q\,v}
 
. Furthermore, one has 
 
 
 
 m
 
 q
 =
 (
 m
 q
 v
 )
 
 q
 
 
 {\displaystyle m\,q=(mqv)\,q}
 
 and by the monomorphism property of 
 
 
 
 m
 q
 
 
 {\displaystyle mq}
 
 one obtains 
 
 
 
 
 
 id
 
 
 E
 
 q
 
 α
 ,
 β
 
 
 
 
 =
 v
 
 q
 
 
 {\displaystyle {\text{id}}_{Eq_{\alpha ,\beta }}=v\,q}
 
.

This means that 
 
 
 
 I
 ≡
 E
 
 q
 
 α
 ,
 β
 
 
 
 
 {\displaystyle I\equiv Eq_{\alpha ,\beta }}
 
 and thus that 
 
 
 
 
 
 id
 
 
 I
 
 
 =
 q
 
 v
 
 
 {\displaystyle {\text{id}}_{I}=q\,v}
 
 equalizes 
 
 
 
 (
 α
 ,
 β
 )
 
 
 {\displaystyle (\alpha ,\beta )}
 
, whence 
 
 
 
 α
 =
 β
 
 
 {\displaystyle \alpha =\beta }
 
.

<h2 id="second-definition">Second definition</h2>
In a category 
 
 
 
 C
 
 
 {\displaystyle C}
 
 with all finite <a href="/facts/Limit_(category_theory)/JoLugLs3">limits</a> and <a href="/facts/Colimits/JoLugLs3">colimits</a>, the image is defined as the <a href="/facts/Equaliser_(mathematics)/4eD7pB8y">equalizer</a> 
 
 
 
 (
 I
 m
 ,
 m
 )
 
 
 {\displaystyle (Im,m)}
 
 of the so-called cokernel pair 
 
 
 
 (
 Y
 
 ⊔
 
 X
 
 
 Y
 ,
 
 i
 
 1
 
 
 ,
 
 i
 
 2
 
 
 )
 
 
 {\displaystyle (Y\sqcup _{X}Y,i_{1},i_{2})}
 
, which is the <a href="/facts/Pushout_(category_theory)/6wuns0Yo">cocartesian</a> of a morphism with itself over its domain, which will result in a pair of morphisms 
 
 
 
 
 i
 
 1
 
 
 ,
 
 i
 
 2
 
 
 :
 Y
 →
 Y
 
 ⊔
 
 X
 
 
 Y
 
 
 {\displaystyle i_{1},i_{2}:Y\to Y\sqcup _{X}Y}
 
, on which the <a href="/facts/Equalizer_(category_theory)/4eD7pB8y">equalizer</a> is taken, i.e. the first of the following diagrams is <a href="/facts/Pushout_(category_theory)/6wuns0Yo">cocartesian</a>, and the second <a href="/facts/Equalizer_(category_theory)/4eD7pB8y">equalizing</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

Remarks:

<ol><li>Finite <a href="/facts/Bicompleteness/wpQzAbpB">bicompleteness</a> of the category ensures that pushouts and equalizers exist.</li>
<li>
 
 
 
 (
 I
 m
 ,
 m
 )
 
 
 {\displaystyle (Im,m)}
 
 can be called regular image as 
 
 
 
 m
 
 
 {\displaystyle m}
 
 is a <a href="/facts/Regular_monomorphism/Q9BrtDTs">regular monomorphism</a>, i.e. the equalizer of a pair of morphisms. (Recall also that an equalizer is automatically a monomorphism).</li>
<li>In an abelian category, the cokernel pair property can be written 
 
 
 
 
 i
 
 1
 
 
 
 f
 =
 
 i
 
 2
 
 
 
 f
  
 ⇔
  
 (
 
 i
 
 1
 
 
 −
 
 i
 
 2
 
 
 )
 
 f
 =
 0
 =
 0
 
 f
 
 
 {\displaystyle i_{1}\,f=i_{2}\,f\ \Leftrightarrow \ (i_{1}-i_{2})\,f=0=0\,f}
 
 and the equalizer condition 
 
 
 
 
 i
 
 1
 
 
 
 m
 =
 
 i
 
 2
 
 
 
 m
  
 ⇔
  
 (
 
 i
 
 1
 
 
 −
 
 i
 
 2
 
 
 )
 
 m
 =
 0
 
 m
 
 
 {\displaystyle i_{1}\,m=i_{2}\,m\ \Leftrightarrow \ (i_{1}-i_{2})\,m=0\,m}
 
. Moreover, all monomorphisms are regular.</li></ol>

Theorem—If 
 
 
 
 f
 
 
 {\displaystyle f}
 
 always factorizes through regular monomorphisms, then the two definitions coincide.

Proof
First definition implies the second: Assume that (1) holds with 
 
 
 
 m
 
 
 {\displaystyle m}
 
 regular monomorphism.

<ul><li>Equalization: one needs to show that 
 
 
 
 
 i
 
 1
 
 
 
 m
 =
 
 i
 
 2
 
 
 
 m
 
 
 {\displaystyle i_{1}\,m=i_{2}\,m}
 
 . As the cokernel pair of 
 
 
 
 f
 ,
  
 
 i
 
 1
 
 
 
 f
 =
 
 i
 
 2
 
 
 
 f
 
 
 {\displaystyle f,\ i_{1}\,f=i_{2}\,f}
 
 and by previous proposition, since 
 
 
 
 C
 
 
 {\displaystyle C}
 
 has all equalizers, the arrow 
 
 
 
 e
 
 
 {\displaystyle e}
 
 in the factorization 
 
 
 
 f
 =
 m
 
 e
 
 
 {\displaystyle f=m\,e}
 
 is an <a href="/facts/Epimorphism/IrzonI01">epimorphism</a>, hence 
 
 
 
 
 i
 
 1
 
 
 
 f
 =
 
 i
 
 2
 
 
 
 f
  
 ⇒
  
 
 i
 
 1
 
 
 
 m
 =
 
 i
 
 2
 
 
 
 m
 
 
 {\displaystyle i_{1}\,f=i_{2}\,f\ \Rightarrow \ i_{1}\,m=i_{2}\,m}
 
.</li>
<li>Universality: in a category with all colimits (or at least all pushouts) 
 
 
 
 m
 
 
 {\displaystyle m}
 
 itself admits a cokernel pair 
 
 
 
 (
 Y
 
 ⊔
 
 I
 
 
 Y
 ,
 
 c
 
 1
 
 
 ,
 
 c
 
 2
 
 
 )
 
 
 {\displaystyle (Y\sqcup _{I}Y,c_{1},c_{2})}
 
</li></ul>

Moreover, as a regular monomorphism, 
 
 
 
 (
 I
 ,
 m
 )
 
 
 {\displaystyle (I,m)}
 
 is the equalizer of a pair of morphisms 
 
 
 
 
 b
 
 1
 
 
 ,
 
 b
 
 2
 
 
 :
 Y
 ⟶
 B
 
 
 {\displaystyle b_{1},b_{2}:Y\longrightarrow B}
 
 but we claim here that it is also the equalizer of 
 
 
 
 
 c
 
 1
 
 
 ,
 
 c
 
 2
 
 
 :
 Y
 ⟶
 Y
 
 ⊔
 
 I
 
 
 Y
 
 
 {\displaystyle c_{1},c_{2}:Y\longrightarrow Y\sqcup _{I}Y}
 
.
Indeed, by construction 
 
 
 
 
 b
 
 1
 
 
 
 m
 =
 
 b
 
 2
 
 
 
 m
 
 
 {\displaystyle b_{1}\,m=b_{2}\,m}
 
 thus the "cokernel pair" diagram for 
 
 
 
 m
 
 
 {\displaystyle m}
 
 yields a unique morphism 
 
 
 
 
 u
 ′
 
 :
 Y
 
 ⊔
 
 I
 
 
 Y
 ⟶
 B
 
 
 {\displaystyle u':Y\sqcup _{I}Y\longrightarrow B}
 
 such that 
 
 
 
 
 b
 
 1
 
 
 =
 
 u
 ′
 
 
 
 c
 
 1
 
 
 ,
  
 
 b
 
 2
 
 
 =
 
 u
 ′
 
 
 
 c
 
 2
 
 
 
 
 {\displaystyle b_{1}=u'\,c_{1},\ b_{2}=u'\,c_{2}}
 
. Now, a map 
 
 
 
 
 m
 ′
 
 :
 
 I
 ′
 
 ⟶
 Y
 
 
 {\displaystyle m':I'\longrightarrow Y}
 
 which equalizes 
 
 
 
 (
 
 c
 
 1
 
 
 ,
 
 c
 
 2
 
 
 )
 
 
 {\displaystyle (c_{1},c_{2})}
 
 also satisfies 
 
 
 
 
 b
 
 1
 
 
 
 
 m
 ′
 
 =
 
 u
 ′
 
 
 
 c
 
 1
 
 
 
 
 m
 ′
 
 =
 
 u
 ′
 
 
 
 c
 
 2
 
 
 
 
 m
 ′
 
 =
 
 b
 
 2
 
 
 
 
 m
 ′
 
 
 
 {\displaystyle b_{1}\,m'=u'\,c_{1}\,m'=u'\,c_{2}\,m'=b_{2}\,m'}
 
, hence by the equalizer diagram for 
 
 
 
 (
 
 b
 
 1
 
 
 ,
 
 b
 
 2
 
 
 )
 
 
 {\displaystyle (b_{1},b_{2})}
 
, there exists a unique map 
 
 
 
 
 h
 ′
 
 :
 
 I
 ′
 
 →
 I
 
 
 {\displaystyle h':I'\to I}
 
 such that 
 
 
 
 
 m
 ′
 
 =
 m
 
 
 h
 ′
 
 
 
 {\displaystyle m'=m\,h'}
 
.
Finally, use the cokernel pair diagram (of 
 
 
 
 f
 
 
 {\displaystyle f}
 
) with 
 
 
 
 
 j
 
 1
 
 
 :=
 
 c
 
 1
 
 
 ,
  
 
 j
 
 2
 
 
 :=
 
 c
 
 2
 
 
 ,
  
 Z
 :=
 Y
 
 ⊔
 
 I
 
 
 Y
 
 
 {\displaystyle j_{1}:=c_{1},\ j_{2}:=c_{2},\ Z:=Y\sqcup _{I}Y}
 
 : there exists a unique 
 
 
 
 u
 :
 Y
 
 ⊔
 
 X
 
 
 Y
 ⟶
 Y
 
 ⊔
 
 I
 
 
 Y
 
 
 {\displaystyle u:Y\sqcup _{X}Y\longrightarrow Y\sqcup _{I}Y}
 
 such that 
 
 
 
 
 c
 
 1
 
 
 =
 u
 
 
 i
 
 1
 
 
 ,
  
 
 c
 
 2
 
 
 =
 u
 
 
 i
 
 2
 
 
 
 
 {\displaystyle c_{1}=u\,i_{1},\ c_{2}=u\,i_{2}}
 
. Therefore, any map 
 
 
 
 g
 
 
 {\displaystyle g}
 
 which equalizes 
 
 
 
 (
 
 i
 
 1
 
 
 ,
 
 i
 
 2
 
 
 )
 
 
 {\displaystyle (i_{1},i_{2})}
 
 also equalizes 
 
 
 
 (
 
 c
 
 1
 
 
 ,
 
 c
 
 2
 
 
 )
 
 
 {\displaystyle (c_{1},c_{2})}
 
 and thus uniquely factorizes as 
 
 
 
 g
 =
 m
 
 
 h
 ′
 
 
 
 {\displaystyle g=m\,h'}
 
. This exactly means that 
 
 
 
 (
 I
 ,
 m
 )
 
 
 {\displaystyle (I,m)}
 
 is the equalizer of 
 
 
 
 (
 
 i
 
 1
 
 
 ,
 
 i
 
 2
 
 
 )
 
 
 {\displaystyle (i_{1},i_{2})}
 
.
Second definition implies the first:

<ul><li>Factorization: taking 
 
 
 
 
 m
 ′
 
 :=
 f
 
 
 {\displaystyle m':=f}
 
 in the equalizer diagram (
 
 
 
 
 m
 ′
 
 
 
 {\displaystyle m'}
 
 corresponds to 
 
 
 
 g
 
 
 {\displaystyle g}
 
), one obtains the factorization 
 
 
 
 f
 =
 m
 
 h
 
 
 {\displaystyle f=m\,h}
 
.</li>
<li>Universality: let 
 
 
 
 f
 =
 
 m
 ′
 
 
 
 e
 ′
 
 
 
 {\displaystyle f=m'\,e'}
 
 be a factorization with 
 
 
 
 
 m
 ′
 
 
 
 {\displaystyle m'}
 
 regular monomorphism, i.e. the equalizer of some pair 
 
 
 
 (
 
 d
 
 1
 
 
 ,
 
 d
 
 2
 
 
 )
 
 
 {\displaystyle (d_{1},d_{2})}
 
.</li></ul>

Then 
 
 
 
 
 d
 
 1
 
 
 
 
 m
 ′
 
 =
 
 d
 
 2
 
 
 
 
 m
 ′
 
  
 ⇒
  
 
 d
 
 1
 
 
 
 f
 =
 
 d
 
 1
 
 
 
 
 m
 ′
 
 
 e
 =
 
 d
 
 2
 
 
 
 
 m
 ′
 
 
 e
 =
 
 d
 
 2
 
 
 
 f
 
 
 {\displaystyle d_{1}\,m'=d_{2}\,m'\ \Rightarrow \ d_{1}\,f=d_{1}\,m'\,e=d_{2}\,m'\,e=d_{2}\,f}
 
 so that by the "cokernel pair" diagram (of 
 
 
 
 f
 
 
 {\displaystyle f}
 
), with 
 
 
 
 
 j
 
 1
 
 
 :=
 
 d
 
 1
 
 
 ,
  
 
 j
 
 2
 
 
 :=
 
 d
 
 2
 
 
 ,
  
 Z
 :=
 D
 
 
 {\displaystyle j_{1}:=d_{1},\ j_{2}:=d_{2},\ Z:=D}
 
, there exists a unique 
 
 
 
 
 u
 ″
 
 :
 Y
 
 ⊔
 
 X
 
 
 Y
 ⟶
 D
 
 
 {\displaystyle u'':Y\sqcup _{X}Y\longrightarrow D}
 
 such that 
 
 
 
 
 d
 
 1
 
 
 =
 
 u
 ″
 
 
 
 i
 
 1
 
 
 ,
  
 
 d
 
 2
 
 
 =
 
 u
 ″
 
 
 
 i
 
 2
 
 
 
 
 {\displaystyle d_{1}=u''\,i_{1},\ d_{2}=u''\,i_{2}}
 
.
Now, from 
 
 
 
 
 i
 
 1
 
 
 
 m
 =
 
 i
 
 2
 
 
 
 m
 
 
 {\displaystyle i_{1}\,m=i_{2}\,m}
 
 (m from the equalizer of (i1, i2) diagram), one obtains 
 
 
 
 
 d
 
 1
 
 
 
 m
 =
 
 u
 ″
 
 
 
 i
 
 1
 
 
 
 m
 =
 
 u
 ″
 
 
 
 i
 
 2
 
 
 
 m
 =
 
 d
 
 2
 
 
 
 m
 
 
 {\displaystyle d_{1}\,m=u''\,i_{1}\,m=u''\,i_{2}\,m=d_{2}\,m}
 
, hence by the universality in the (equalizer of (d1, d2) diagram, with f replaced by m), there exists a unique 
 
 
 
 v
 :
 I
 m
 ⟶
 
 I
 ′
 
 
 
 {\displaystyle v:Im\longrightarrow I'}
 
 such that 
 
 
 
 m
 =
 
 m
 ′
 
 
 v
 
 
 {\displaystyle m=m'\,v}
 
.

<h2 id="examples">Examples</h2>
In the <a href="/facts/Category_of_sets/LF5ykYl0">category of sets</a> the image of a morphism 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f\colon X\to Y}
 
 is the <a href="/facts/Inclusion_map/VUFNdaDH">inclusion</a> from the ordinary <a href="/facts/Image_(mathematics)/KANefMAl">image</a> 
 
 
 
 {
 f
 (
 x
 )
  
 
 |
 
  
 x
 ∈
 X
 }
 
 
 {\displaystyle \{f(x)~|~x\in X\}}
 
 to 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
. In many <a href="/facts/Concrete_category/4vYnbnjq">concrete categories</a> such as <a href="/facts/Category_of_groups/sxSd9Kbc">groups</a>, <a href="/facts/Category_of_abelian_groups/jF3sXRFn">abelian groups</a> and (left- or right) <a href="/facts/Module_(mathematics)/jP0LIhFL">modules</a>, the image of a morphism is the image of the correspondent morphism in the category of sets.
In any <a href="/facts/Normal_category/vSYyev04">normal category</a> with a <a href="/facts/Zero_object/oeckvXRj">zero object</a> and <a href="/facts/Kernel_(category_theory)/Tsn0uLfc">kernels</a> and <a href="/facts/Cokernel_(category_theory)/vYFQRPoh">cokernels</a> for every morphism, the image of a morphism 
 
 
 
 f
 
 
 {\displaystyle f}
 
 can be expressed as follows:

im f = ker coker f
In an <a href="/facts/Abelian_category/4WN473Yc">abelian category</a> (which is in particular binormal), if f is a monomorphism then f = ker coker f, and so f = im f.

<h2 id="essential-image">Essential Image</h2>
A related notion to image is essential image.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>
A subcategory 
 
 
 
 C
 ⊂
 B
 
 
 {\displaystyle C\subset B}
 
 of a (strict) category is said to be replete if for every 
 
 
 
 x
 ∈
 C
 
 
 {\displaystyle x\in C}
 
, and for every isomorphism 
 
 
 
 ι
 :
 x
 →
 y
 
 
 {\displaystyle \iota :x\to y}
 
, both 
 
 
 
 ι
 
 
 {\displaystyle \iota }
 
 and 
 
 
 
 y
 
 
 {\displaystyle y}
 
 belong to C. 
Given a functor 
 
 
 
 F
 :
 A
 →
 B
 
 
 {\displaystyle F\colon A\to B}
 
 between categories, the smallest <a href="/facts/Replete_subcategory/QoWpVgeD">replete subcategory</a> of the target n-category B containing the image of A under F. 

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Subobject/lEcCIhrP">Subobject</a></li>
<li><a href="/facts/Coimage/khlveI8X">Coimage</a></li>
<li><a href="/facts/Image_(mathematics)/KANefMAl">Image (mathematics)</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn: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 <a href="978-0-12-499250-4" target="_blank">978-0-12-499250-4</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn: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 <a href="978-0-12-499250-4" target="_blank">978-0-12-499250-4</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn: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 <a href="/wiki/Masaki_Kashiwara" target="_blank">/wiki/Masaki_Kashiwara</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">"essential image in nLab". ncatlab.org. Retrieved 2024-11-15. <a href="https://ncatlab.org/nlab/show/essential+image" target="_blank">https://ncatlab.org/nlab/show/essential+image</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
</ol>

Image (category theory) open-in-new

Image (category theory)