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Image (category theory) - Wikipedia

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

Given a category C {\displaystyle C} and a morphism f : X → Y {\displaystyle f\colon X\to Y} in C {\displaystyle C} , the image^[1] of f {\displaystyle f} is a monomorphism m : I → Y {\displaystyle m\colon I\to Y} satisfying the following universal property:

1. There exists a morphism e : X → I {\displaystyle e\colon X\to I} such that f = m e {\displaystyle f=m\,e} .
2. For any object I ′ {\displaystyle I'} with a morphism e ′ : X → I ′ {\displaystyle e'\colon X\to I'} and a monomorphism m ′ : I ′ → Y {\displaystyle m'\colon I'\to Y} such that f = m ′ e ′ {\displaystyle f=m'\,e'} , there exists a unique morphism v : I → I ′ {\displaystyle v\colon I\to I'} such that m = m ′ v {\displaystyle m=m'\,v} .

Remarks:

1. such a factorization does not necessarily exist.
2. e {\displaystyle e} is unique by definition of m {\displaystyle m} monic.
3. m ′ e ′ = f = m e = m ′ v e {\displaystyle m'e'=f=me=m've} , therefore e ′ = v e {\displaystyle e'=ve} by m ′ {\displaystyle m'} monic.
4. v {\displaystyle v} is monic.
5. m = m ′ v {\displaystyle m=m'\,v} already implies that v {\displaystyle v} is unique.

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 equalizers then the e {\displaystyle e} in the factorization f = m e {\displaystyle f=m\,e} of (1) is an epimorphism.^[2]

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 } .

In a category C {\displaystyle C} with all finite limits and colimits, the image is defined as the equalizer ( 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 cocartesian 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 equalizer is taken, i.e. the first of the following diagrams is cocartesian, and the second equalizing.^[3]

Remarks:

1. Finite bicompleteness of the category ensures that pushouts and equalizers exist.
2. ( I m , m ) {\displaystyle (Im,m)} can be called regular image as m {\displaystyle m} is a regular monomorphism, i.e. the equalizer of a pair of morphisms. (Recall also that an equalizer is automatically a monomorphism).
3. 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.

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 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 epimorphism, 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} .
• 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})}

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:

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} .

In the category of sets the image of a morphism f : X → Y {\displaystyle f\colon X\to Y} is the inclusion from the ordinary image { f ( x )   |   x ∈ X } {\displaystyle \{f(x)~|~x\in X\}} to 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.

A related notion to image is essential image.^[4]

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 replete subcategory of the target n-category B containing the image of A under F.

• Subobject
• Coimage
• Image (mathematics)

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
4. ^ "essential image in nLab". ncatlab.org. Retrieved 2024-11-15.

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