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}

$C$ and a morphism

{displaystyle fcolon Xto Y}

$fcolon Xto Y$ in

{displaystyle C}

$C$ , the image^[1]
of

{displaystyle f}

$f$ is a monomorphism

{displaystyle mcolon Ito Y}

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

There exists a morphism ${displaystyle ecolon Xto I}$
For any object ${displaystyle I’}$

Remarks:

such a factorization does not necessarily exist.
${displaystyle e}$
${displaystyle m’e’=f=me=m’ve}$
${displaystyle v}$
${displaystyle m=m’,v}$

The image of

{displaystyle f}

$f$ is often denoted by

{displaystyle {text{Im}}f}

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

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

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

Proposition: If

{displaystyle C}

$C$ has all equalizers then the

{displaystyle e}

$e$ in the factorization

{displaystyle f=m,e}

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

Proof

Let

{displaystyle alpha ,,beta }

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

{displaystyle alpha ,e=beta ,e}

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

{displaystyle alpha =beta }

$alpha=beta$ . Since the equalizer of

{displaystyle (alpha ,beta )}

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

{displaystyle e}

$e$ factorizes as

{displaystyle e=q,e’}

${displaystyle e=q,e'}$ with

{displaystyle q}

$q$ monic. But then

{displaystyle f=(m,q),e’}

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

{displaystyle f}

$f$ with

{displaystyle (m,q)}

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

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

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

{displaystyle m=m,q,v}

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

{displaystyle m}

$m$ is monic

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

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

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

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

{displaystyle mq}

${displaystyle mq}$ one obtains

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

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

This means that

{displaystyle Iequiv Eq_{alpha ,beta }}

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

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

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

{displaystyle (alpha ,beta )}

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

{displaystyle alpha =beta }

${displaystyle alpha =beta }$ .

Second definition[edit]

In a category

{displaystyle C}

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

{displaystyle (Im,m)}

${displaystyle (Im,m)}$ of the so-called cokernel pair

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

${displaystyle (Ysqcup _{X}Y,i_{1},i_{2})}$ .^[3]

^{[clarification needed]}

Remarks:

Finite bicompleteness of the category ensures that pushouts and equalizers exist.
${displaystyle (Im,m)}$
In an abelian category, the cokernel pair property can be written ${displaystyle i_{1},f=i_{2},f Leftrightarrow (i_{1}-i_{2}),f=0=0,f}$

Theorem — If

{displaystyle f}

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

Proof

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

{displaystyle m}

$m$ regular monomorphism.

Equalization: one needs to show that ${displaystyle i_{1},m=i_{2},m}$
Universality: in a category with all colimits (or at least all pushouts) ${displaystyle m}$

Moreover, as a regular monomorphism,

{displaystyle (I,m)}

Indeed, by construction

{displaystyle b_{1},m=b_{2},m}

Finally, use the cokernel pair diagram (of

{displaystyle f}

Second definition implies the first:

Then

{displaystyle d_{1},m’=d_{2},m’ Rightarrow d_{1},f=d_{1},m’,e=d_{2},m’,e=d_{2},f}

Now, from

{displaystyle i_{1},m=i_{2},m}

Examples[edit]

In the category of sets the image of a morphism

{displaystyle fcolon Xto Y}

$fcolon Xto Y$ is the inclusion from the ordinary image

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

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

{displaystyle Y}

$Y$ . 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}

$f$ 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.

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

[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

Image (category theory) – Wikipedia

General definition[edit]

Second definition[edit]

Examples[edit]

See also[edit]

References[edit]

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