Image (category theory)#Essential Image

Given a category $C$ and a morphism $f\backslash colon\; X\backslash to\; Y$ in $C$, the image{{Citation| last=Mitchell| first=Barry| title=Theory of categories|publisher=Academic Press| series=Pure and applied mathematics| isbn=978-0-12-499250-4| mr=0202787| year=1965| volume=17}} Section I.10 p.12

of $f$ is a monomorphism $m\backslash colon\; I\backslash to\; Y$ satisfying the following universal property:

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

Remarks:

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

File:Image Theorie des catégories.png

The image of $f$ is often denoted by $\backslash text\{Im\}\; f$ or $\backslash text\{Im\}\; (f)$.

Proposition: If $C$ has all equalizers then the $e$ in the factorization $f=\; m\backslash ,\; e$ of (1) is an epimorphism.{{Citation| last=Mitchell| first=Barry| title=Theory of categories|publisher=Academic Press| series=Pure and applied mathematics| isbn=978-0-12-499250-4| mr=0202787| year=1965| volume=17}} Proposition 10.1 p.12

{{Math proof|

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

File:E epimorphism.png

This means that $I\; \backslash equiv\; Eq\_\{\backslash alpha,\backslash beta\}$ and thus that $\backslash text\{id\}\_I\; =\; q\backslash ,\; v$ equalizes $(\backslash alpha,\; \backslash beta)$, whence $\backslash alpha\; =\; \backslash beta$.

}}

In a category $C$ with all finite limits and colimits, the image is defined as the equalizer $(Im,m)$ of the so-called cokernel pair $(Y\; \backslash sqcup\_X\; Y,\; i\_1,\; i\_2)$, which is the cocartesian of a morphism with itself over its domain, which will result in a pair of morphisms $i\_1,i\_2:Y\backslash to\; Y\backslash sqcup\_X\; Y$, on which the equalizer is taken, i.e. the first of the following diagrams is cocartesian, and the second equalizing.{{Citation|last= Kashiwara|first= Masaki| author1link = Masaki Kashiwara|author2link = Pierre Schapira (mathematician)|last2 = Schapira |first2= Pierre|title="Categories and Sheaves"|year=2006|publisher=Springer| series = Grundlehren der Mathematischen Wissenschaften| place= Berlin Heidelberg|pages = 113–114|volume=332}} Definition 5.1.1

File:Cokernel pair.png

File:Equalizer of the cokernel pair, diagram.png

Remarks:

1. Finite bicompleteness of the category ensures that pushouts and equalizers exist.
2. $(Im,m)$ can be called regular image as $m$ 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 $i\_1\backslash ,\; f\; =\; i\_2\backslash ,\; f\backslash \; \backslash Leftrightarrow\backslash \; (i\_1\; -\; i\_2)\backslash ,\; f\; =\; 0\; =\; 0\backslash ,\; f$ and the equalizer condition $i\_1\backslash ,\; m\; =\; i\_2\backslash ,\; m\backslash \; \backslash Leftrightarrow\backslash \; (i\_1\; -\; i\_2)\backslash ,\; m\; =\; 0\; \backslash ,\; m$. Moreover, all monomorphisms are regular.

{{Math theorem|If $f$ always factorizes through regular monomorphisms, then the two definitions coincide.

}}

{{Math proof|

First definition implies the second: Assume that (1) holds with $m$ regular monomorphism.

• Equalization: one needs to show that $i\_1\backslash ,\; m=\; i\_2\backslash ,\; m$ . As the cokernel pair of $f,\backslash \; i\_1\backslash ,\; f=\; i\_2\backslash ,\; f$ and by previous proposition, since $C$ has all equalizers, the arrow $e$ in the factorization $f=\; m\backslash ,\; e$ is an epimorphism, hence $i\_1\backslash ,\; f=\; i\_2\backslash ,\; f\backslash \; \backslash Rightarrow\; \backslash \; i\_1\backslash ,\; m=\; i\_2\backslash ,\; m$.
• Universality: in a category with all colimits (or at least all pushouts) $m$ itself admits a cokernel pair $(Y\; \backslash sqcup\_\{I\}Y,\; c\_1,\; c\_2)$

File:Cokernel pair m.png

:Moreover, as a regular monomorphism, $(I,m)$ is the equalizer of a pair of morphisms $b\_1,\; b\_2:\; Y\; \backslash longrightarrow\; B$ but we claim here that it is also the equalizer of $c\_1,\; c\_2:\; Y\; \backslash longrightarrow\; Y\; \backslash sqcup\_\{I\}Y$.

:Indeed, by construction $b\_1\backslash ,\; m\; =\; b\_2\backslash ,\; m$ thus the "cokernel pair" diagram for $m$ yields a unique morphism $u\text{'}:\; Y\; \backslash sqcup\_\{I\}Y\; \backslash longrightarrow\; B$ such that $b\_1\; =\; u\text{'}\backslash ,\; c\_1,\backslash \; b\_2\; =\; u\text{'}\backslash ,\; c\_2$. Now, a map $m\text{'}:\; I\text{'}\backslash longrightarrow\; Y$ which equalizes $(c\_1,\; c\_2)$ also satisfies $b\_1\backslash ,\; m\text{'}=\; u\text{'}\backslash ,\; c\_1\; \backslash ,\; m\text{'}=\; u\text{'}\backslash ,\; c\_2\backslash ,\; m\text{'}=\; b\_2\backslash ,\; m\text{'}$, hence by the equalizer diagram for $(b\_1,\; b\_2)$, there exists a unique map $h\text{'}:\; I\text{'}\backslash to\; I$ such that $m\text{'}=\; m\backslash ,\; h\text{'}$.

:Finally, use the cokernel pair diagram (of $f$) with $j\_1\; :=\; c\_1,\backslash \; j\_2\; :=\; c\_2,\backslash \; Z:=\; Y\backslash sqcup\_I\; Y$ : there exists a unique $u:\; Y\; \backslash sqcup\_\{X\}Y\; \backslash longrightarrow\; Y\backslash sqcup\_I\; Y$ such that $c\_1\; =\; u\backslash ,\; i\_1,\backslash \; c\_2\; =\; u\backslash ,\; i\_2$. Therefore, any map $g$ which equalizes $(i\_1,\; i\_2)$ also equalizes $(c\_1,\; c\_2)$ and thus uniquely factorizes as $g=\; m\backslash ,\; h\text{'}$. This exactly means that $(I,m)$ is the equalizer of $(i\_1,\; i\_2)$.

Second definition implies the first:

• Factorization: taking $m\text{'}\; :=\; f$ in the equalizer diagram ($m\text{'}$ corresponds to $g$), one obtains the factorization $f\; =\; m\backslash ,\; h$.
• Universality: let $f\; =\; m\text{'}\backslash ,\; e\text{'}$ be a factorization with $m\text{'}$ regular monomorphism, i.e. the equalizer of some pair $(d\_1,\; d\_2)$.

File:Equalizerd1d2.png

:Then $d\_1\backslash ,\; m\text{'}=\; d\_2\backslash ,\; m\text{'}\backslash \; \backslash Rightarrow\; \backslash \; d\_1\backslash ,\; f=d\_1\backslash ,\; m\text{'}\backslash ,\; e=\; d\_2\backslash ,\; m\text{'}\backslash ,\; e=\; d\_2\backslash ,\; f$ so that by the "cokernel pair" diagram (of $f$), with $j\_1\; :=\; d\_1,\backslash \; j\_2\; :=\; d\_2,\backslash \; Z:=\; D$, there exists a unique $u$: Y \sqcup_{X}Y \longrightarrow D such that $d\_1\; =\; u$\, i_1,\ d_2 = u''\, i_2.

:Now, from $i\_1\backslash ,\; m=\; i\_2\backslash ,\; m$ (m from the equalizer of (i₁, i₂) diagram), one obtains $d\_1\backslash ,\; m=\; u$\, 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 $v:\; Im\; \backslash longrightarrow\; I\text{'}$ such that $m\; =\; m\text{'}\backslash ,\; v$.

}}

In the category of sets the image of a morphism $f\backslash colon\; X\; \backslash to\; Y$ is the inclusion from the ordinary image $\backslash \{f(x)\; ~|~\; x\; \backslash in\; X\backslash \}$ to $Y$. 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 $f$ 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.{{Cite web |title=essential image in nLab |url=https://ncatlab.org/nlab/show/essential+image |access-date=2024-11-15 |website=ncatlab.org}}

A subcategory $C\; \backslash subset\; B$ of a (strict) category is said to be replete if for every $x\; \backslash in\; C$, and for every isomorphism $\backslash iota:\; x\; \backslash to\; y$, both $\backslash iota$ and $y$ belong to C.

Given a functor $F\; \backslash colon\; A\; \backslash to\; B$ between categories, the smallest replete subcategory of the target n-category B containing the image of A under F.

• Subobject
• Coimage
• Image (mathematics)

{{Reflist}}

Category:Category theory