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Image (category theory)
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In [[category theory]], a branch of [[mathematics]], the '''image''' of a [[morphism]] is a generalization of the [[image (mathematics)|image]] of a [[Function (mathematics)|function]]. == General definition == Given a [[Category (mathematics)|category]] <math> C</math> and a [[morphism]] <math>f\colon X\to Y</math> in <math> C </math>, the '''image'''<ref>{{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</ref> of <math> f</math> is a [[monomorphism]] <math>m\colon I\to Y</math> satisfying the following [[universal property]]: #There exists a morphism <math>e\colon X\to I</math> such that <math>f = m\, e</math>. #For any object <math> I' </math> with a morphism <math>e'\colon X\to I'</math> and a monomorphism <math>m'\colon I'\to Y</math> such that <math>f = m'\, e'</math>, there exists a unique morphism <math>v\colon I\to I'</math> such that <math>m = m'\, v</math>. '''Remarks:''' # such a factorization does not necessarily exist. # <math> e</math> is unique by definition of <math> m</math> [[Monomorphism|monic]]. # <math>m'e'=f=me=m've</math>, therefore <math>e'=ve</math> by <math>m'</math> monic. # <math> v</math> is monic. # <math>m = m'\, v</math> already implies that <math> v</math> is unique. <!-- [[File:Image diagram category theory.svg]] great diagram but I needed to change the notations --> [[File:Image Theorie des catégories.png|400px|center]] The image of <math> f</math> is often denoted by <math>\text{Im} f</math> or <math>\text{Im} (f)</math>. '''Proposition:''' If <math> C</math> has all [[Equaliser (mathematics)|equalizers]] then the <math> e</math> in the factorization <math> f= m\, e</math> of (1) is an [[epimorphism]].<ref>{{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</ref> {{Math proof| Let <math> \alpha,\, \beta</math> be such that <math> \alpha\, e =\beta\, e</math>, one needs to show that <math> \alpha=\beta</math>. Since the equalizer of <math> (\alpha, \beta)</math> exists, <math> e</math> factorizes as <math> e= q\, e'</math> with <math> q</math> monic. But then <math> f= (m\, q)\, e'</math> is a factorization of <math> f</math> with <math> (m\, q)</math> monomorphism. Hence by the universal property of the image there exists a unique arrow <math> v: I \to Eq_{\alpha,\beta}</math> such that <math> m = m\,q\, v</math> and since <math> m</math> is monic <math> \text{id}_I = q\, v</math>. Furthermore, one has <math> m\, q = (m q v)\,q </math> and by the monomorphism property of <math> mq</math> one obtains <math> \text{id}_{Eq_{\alpha,\beta}}= v\, q</math>. [[File:E epimorphism.png|300px|center]] This means that <math> I \equiv Eq_{\alpha,\beta}</math> and thus that <math> \text{id}_I = q\, v</math> equalizes <math> (\alpha, \beta)</math>, whence <math> \alpha = \beta</math>. }} == Second definition == In a category <math> C</math> with all finite [[Limit (category theory)|limits]] and [[colimits]], the '''image''' is defined as the [[Equaliser (mathematics)#In category theory|equalizer]] <math>(Im,m)</math> of the so-called '''cokernel pair''' <math> (Y \sqcup_X Y, i_1, i_2)</math>, which is the [[Pushout (category theory)|cocartesian]] of a morphism with itself over its domain, which will result in a pair of morphisms <math>i_1,i_2:Y\to Y\sqcup_X Y</math>, on which the [[Equalizer (category theory)|equalizer]] is taken, i.e. the first of the following diagrams is [[Pushout (category theory)|cocartesian]], and the second [[Equalizer (category theory)|equalizing]].<ref>{{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</ref> [[File:Cokernel pair.png|500px|center]] [[File:Equalizer of the cokernel pair, diagram.png|500px|center]] '''Remarks:''' # Finite [[bicompleteness]] of the category ensures that pushouts and equalizers exist. # <math>(Im,m)</math> can be called '''regular image''' as <math>m</math> is a [[regular monomorphism]], i.e. the equalizer of a pair of morphisms. (Recall also that an equalizer is automatically a monomorphism). # In an abelian category, the cokernel pair property can be written <math>i_1\, f = i_2\, f\ \Leftrightarrow\ (i_1 - i_2)\, f = 0 = 0\, f </math> and the equalizer condition <math> i_1\, m = i_2\, m\ \Leftrightarrow\ (i_1 - i_2)\, m = 0 \, m</math>. Moreover, all monomorphisms are regular. {{Math theorem|If <math>f</math> always factorizes through regular monomorphisms, then the two definitions coincide. }} {{Math proof| '''First definition implies the second:''' Assume that ''(1)'' holds with <math>m</math> regular monomorphism. <!-- I got the proof from this site, dual statement https://qchu.wordpress.com/2012/11/03/regular-and-effective-monomorphisms-and-epimorphisms/ "Proposition: Let f : a \to b be a morphism with a kernel pair a \times_b a. Then f is a regular epimorphism if and only if it is an effective epimorphism." --> * '''Equalization:''' one needs to show that <math> i_1\, m= i_2\, m</math> . As the cokernel pair of <math>f,\ i_1\, f= i_2\, f</math> and by previous proposition, since <math>C</math> has all equalizers, the arrow <math> e </math> in the factorization <math> f= m\, e</math> is an [[epimorphism]], hence <math> i_1\, f= i_2\, f\ \Rightarrow \ i_1\, m= i_2\, m</math>. * '''Universality:''' in a category with all colimits (or at least all pushouts) <math>m</math> itself admits a cokernel pair <math> (Y \sqcup_{I}Y, c_1, c_2) </math> [[File:Cokernel pair m.png|450px|center]] :Moreover, as a regular monomorphism, <math>(I,m)</math> is the equalizer of a pair of morphisms <math> b_1, b_2: Y \longrightarrow B</math> but we claim here that it is also the equalizer of <math> c_1, c_2: Y \longrightarrow Y \sqcup_{I}Y</math>. :Indeed, by construction <math> b_1\, m = b_2\, m</math> thus the "cokernel pair" diagram for <math> m</math> yields a unique morphism <math> u': Y \sqcup_{I}Y \longrightarrow B </math> such that <math> b_1 = u'\, c_1,\ b_2 = u'\, c_2</math>. Now, a map <math> m': I'\longrightarrow Y</math> which equalizes <math> (c_1, c_2)</math> also satisfies <math> b_1\, m'= u'\, c_1 \, m'= u'\, c_2\, m'= b_2\, m'</math>, hence by the equalizer diagram for <math> (b_1, b_2)</math>, there exists a unique map <math> h': I'\to I </math> such that <math> m'= m\, h'</math>. :Finally, use the cokernel pair diagram (of <math>f</math>) with <math>j_1 := c_1,\ j_2 := c_2,\ Z:= Y\sqcup_I Y</math> : there exists a unique <math> u: Y \sqcup_{X}Y \longrightarrow Y\sqcup_I Y </math> such that <math> c_1 = u\, i_1,\ c_2 = u\, i_2</math>. Therefore, any map <math>g</math> which equalizes <math> (i_1, i_2)</math> also equalizes <math> (c_1, c_2)</math> and thus uniquely factorizes as <math>g= m\, h' </math>. This exactly means that <math> (I,m) </math> is the equalizer of <math> (i_1, i_2)</math>. '''Second definition implies the first:''' <!-- Again, I dualized the proof from http://math.stackexchange.com/questions/1783932/regular-coimages-in-a-finitely-complete-and-finitely-cocomplete-category --> * '''Factorization:''' taking <math> m' := f</math> in the equalizer diagram (<math> m' </math> corresponds to <math> g </math>), one obtains the factorization <math> f = m\, h </math>. * '''Universality:''' let <math> f = m'\, e' </math> be a factorization with <math>m' </math> regular monomorphism, i.e. the equalizer of some pair <math> (d_1, d_2) </math>. [[File:Equalizerd1d2.png|450px|center]] :Then <math> d_1\, m'= d_2\, m'\ \Rightarrow \ d_1\, f=d_1\, m'\, e= d_2\, m'\, e= d_2\, f</math> so that by the "cokernel pair" diagram (of <math>f</math>), with <math>j_1 := d_1,\ j_2 := d_2,\ Z:= D</math>, there exists a unique <math> u'': Y \sqcup_{X}Y \longrightarrow D </math> such that <math> d_1 = u''\, i_1,\ d_2 = u''\, i_2</math>. :Now, from <math> i_1\, m= i_2\, m</math> (''m'' from the equalizer of (''i<sub>1</sub>, i<sub>2</sub>'') diagram), one obtains <math> d_1\, m= u''\, i_1\, m = u''\, i_2\, m = d_2\, m</math>, hence by the universality in the (equalizer of (''d<sub>1</sub>, d<sub>2</sub>'') diagram, with ''f'' replaced by ''m''), there exists a unique <math> v: Im \longrightarrow I'</math> such that <math> m = m'\, v</math>. }} == Examples == In the [[category of sets]] the image of a morphism <math>f\colon X \to Y</math> is the [[inclusion map|inclusion]] from the ordinary [[image (mathematics)|image]] <math>\{f(x) ~|~ x \in X\}</math> to <math>Y</math>. In many [[Concrete category|concrete categories]] such as [[Category of groups|groups]], [[Category of abelian groups|abelian groups]] and (left- or right) [[Module (mathematics)|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 [[Kernel (category theory)|kernels]] and [[Cokernel (category theory)|cokernels]] for every morphism, the image of a morphism <math>f</math> 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''. == Essential Image == A related notion to image is ''essential image.''<ref>{{Cite web |title=essential image in nLab |url=https://ncatlab.org/nlab/show/essential+image |access-date=2024-11-15 |website=ncatlab.org}}</ref> A subcategory <math>C \subset B</math> of a (strict) category is said to be '''replete''' if for every <math>x \in C</math>, and for every isomorphism <math>\iota: x \to y</math>, both <math>\iota </math> and <math>y</math> belong to C. Given a functor <math>F \colon A \to B</math> between categories, the smallest [[replete subcategory]] of the target n-category B containing the image of A under F. == See also == *[[Subobject]] *[[Coimage]] *[[Image (mathematics)]] == References == {{Reflist}} [[Category:Category theory]]
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