Image (category theory)
In category theory, a branch of mathematics, the image of a morphism is a generalization of the image of a function.
General definitionEdit
Given a category <math> C</math> and a morphism <math>f\colon X\to Y</math> in <math> C </math>, the image<ref>Template:Citation 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> 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.
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 equalizers then the <math> e</math> in the factorization <math> f= m\, e</math> of (1) is an epimorphism.<ref>Template:Citation Proposition 10.1 p.12</ref>
Template:Math proof= v\, q</math>.
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 definitionEdit
In a category <math> C</math> with all finite limits and colimits, the image is defined as the equalizer <math>(Im,m)</math> of the so-called cokernel pair <math> (Y \sqcup_X Y, i_1, i_2)</math>, which is the 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 is taken, i.e. the first of the following diagrams is cocartesian, and the second equalizing.<ref>Template:Citation Definition 5.1.1</ref>
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.
ExamplesEdit
In the category of sets the image of a morphism <math>f\colon X \to Y</math> is the inclusion from the ordinary image <math>\{f(x) ~|~ x \in X\}</math> to <math>Y</math>. 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 <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 ImageEdit
A related notion to image is essential image.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</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.