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 definition
Given a category $$ C$$ and a morphism $$f\colon X\to Y$$ in $$ C $$, the image of $$ f$$ is a monomorphism $$m\colon I\to Y$$ satisfying the following universal property:
 * 1) There exists a morphism $$e\colon X\to I$$ such that $$f = m\, e$$.
 * 2) For any object $$ I' $$ with a morphism $$e'\colon X\to I'$$ and a monomorphism $$m'\colon I'\to Y$$ such that $$f = m'\, e'$$, there exists a unique morphism $$v\colon I\to I'$$ such that $$m = m'\, v$$.

Remarks:
 * 1) such a factorization does not necessarily exist.
 * 2) $$ e$$ is unique by definition of $$ m$$ monic.
 * 3) $$m'e'=f=me=m've$$, therefore $$e'=ve$$ by $$m'$$ monic.
 * 4) $$ v$$ is monic.
 * 5) $$m = m'\, v$$ already implies that $$ v$$ is unique.



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

Proposition: If $$ C$$ has all equalizers then the $$ e$$ in the factorization $$ f= m\, e$$ of (1) is an epimorphism.

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Second definition
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 $$ (Y \sqcup_X Y, i_1, i_2)$$, which is the cocartesian of a morphism with itself over its domain, which will result in a pair of morphisms $$i_1,i_2:Y\to Y\sqcup_X Y$$, on which the equalizer is taken, i.e. the first of the following diagrams is cocartesian, and the second equalizing.



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 $$i_1\, f = i_2\, f\ \Leftrightarrow\ (i_1 - i_2)\, f = 0 = 0\, f $$ and the equalizer condition $$ i_1\, m = i_2\, m\ \Leftrightarrow\ (i_1 - i_2)\, m = 0 \, m$$. Moreover, all monomorphisms are regular.

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