Image (category theory) explained

In category theory, a branch of mathematics, the image of a morphism is a generalization of the image of a function.

General definition

C and a morphism f\colon X\to Y in C , the image^[1] 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 .

1. 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.

e is unique by definition of m monic.m'e' = f = me = m've , therefore e' = ve by m' monic.v is monic.m = m' v already implies that v is unique.

The image of

f is often denoted by Im f or Im (f) .

Proposition: If

C has all equalizers then the e in the factorization f = m e of (1) is an epimorphism.^[2]

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.^[3]

Remarks:

1. Finite bicompleteness of the category ensures that pushouts and equalizers exist.

(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).

1. 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.

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.

See also

• Subobject
• Coimage
• Image (mathematics)

Notes and References

1. Section I.10 p.12
2. Proposition 10.1 p.12
3. Definition 5.1.1