Image (category theory)

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

## Contents

* 1 General definition
* 2 Second definition
* 3 Examples
* 4 See also
* 5 References

## General definition

Given a category C and a morphism f:X->Y in C, the image[1] of f is a monomorphism m:I->Y satisfying the following universal property:

1. There exists a morphism e:X->I such that f=me.
2. For any object I′ with a morphism e′:X->I′ and a monomorphism m′:I′->Y such that f=m′e′, there exists a unique morphism v:I->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 Imf or Im(f).

Proposition: If C has all equalizers then the e in the factorization f=me of (1) is an epimorphism.[2]

Proof

Let α,β be such that αe=βe, one needs to show that α=β. Since the equalizer of (α,β) exists, e factorizes as e=qe′ with q monic. But then f=(mq)e′ is a factorization of f with (mq) monomorphism. Hence by the universal property of the image there exists a unique arrow v:I->Eqα,β such that m=mqv and since m is monic idI=qv. Furthermore, one has mq=(mqv)q and by the monomorphism property of mq one obtains idEqα,β=vq.

This means that I≡Eqα,β and thus that idI=qv equalizes (α,β), whence α=β.

## 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⊔XY,i1,i2), which is the cocartesian of a morphism with itself over its domain, which will result in a pair of morphisms i1,i2:Y->Y⊔XY, 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.
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 i1f=i2f ⇔ (i1−i2)f=0=0f and the equalizer condition i1m=i2m ⇔ (i1−i2)m=0m. Moreover, all monomorphisms are regular.

Theorem -- If f always factorizes through regular monomorphisms, then the two definitions coincide.

Proof

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

* Equalization: one needs to show that i1m=i2m . As the cokernel pair of f, i1f=i2f and by previous proposition, since C has all equalizers, the arrow e in the factorization f=me is an epimorphism, hence i1f=i2f ⇒ i1m=i2m.
* Universality: in a category with all colimits (or at least all pushouts) m itself admits a cokernel pair (Y⊔IY,c1,c2)

Moreover, as a regular monomorphism, (I,m) is the equalizer of a pair of morphisms b1,b2:Y⟶B but we claim here that it is also the equalizer of c1,c2:Y⟶Y⊔IY.
Indeed, by construction b1m=b2m thus the "cokernel pair" diagram for m yields a unique morphism u′:Y⊔IY⟶B such that b1=u′c1, b2=u′c2. Now, a map m′:I′⟶Y which equalizes (c1,c2) also satisfies b1m′=u′c1m′=u′c2m′=b2m′, hence by the equalizer diagram for (b1,b2), there exists a unique map h′:I′->I such that m′=mh′.
Finally, use the cokernel pair diagram (of f) with j1:=c1, j2:=c2, Z:=Y⊔IY : there exists a unique u:Y⊔XY⟶Y⊔IY such that c1=ui1, c2=ui2. Therefore, any map g which equalizes (i1,i2) also equalizes (c1,c2) and thus uniquely factorizes as g=mh′. This exactly means that (I,m) is the equalizer of (i1,i2).

Second definition implies the first:

* Factorization: taking m′:=f in the equalizer diagram (m′ corresponds to g), one obtains the factorization f=mh.
* Universality: let f=m′e′ be a factorization with m′ regular monomorphism, i.e. the equalizer of some pair (d1,d2).

Then d1m′=d2m′ ⇒ d1f=d1m′e=d2m′e=d2f so that by the "cokernel pair" diagram (of f), with j1:=d1, j2:=d2, Z:=D, there exists a unique u″:Y⊔XY⟶D such that d1=u″i1, d2=u″i2.
Now, from i1m=i2m (m from the equalizer of (i1, i2) diagram), one obtains d1m=u″i1m=u″i2m=d2m, hence by the universality in the (equalizer of (d1, d2) diagram, with f replaced by m), there exists a unique v:Im⟶I′ such that m=m′v.

## Examples

In the category of sets the image of a morphism f:X->Y is the inclusion from the ordinary image {f(x)|x∈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)

## References

1. ↑ Mitchell, Barry (1965), Theory of categories, Pure and applied mathematics, 17, Academic Press, ISBN 978-0-12-499250-4 Section I.10 p.12
2. ↑ Mitchell, Barry (1965), Theory of categories, Pure and applied mathematics, 17, Academic Press, ISBN 978-0-12-499250-4 Proposition 10.1 p.12
3. ↑ Kashiwara, Masaki; Schapira, Pierre (2006), "Categories and Sheaves", Grundlehren der Mathematischen Wissenschaften, 332, Berlin Heidelberg: Springer, pp. 113–114 Definition 5.1.1

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