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 and a morphism in , the image
of is a monomorphism satisfying the following universal property:
  1. There exists a morphism such that.
  2. For any object with a morphism and a monomorphism such that, there exists a unique morphism such that.
Remarks:
  1. such a factorization does not necessarily exist.
  2. is unique by definition of monic.
  3. by monic.
  4. is monic.
  5. already implies that is unique.


The image of is often denoted by or.
Proposition: If has all equalizers then the in the factorization of is an epimorphism.

Second definition

In a category with all finite limits and colimits, the image is defined as the equalizer of the so-called cokernel pair
Remarks:
  1. Finite bicompleteness of the category ensures that pushouts and equalizers exist.
  2. can be called regular image as is a regular monomorphism, i.e. the equalizer of a pair of morphism..
  3. In an abelian category, the cokernel pair property can be written and the equalizer condition. Moreover, all monomorphisms are regular.

    Examples

In the category of sets the image of a morphism is the inclusion from the ordinary image to. In many concrete categories such as groups, abelian groups and 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 can be expressed as follows:

In an abelian category, if f is a monomorphism then f = ker coker f, and so f = im f.