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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. , therefore 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 (1) 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 , which is the cocartesian of a morphism with itself over its domain, which will result in a pair of morphisms , 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. can be called regular image as 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 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 (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 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 Image

A related notion to image is essential image.

A subcategory of a (strict) category is said to be replete if for every , and for every isomorphism , both and belong to C.

Given a functor between categories, the smallest replete subcategory of the target n-category B containing the image of A under F.

See also

References