Image (category theory)

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

General Definition[edit]

Given a category and a morphism in , the image [1] 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. is monic.
  4. already implies that is unique.
Image Theorie des catégories.pngNumérotation (1).png

The image of is often denoted by or .

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

Second definition[edit]

In a category with all finite limits and colimits, the image is defined as the equalizer of the so-called cokernel pair .[3]

Cokernel pair.png
Image, Equalizer of the cokernel pair.png

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. (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.

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

Examples[edit]

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.

See also[edit]

References[edit]

  1. ^ Mitchell, Barry (1965), Theory of categories, Pure and applied mathematics, 17, Academic Press, ISBN 978-0-124-99250-4, MR 0202787  Section I.10 p.12
  2. ^ Mitchell, Barry (1965), Theory of categories, Pure and applied mathematics, 17, Academic Press, ISBN 978-0-124-99250-4, MR 0202787  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