Image (category theory)

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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 m 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 will 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