Image (category Theory)
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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[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. by monic.
  4. is monic.
  5. 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]

Proof —

Let be such that , one needs to show that . Since the equalizer of exists, factorizes as with monic. But then is a factorization of with monomorphism. Hence by the universal property of the image there exists a unique arrow such that and since is monic . Furthermore, one has and by the monomorphism property of one obtains .

E epimorphism.png

This means that and thus that equalizes , whence .

Second definition

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
Equalizer of the cokernel pair, diagram.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.

Proof —

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

  • Equalization: one needs to show that . As the cokernel pair of and by previous proposition, since has all equalizers, the arrow in the factorization is an epimorphism, hence .
  • Universality: in a category with all colimits (or at least all pushouts) itself admits a cokernel pair
Cokernel pair m.png
Moreover, as a regular monomorphism, is the equalizer of a pair of morphisms but we claim here that it is also the equalizer of .
Indeed, by construction thus the "cokernel pair" diagram for yields a unique morphism such that . Now, a map which equalizes also satisfies , hence by the equalizer diagram for , there exists a unique map such that .
Finally, use the cokernel pair diagram (of ) with  : there exists a unique such that . Therefore, any map which equalizes also equalizes and thus uniquely factorizes as . This exactly means that is the equalizer of .

Second definition implies the first:

  • Factorization: taking in the equalizer diagram ( corresponds to ), one obtains the factorization .
  • Universality: let be a factorization with regular monomorphism, i.e. the equalizer of some pair .
Equalizerd1d2.png
Then so that by the "cokernel pair" diagram (of ), with , there exists a unique such that .
Now, from (m from the equalizer of (i1, i2) diagram), one obtains , hence by the universality in the (equalizer of (d1, d2) diagram, with f replaced by m), there exists a unique such that .

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.

See also

References

  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

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Image_(category_theory)
 



 



 
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