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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

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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. , 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.[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 .

This means that and thus that equalizes , whence .

Second definition

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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.[3]

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.

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
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 .
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

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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

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References

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  1. ^ Mitchell, Barry (1965), Theory of categories, Pure and applied mathematics, vol. 17, Academic Press, ISBN 978-0-12-499250-4, MR 0202787 Section I.10 p.12
  2. ^ Mitchell, Barry (1965), Theory of categories, Pure and applied mathematics, vol. 17, Academic Press, ISBN 978-0-12-499250-4, MR 0202787 Proposition 10.1 p.12
  3. ^ Kashiwara, Masaki; Schapira, Pierre (2006), "Categories and Sheaves", Grundlehren der Mathematischen Wissenschaften, vol. 332, Berlin Heidelberg: Springer, pp. 113–114 Definition 5.1.1