Image (category theory)

Summary

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.

Essential Image

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A related notion to image is essential image.[4]

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

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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
  4. ^ "essential image in nLab". ncatlab.org. Retrieved 2024-11-15.