Image (category theory) – Wikipedia
In category theory, a branch of mathematics, the image of a morphism is a generalization of the image of a function.
Given a category
and a morphism
, the image
is a monomorphism
satisfying the following universal property:
- There exists a morphism
such that .
- For any object
with a morphism and a monomorphism such that , there exists a unique morphism such that .
- such a factorization does not necessarily exist.
is unique by definition of monic.
, therefore by monic.
already implies that is unique.
The image of
is often denoted by
has all equalizers then the
in the factorization
of (1) is an epimorphism.
be such that
, one needs to show that
. Since the equalizer of
monic. But then
is a factorization of
monomorphism. Hence by the universal property of the image there exists a unique arrow
. Furthermore, one has
and by the monomorphism property of
This means that
and thus that
In a category
with all finite limits and colimits, the image is defined as the equalizer
of the so-called cokernel pair
- Finite bicompleteness of the category ensures that pushouts and equalizers exist.
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).
- 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.
First definition implies the second: Assume that (1) holds with
- 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:
- 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 .
In the category of sets the image of a morphism
is the inclusion from the ordinary image
. 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.
- ^ 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
- ^ 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
- ^ Kashiwara, Masaki; Schapira, Pierre (2006), “Categories and Sheaves”, Grundlehren der Mathematischen Wissenschaften, vol. 332, Berlin Heidelberg: Springer, pp. 113–114 Definition 5.1.1