Image (category theory)
From Wikipedia, the free encyclopedia
Given a category C and a morphism
in C, the image of f is a monomorphism
satisfying the following universal property:
- There exists a morphism
such that f = hg. - For any object Z with a morphism
and a monomorphism
such that f = lk, there exists a unique morphism
such that h = lm.
Note the following:
- g is unique.
- m is monic.
- h=lm already implies that m is unique.
- k=mg
The image of f is often denoted by im f or Im(f).
One can show that a morphism f is monic if and only if f = im f.
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
This holds especially in abelian categories.
See also [edit]
References [edit]
- Section I.10 of Mitchell, Barry (1965), Theory of categories, Pure and applied mathematics 17, Academic Press, ISBN 978-0-124-99250-4, MR 0202787
such that f = hg.
and a monomorphism
such that f = lk, there exists a unique morphism
such that h = lm.