Image (category theory)
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 .
- For any object Z with a morphism and a monomorphism such that , there exists a unique morphism such that .
Remarks:
- such a factorization does not necessarily exist
- g is unique by definition of monic (= left invertible, abstraction of injectivity)
- m is monic.
- h=lm already implies that m is unique.
The image of f is often denoted by im f or Im(f).
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
- 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.
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