Image (category theory)

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Given a category C and a morphism in C, the image of f is a monomorphism satisfying the following universal property:

  1. There exists a morphism such that .
  2. For any object Z 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. g is unique by definition of monic (= left invertible, abstraction of injectivity)
  3. m is monic.
  4. h=lm already implies that m is unique.
  5. k=mg



Image diagram category theory.svg

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

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

References