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 .


  1. such a factorization does not necessarily exist
  2. g is unique by definition of monic
  3. m is monic.
  4. h=lm already implies that m is unique.

Image diagram category theory.svg

The image of f is often denoted by im f or Im(f).


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[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.