Image (category theory)

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Given a category C and a morphism f\colon X\to Y in C, the image of f is a monomorphism h\colon I\to Y satisfying the following universal property:

  1. There exists a morphism g\colon X\to I such that f = gh (using postfix notation, i.e. the composition reads left to right).
  2. For any object Z with a morphism k\colon X\to Z and a monomorphism l\colon Z\to Y such that f = kl, there exists a unique morphism m\colon I\to Z such that h = ml.

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=ml already implies that m is unique.
  5. k=gm



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[edit]

In the category of sets the image of a morphism f\colon X \to Y is the inclusion from the ordinary image \{f(x) ~|~ x \in X\} to Y. 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 f 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