Image (category theory)

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

  1. There exists a morphism g:X\rightarrow I such that f = hg.
  2. For any object Z with a morphism k:X\rightarrow Z and a monomorphism l:Z\rightarrow Y such that f = lk, there exists a unique morphism m:I\rightarrow Z such that k = mg and h = lm.

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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.

[edit] Examples

In the category of sets the image of a morphism f : 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.

[edit] See also

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