In category theory, a branch of mathematics, the image of a morphism is a generalization of the image of a function.

General definition

Given a category C and a morphism in C, the image of f is a monomorphism satisfying the following universal property: Remarks: The image of f is often denoted by \text{Im} f or. Proposition: If C has all equalizers then the e in the factorization f= m, e of (1) is an epimorphism.

Second definition

In a category C with all finite limits and colimits, the image is defined as the equalizer (Im,m) of the so-called cokernel pair, which is the cocartesian of a morphism with itself over its domain, which will result in a pair of morphisms , on which the equalizer is taken, i.e. the first of the following diagrams is cocartesian, and the second equalizing. Remarks:

Examples

In the category of sets the image of a morphism is the inclusion from the ordinary image 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: In an abelian category (which is in particular binormal), if f is a monomorphism then f = ker coker f, and so f = im f.

Essential Image

A related notion to image is essential image. A subcategory C \subset B of a (strict) category is said to be replete if for every x \in C, and for every isomorphism, both \iota and y belong to C. Given a functor between categories, the smallest replete subcategory of the target n-category B containing the image of A under F.