Image (category theory)

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

and a morphism

, the image[1] of

satisfying the following universal property: Remarks: The image of

α = β

( α , β )

Hence by the universal property of the image there exists a unique arrow

α , β

and by the monomorphism property of

α , β

α , β

( α , β )

α = β

In a category

with all finite limits and colimits, the image is defined as the equalizer

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.

[3] Remarks: Theorem — If

always factorizes through regular monomorphisms, then the two definitions coincide.

First definition implies the second: Assume that (1) holds with

regular monomorphism.

Second definition implies the first: In the category of sets the image of a morphism

is the inclusion from the ordinary image

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: In an abelian category (which is in particular binormal), if f is a monomorphism then f = ker coker f, and so f = im f. A related notion to image is essential image.

of a (strict) category is said to be replete if for every

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.