Image (category theory)

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 formula_1 and a morphism formula_2 in formula_3, the image
of formula_4 is a monomorphism formula_5 satisfying the following universal property:
Remarks:
The image of formula_4 is often denoted by formula_22 or formula_23.
Proposition: If formula_1 has all equalizers then the formula_14 in the factorization formula_26 of "(1)" is an epimorphism.
= v\, q</math>.
This means that formula_27 and thus that formula_28 equalizes formula_29, whence formula_30.
Second definition.
In a category formula_1 with all finite limits and colimits, the image is defined as the equalizer formula_32 of the so-called cokernel pair formula_33.
Remarks: 
Examples.
In the category of sets the image of a morphism formula_38 is the inclusion from the ordinary image formula_39 to formula_40. 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 formula_41 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".