Image (category theory)

In , a branch of mathematics, the image of a morphism is a generalization of the of a function. Given a and a morphism in , the image of is a monomorphism satisfying the following universal property: There exists a morphism such that . For any object with a morphism and a monomorphism such that , there exists a unique morphism such that . Remarks: such a factorization does not necessarily exist. is unique by definition of monic. therefore by monic. is monic. already implies that is unique. The image of is often denoted by or . Proposition: If has all equalizers then the in the factorization of (1) is an epimorphism. Let be such that , one needs to show that . Since the equalizer of exists, factorizes as with monic. But then is a factorization of with monomorphism. Hence by the universal property of the image there exists a unique arrow such that and since is monic . Furthermore, one has and by the monomorphism property of one obtains . This means that and thus that equalizes , whence . In a category with all finite and colimits, the image is defined as the of the so-called cokernel pair , which is the of a morphism with itself over its domain, which will result in a pair of morphisms , on which the is taken, i.e. the first of the following diagrams is , and the second . Remarks: Finite bicompleteness of the category ensures that pushouts and equalizers exist. can be called regular image as is a regular monomorphism, i.e. the equalizer of a pair of morphisms. (Recall also that an equalizer is automatically a monomorphism). In an abelian category, the cokernel pair property can be written and the equalizer condition . Moreover, all monomorphisms are regular. If always factorizes through regular monomorphisms, then the two definitions coincide. First definition implies the second: Assume that (1) holds with regular monomorphism. Equalization: one needs to show that . As the cokernel pair of and by previous proposition, since has all equalizers, the arrow in the factorization is an epimorphism, hence .