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Given a category C and a morphism in C, the image of f is a monomorphism satisfying the following universal property: The word category (plural categories; from Greek κατηγορια meaning assertion or accusation, hence categorical denial) has several meanings: it is used informally to mean a class of things, as in the category of all living things. See categorization. ...
In mathematics, a morphism is an abstraction of a function or mapping between two spaces. ...
In the context of abstract algebra or universal algebra, a monomorphism is simply an injective homomorphism. ...
- There exists a morphism such that f = hg.
- For any object Z with a morphism and a monomorphism such that f = lk, there exists a unique morphism such that k = mg and h = lm.
The image of f is often denoted by im f.
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 mathematics, the category of sets is the category whose objects are all sets and whose morphisms are all functions. ...
In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. ...
In mathematics, a concrete category is a category in which, roughly speaking, all objects are sets possibly carrying some additional structure, all morphisms are functions between those sets, and the composition of morphisms is the composition of functions. ...
In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. ...
In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. ...
In abstract algebra, a module is a generalization of a vector space. ...
In any normal category with a zero object and kernels and kernels for every morphism, the image of a morphism f can be expressed as follows: In category theory and its applications to mathematics, a normal monomorphism or normal epimorphism is a particularly well-behaved type of morphism. ...
In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object: T is terminal, if to every object X in C there...
In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms and the kernels of module homomorphisms and certain other kernels from algebra. ...
In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms and the kernels of module homomorphisms and certain other kernels from algebra. ...
- im f = ker coker f
This holds especially in abelian categories. In mathematics, an abelian category is a certain kind of category in which morphisms and objects can be added and in which kernels and cokernels exist and have nice properties. ...
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