Image of a morphism

of a category

A concept similar to that of the image of a mapping of one set into another. However, in category theory there are several approaches to defining this concept. The simplest approach is closely related to the concept of a bicategory. Suppose that in the category $ \mathfrak K $ there exists a bicategory structure $ \mathfrak K = ( \mathfrak K , \mathfrak E , \mathfrak M ) $, where $ \mathfrak E $ is the class of admissible epimorphisms and $ \mathfrak M $ the class of admissible monomorphisms. If $ \alpha : A \rightarrow B $ is an arbitrary morphism in $ \mathfrak K $ and $ \alpha = \nu \mu $ is an admissible factorization of $ \alpha $, that is, $ \nu \in \mathfrak E $, $ \mu \in \mathfrak M $, then the subobject $ ( \mu ] $ of the object $ B $ defined by the monomorphism $ \mu $ is called the (admissible) image of the morphism $ \alpha $( relative to the given bicategory structure). If there is a unique bicategory structure in $ \mathfrak K $, then one may speak of the image of the morphism $ \alpha $. In particular, in the categories of sets, groups or vector spaces over a field, the definition stated above reduces to the usual definition of the image of a mapping or homomorphism.

On the other hand, if there are several bicategory structures in $ \mathfrak K $, then a given morphism may have different images relative to the different structures. This occurs, for example, in the categories of topological spaces and associative rings.

The following is another approach to the definition of the image of a morphism. One says that a morphism $ \alpha : A \rightarrow B $ factors through a subobject $ ( \mu ] $ of the object $ B $ if $ \alpha $ can be written in the form $ \alpha = \alpha ^ \prime \mu $. The smallest subobject of $ B $ through which $ \alpha $ can be factored, if it exists, is called the image of $ \alpha $. If $ \mathfrak K $ is well-powered and has limits of families of monomorphisms with a common codomain, then every morphism in $ \mathfrak K $ has an image.

If there is a bicategory structure in $ \mathfrak K $ in which all monomorphisms are admissible, then the second definition of the image of a morphism is equivalent to the definition by means of the given bicategory structure.

The image of a morphism is usually denoted by $ \mathop{\rm Im} \alpha $; $ \mathop{\rm im} \alpha $ denotes any representative of the subobject $ \mathop{\rm Im} \alpha $.

Comments

If the category $ \mathfrak K $ has pull-backs, the converse of the assertion in the penultimate paragraph above holds: The existence of images (in the second sense) implies that the class of all monomorphisms forms one half of a factorization ( "bicategory" ) structure on $ \mathfrak K $, the other half being the class of extremal epimorphisms (i.e. those which do not factor through any proper subobject of their codomain). Image factorizations (in the second sense) play a role in the theory of regular categories [a1]; indeed, the simplest definition of a regular category is as a category with finite limits and images in which image factorizations are stable under pull-back.

References

[a1]	P.A. Grillet, "Regular categories" , Exact categories and categories of sheaves , Lect. notes in math. , 236 , Springer pp. 121–222