Subcategory

In mathematics, a subcategory of a category "C" is a category "S" whose objects are objects in "C" and whose morphisms are morphisms in "C" with the same identities and composition of morphisms. Intuitively, a subcategory of "C" is a category obtained from "C" by "removing" objects and arrows.

Formal definition

Let "C" be a category. A subcategory "S" of "C" is given by
*a subclass of objects of "C", denoted ob("S"),
*a subclass of morphisms of "C", denoted hom("S").such that
*for every "X" in ob("S"), the identity morphism id_"X" is in hom("S"),
*for every morphism "f" : "X" → "Y" is hom("S"), both the source "X" and the target "Y" are in ob("S"),
*for every pair of morphisms "f" and "g" in hom("S") the composite "f" o "g" is in hom("S") whenever it is defined.

These conditions ensure that "S" is a category in its own right. There is a natural functor "I" : "S" → "C", called the inclusion functor which is just the identity on objects and morphisms.

A full subcategory of a category "C" is a subcategory "S" of "C" such that for each pair of objects "X" and "Y" of "S":$mathrm\{Hom\}\_mathcal\{S\}(X,Y)=mathrm\{Hom\}\_mathcal\{C\}(X,Y).$A full subcategory is one that includes "all" morphisms between objects of "S". For any collection of objects "A" in "C", there is a unique full subcategory of "C" whose objects are those in "A".

Embeddings

Given a subcategory "S" of "C" the inclusion functor "I" : "S" → "C" is both faithful and injective on objects. It is full if and only if "S" is a full subcategory.

A functor "F" : "B" → "C" is called an embedding if it is
*a faithful functor, and
*injective on objects.Equivalently, "F" is an embedding if it is injective on morphisms. A functor "F" is called full embedding if it is a full functor and an embedding.

For any (full) embedding "F" : "B" → "C" the image of "F" is a (full) subcategory "S" of "C" and "F" induces a isomorphism of categories between "B" and "S".

Types of subcategories

A subcategory "S" of "C" is said to be isomorphism-closed or replete if every isomorphism "k" : "X" → "Y" in "C" such that "Y" is in "S" also belongs to "S". A isomorphism-closed full subcategory is said to be strictly full.

A subcategory of "C" is wide or lluf (a term first posed by P. Freyd [cite journal |last= Freyd|first= Peter|authorlink=Peter J. Freyd |year= 1990|month= |title= Algebraically complete categories|journal=LNCS |volume= 1488|quote=Proc. Category Theory, Como ] ) if it contains all the objects of "C". A lluf subcategory is typically not full: the only full lluf subcategory of a category is that category itself.

A Serre subcategory is a non-empty full subcategory "S" of an abelian category "C" such that for all short exact sequences

:$0\; o\; M\text{'}\; o\; M\; o\; M"\; o\; 0$

in "C", "M" belongs to "S" if and only if both $M\text{'}$ and $M"$ do. This notion arises from Serre's C-theory.

References

See also

*Reflective subcategory