Subcategory

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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.

1 Formal definition
2 Embeddings
3 Types of subcategories
4 References
5 See also

[edit] 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.

[edit] 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.

[edit] 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^[1]) 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

$0\to M'\to M\to M''\to 0$