In mathematics, specifically category theory, a subcategory of a category C {\displaystyle {\mathcal {C}}} is a category S {\displaystyle {\mathcal {S}}} whose objects are objects in C {\displaystyle {\mathcal {C}}} and whose morphisms are morphisms in C {\displaystyle {\mathcal {C}}} with the same identities and composition of morphisms. Intuitively, a subcategory of C {\displaystyle {\mathcal {C}}} is a category obtained from C {\displaystyle {\mathcal {C}}} by "removing" some of its objects and arrows.

Formal definition

Let C {\displaystyle {\mathcal {C}}} be a category. A subcategory S {\displaystyle {\mathcal {S}}} of C {\displaystyle {\mathcal {C}}} is given by

  • a subcollection of objects of C {\displaystyle {\mathcal {C}}}, denoted ob ⁡ ( S ) {\displaystyle \operatorname {ob} ({\mathcal {S}})},
  • a subcollection of morphisms of C {\displaystyle {\mathcal {C}}}, denoted mor ⁡ ( S ) {\displaystyle \operatorname {mor} ({\mathcal {S}})}.

such that

  • for every X {\displaystyle X} in ob ⁡ ( S ) {\displaystyle \operatorname {ob} ({\mathcal {S}})}, the identity morphism idX {\displaystyle X} is in mor ⁡ ( S ) {\displaystyle \operatorname {mor} ({\mathcal {S}})},
  • for every morphism f : X → Y {\displaystyle f:X\to Y} in mor ⁡ ( S ) {\displaystyle \operatorname {mor} ({\mathcal {S}})}, both the source X {\displaystyle X} and the target Y {\displaystyle Y} are in ob ⁡ ( S ) {\displaystyle \operatorname {ob} ({\mathcal {S}})},
  • for every pair of morphisms f {\displaystyle f} and g {\displaystyle g} in mor ⁡ ( S ) {\displaystyle \operatorname {mor} ({\mathcal {S}})} the composite f ∘ g {\displaystyle f\circ g} is in mor ⁡ ( S ) {\displaystyle \operatorname {mor} ({\mathcal {S}})} whenever it is defined.

These conditions ensure that S {\displaystyle {\mathcal {S}}} is a category in its own right: its collection of objects is ob ⁡ ( S ) {\displaystyle \operatorname {ob} ({\mathcal {S}})}, its collection of morphisms is mor ⁡ ( S ) {\displaystyle \operatorname {mor} ({\mathcal {S}})}, and its identities and composition are as in C {\displaystyle {\mathcal {C}}}. There is an obvious faithful functor I : S → C {\displaystyle I:{\mathcal {S}}\to {\mathcal {C}}}, called the inclusion functor which takes objects and morphisms to themselves.

Let S {\displaystyle {\mathcal {S}}} be a subcategory of a category C {\displaystyle {\mathcal {C}}}. We say that S {\displaystyle {\mathcal {S}}} is a full subcategory of C {\displaystyle {\mathcal {C}}} if for each pair of objects X {\displaystyle X} and Y {\displaystyle Y} of S {\displaystyle {\mathcal {S}}},

H o m S ( X , Y ) = H o m C ( X , Y ) . {\displaystyle \mathrm {Hom} _{\mathcal {S}}(X,Y)=\mathrm {Hom} _{\mathcal {C}}(X,Y).}

A full subcategory is one that includes all morphisms in C {\displaystyle {\mathcal {C}}} between objects of S {\displaystyle {\mathcal {S}}}. For any collection of objects A {\displaystyle A} in C {\displaystyle {\mathcal {C}}}, there is a unique full subcategory of C {\displaystyle {\mathcal {C}}} whose objects are those in A {\displaystyle A}.

Examples

Embeddings

Given a subcategory S {\displaystyle {\mathcal {S}}} of C {\displaystyle {\mathcal {C}}}, the inclusion functor I : S → C {\displaystyle I:{\mathcal {S}}\to {\mathcal {C}}} is both a faithful functor and injective on objects. It is full if and only if S {\displaystyle {\mathcal {S}}} is a full subcategory.

Some authors define an embedding to be a full and faithful functor. Such a functor is necessarily injective on objects up to isomorphism. For instance, the Yoneda embedding is an embedding in this sense.

Some authors define an embedding to be a full and faithful functor that is injective on objects.

Other authors define a functor to be an embedding if it is faithful and injective on objects. Equivalently, F {\displaystyle F} is an embedding if it is injective on morphisms. A functor F {\displaystyle F} is then called a full embedding if it is a full functor and an embedding.

With the definitions of the previous paragraph, for any (full) embedding F : B → C {\displaystyle F:{\mathcal {B}}\to {\mathcal {C}}} the image of F {\displaystyle F} is a (full) subcategory S {\displaystyle {\mathcal {S}}} of C {\displaystyle {\mathcal {C}}}, and F {\displaystyle F} induces an isomorphism of categories between B {\displaystyle {\mathcal {B}}} and S {\displaystyle {\mathcal {S}}}. If F {\displaystyle F} is a full and faithful functor but not necessarily injective on objects, then the image of F {\displaystyle F} is equivalent to B {\displaystyle {\mathcal {B}}}.

In some categories, one can also speak of morphisms of the category being embeddings.

Types of subcategories

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

A subcategory of C {\displaystyle {\mathcal {C}}} is wide or lluf (a term first posed by Peter Freyd) if it contains all the objects of C {\displaystyle {\mathcal {C}}}. A wide subcategory is typically not full: the only wide full subcategory of a category is that category itself.

A Serre subcategory is a non-empty full subcategory S {\displaystyle {\mathcal {S}}} of an abelian category C {\displaystyle {\mathcal {C}}} such that for all short exact sequences

0 → M ′ → M → M ″ → 0 {\displaystyle 0\to M'\to M\to M''\to 0}

in C {\displaystyle {\mathcal {C}}}, M {\displaystyle M} belongs to S {\displaystyle {\mathcal {S}}} if and only if both M ′ {\displaystyle M'} and M ″ {\displaystyle M''} do. This notion arises from Serre's C-theory.

See also