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 id X {\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}}} ,
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