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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" some of its objects and arrows.

Formal definition[edit]

Let C be a category. A subcategory S of C is given by

  • a subcollection of objects of C, denoted ob(S),
  • a subcollection of morphisms of C, denoted hom(S).

such that

  • for every X in ob(S), the identity morphism idX is in hom(S),
  • for every morphism f : XY in 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: the collection of objects is ob(S), the collection of morphisms is hom(S), and the identities and composition are as in C. There is an obvious faithful functor I : SC, called the inclusion functor which takes objects and morphisms to themselves.

Let S be a subcategory of a category C. We say that S is a full subcategory of C if for each pair of objects X and Y of S

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.



Given a subcategory S of C the inclusion functor I : SC is both a faithful functor and injective on objects. It is full if and only if 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.[1]

Other authors define a functor to be an embedding if it is faithful and injective on objects. Equivalently, F is an embedding if it is injective on morphisms. A functor 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 : BC the image of F is a (full) subcategory S of C and F induces an isomorphism of categories between B and S. If F is not injective on objects, the image of F is equivalent to B.

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

Types of subcategories[edit]

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

A subcategory of C is wide or lluf (a term first posed by Peter Freyd[2]) if it contains all the objects of C.[3] 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 of an abelian category C such that for all short exact sequences

in C, M belongs to S if and only if both and do. This notion arises from Serre's C-theory.

See also[edit]


  1. ^ Jaap van Oosten. "Basic category theory" (PDF). 
  2. ^ Freyd, Peter (1991). "Algebraically complete categories". Proceedings of the International Conference on Category Theory, Como, Italy (CT 1990). Lecture Notes in Mathematics. 1488. Springer. pp. 95–104. doi:10.1007/BFb0084215. 
  3. ^ Wide subcategory in nLab