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

Let be a category. A subcategory of is given by

such that

These conditions ensure that is a category in its own right: its collection of objects is , its collection of morphisms is , and its identities and composition are as in . There is an obvious faithful functor , called the inclusion functor which takes objects and morphisms to themselves.

Let be a subcategory of a category . We say that is a full subcategory of if for each pair of objects and of ,

A full subcategory is one that includes all morphisms in between objects of . For any collection of objects in , there is a unique full subcategory of whose objects are those in .

Given a subcategory of , the inclusion functor is both a faithful functor and injective on objects. It is full if and only if 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, is an embedding if it is injective on morphisms. A functor 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 the image of is a (full) subcategory of , and induces an isomorphism of categories between and . If is not injective on objects then the image of is equivalent to .

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

A subcategory of is said to be isomorphism-closed or replete if every isomorphism in such that is in also belongs to . An isomorphism-closed full subcategory is said to be strictly full.

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

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

• Reflective subcategory
• Exact category, a full subcategory closed under extensions.