Subcategory

In mathematics, specifically category theory, 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.
Let "C" be a category. A subcategory "S" of "C" is given by
such that
These conditions ensure that "S" is a category in its own right: its collection of objects is ob("S"), its collection of morphisms is hom("S"), and its identities and composition are as in "C". There is an obvious faithful functor "I" : "S" → "C", 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 in "C" 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".
Embeddings.
Given a subcategory "S" of "C", the inclusion functor "I" : "S" → "C" 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.
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" : "B" → "C" 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 then 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.
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". 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) if it contains all the objects of "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" of an abelian category "C" such that for all short exact sequences
in "C", "M" belongs to "S" if and only if both formula_3 and formula_4 do. This notion arises from Serre's C-theory.