Subcategory
Encyclopedia
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.
such that
These conditions ensure that S is a category in its own right. There is an obvious faithful
functor
I : S → C, called the inclusion functor which is just the identity on objects and morphisms.
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.
I : S → C is both faithful and injective on objects. It is full if and only if S is a full subcategory.
Many authors define an embedding to be a full and faithful functor.
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.
For any (full) embedding F : B → C the image of F is a (full) subcategory S of C and F induces a isomorphism of categories
between B and S.
In some categories, one can also speak of morphisms of the category being embeddings.
k : X → Y in C such that Y is in S also belongs to S. A isomorphism-closed full subcategory is said to be strictly full.
A subcategory of C is wide or lluf (a term first posed by P. Freyd) if it contains all the objects of C. A lluf subcategory is typically not full: the only full lluf 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 sequence
s
in C, M belongs to S if and only if both
and 
do. This notion arises from Serre
's C-theory.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a subcategory of a category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
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- 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 : X → Y 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. There is an obvious faithful
Full and faithful functors
In category theory, a faithful functor is a functor which is injective when restricted to each set of morphisms that have a given source and target....
functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
I : S → C, called the inclusion functor which is just the identity on objects and morphisms.
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.
Embeddings
Given a subcategory S of C the inclusion functorFunctor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
I : S → C is both faithful and injective on objects. It is full if and only if S is a full subcategory.
Many authors define an embedding to be a full and faithful functor.
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.
For any (full) embedding F : B → C the image of F is a (full) subcategory S of C and F induces a isomorphism of categories
Isomorphism of categories
In category theory, two categories C and D are isomorphic if there exist functors F : C → D and G : D → C which are mutually inverse to each other, i.e. FG = 1D and GF = 1C. This means that both the objects and the morphisms of C and D stand in a one to one correspondence to each other...
between B and S.
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 isomorphismIsomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
k : X → Y in C such that Y is in S also belongs to S. A isomorphism-closed full subcategory is said to be strictly full.
A subcategory of C is wide or lluf (a term first posed by P. Freyd) if it contains all the objects of C. A lluf subcategory is typically not full: the only full lluf subcategory of a category is that category itself.
A Serre subcategory is a non-empty full subcategory S of an abelian category
Abelian category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...
C such that for all short exact sequence
Exact sequence
An exact sequence is a concept in mathematics, especially in homological algebra and other applications of abelian category theory, as well as in differential geometry and group theory...
s
in C, M belongs to S if and only if both
and do. This notion arises from SerreJean-Pierre Serre
Jean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...
's C-theory.