In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, specifically
category theory
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
, a subcategory of a
category
Category, plural categories, may refer to:
General uses
*Classification, the general act of allocating things to classes/categories Philosophy
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce)
* Category ( ...
''C'' is a category ''S'' whose
objects are objects in ''C'' and whose
morphism
In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...
s 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 id
''X'' 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: 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
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
''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''.
Examples
* The category of
finite sets forms a full subcategory of the
category of sets
In the mathematical field of category theory, the category of sets, denoted by Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the functions from ''A'' to ''B'', and the composition of mor ...
.
* The category whose objects are sets and whose morphisms are
bijections forms a non-full subcategory of the category of sets.
* The
category of abelian groups
In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in Ab.
Properties
The zero object o ...
forms a full subcategory of the
category of groups
In mathematics, the category Grp (or Gp) has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory.
Relation to other categories
The ...
.
* The category of
rings (whose morphisms are
unit-preserving
ring homomorphism
In mathematics, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function that preserves addition, multiplication and multiplicative identity ...
s) forms a non-full subcategory of the category of
rngs.
* For a
field ''K'', the category of ''K''-
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
s forms a full subcategory of the category of (left or right) ''K''-
modules.
Embeddings
Given a subcategory ''S'' of ''C'', the inclusion functor is both a faithful functor and
injective
In mathematics, an injective function (also known as injection, or one-to-one function ) is a function that maps distinct elements of its domain to distinct elements of its codomain; that is, implies (equivalently by contraposition, impl ...
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
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
. For instance, the
Yoneda embedding
In mathematics, the Yoneda lemma is a fundamental result in category theory. It is an abstract result on functors of the type ''morphisms into a fixed object''. It is a vast generalisation of Cayley's theorem from group theory (viewing a group as a ...
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
An image or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...
of ''F'' is a (full) subcategory ''S'' of ''C'', and ''F'' induces an
isomorphism of categories
In category theory, two categories ''C'' and ''D'' are isomorphic if there exist functors ''F'' : ''C'' → ''D'' and ''G'' : ''D'' → ''C'' that are mutually inverse to each other, i.e. ''FG'' = 1''D'' (the identity functor on ''D'') and ''GF'' ...
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
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 prototypical example of an abelian category is the category o ...
''C'' such that for all
short exact sequence
In mathematics, an exact sequence is a sequence of morphisms between objects (for example, Group (mathematics), groups, Ring (mathematics), rings, Module (mathematics), modules, and, more generally, objects of an abelian category) such that the Im ...
s
:
in ''C'', ''M'' belongs to ''S'' if and only if both
and
do. This notion arises from
Serre's C-theory.
See also
*
Reflective subcategory
In mathematics, a full subcategory ''A'' of a category ''B'' is said to be reflective in ''B'' when the inclusion functor from ''A'' to ''B'' has a left adjoint. This adjoint is sometimes called a ''reflector'', or ''localization''. Dually, ''A'' ...
*
Exact category, a full subcategory closed under extensions.
References
{{Category theory
Category theory
Hierarchy