Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, a subcategory of a category
''C'' is a category ''S'' whose
Object may refer to:
* Object (philosophy), a thing, being, or concept
** Object (abstract), an object which does not exist at any particular time or place
** Physical object, an identifiable collection of matter
* Goal, an ...
are objects in ''C'' and whose
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
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.
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'').
*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
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
''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''.
* 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 as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition ...
* The category whose objects are sets and whose morphisms are
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
forms a non-full subcategory of the category of sets.
* The category of abelian groups
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 category of
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
(whose morphisms are
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
* ''Unit'' (a ...
-preserving ring homomorphism
s) forms a non-full subcategory of the category of rngs
* For a
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Lawn, an area of mowed grass
* Meadow, a gras ...
''K'', the category of ''K''- vector space
s forms a full subcategory of the category of (left or right) ''K''- modules
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
In category theory, a faithful functor is a functor that is injective on hom-sets, and a full functor is surjective on hom-sets. A functor that has both properties is called a full and faithful functor.
Explicitly, let ''C'' an ...
. Such a functor is necessarily injective on objects up to isomorphism
. For instance, the
In mathematics, the Yoneda lemma is arguably the most important 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 (viewi ...
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
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
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimension ...
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'' which 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
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 In category theory, a branch of mathematics, a subcategory \mathcal of a category \mathcal is said to be isomorphism closed or replete if every \mathcal- isomorphism h:A\to B with A\in\mathcal belongs to \mathcal. This implies that both B and h^ ...
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 John Freyd (; born February 5, 1936) is an American mathematician, a professor at the University of Pennsylvania, known for work in category theory and for founding the False Memory Syndrome Foundation.
Freyd obtained his ...
) 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
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 of ab ...
''C'' such that for all short exact sequence
in ''C'', ''M'' belongs to ''S'' if and only if both
do. This notion arises from Serre's C-theory
* Reflective subcategory
* Exact category
, a full subcategory closed under extensions.