mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, specifically

category theory 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 Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) * ...

''C'' is a category ''S'' whose

objects Object may refer to: General meanings * 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

morphism 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 a ...

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'', :

\mathrm_\mathcal(X,Y)=\mathrm_\mathcal(X,Y).

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 In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, :\ is a finite set with five elements. Th ...

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 of m ...

. * The category whose objects are sets and whose morphisms are

bijections 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 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 of Ab is ...

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 There a ...

. * The category of

rings 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 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 Music * ''Unit'' (alb ...

-preserving

ring homomorphism In ring theory, a branch of abstract algebra, 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 such that ''f'' is: :addition preservi ...

s) forms a non-full subcategory of the category of rngs. * For a

field 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 * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

''K'', the category of ''K''-

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...

s forms a full subcategory of the category of (left or right) ''K''-

modules Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...

Embeddings

Given a subcategory ''S'' of ''C'', the inclusion functor ''I'' : ''S'' → ''C'' 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; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...

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. Formal definitions Explicitly, let ''C'' ...

. Such a functor is necessarily injective on objects up to

isomorphism In mathematics, an isomorphism is a structure-preserving mapping 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 them. The word is ...

. For instance, the

Yoneda embedding 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 (vie ...

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 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-dimensiona ...

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 Equivalence or Equivalent may refer to: Arts and entertainment *Album-equivalent unit, a measurement unit in the music industry * Equivalence class (music) *'' Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre *''Equiva ...

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 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^:B\ ...

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 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. Mathematics Freyd obtained his P ...

) 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 of ab ...

''C'' such that for all

short exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context o ...

s :

0\to M'\to M\to M''\to 0

in ''C'', ''M'' belongs to ''S'' if and only if both

M'

and

M''

do. This notion arises from Serre's C-theory.

References

{{Category theory Category theory Hierarchy

Formal definition

Examples

Embeddings

Types of subcategories

See also

References