# Difference between revisions of "Subcategory"

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In [[mathematics]], a '''subcategory''' of a [[category (mathematics)|category]] ''C'' is a category ''S'' whose objects are objects in ''C'' and whose [[morphism]]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. | In [[mathematics]], a '''subcategory''' of a [[category (mathematics)|category]] ''C'' is a category ''S'' whose objects are objects in ''C'' and whose [[morphism]]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== | ==Formal definition== | ||

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Let ''C'' be a category. A '''subcategory''' ''S'' of ''C'' is given by | 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 objects of ''C'', denoted ob(''S''), | ||

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*for every pair of morphisms ''f'' and ''g'' in hom(''S'') the composite ''f'' o ''g'' is in hom(''S'') whenever it is defined. | *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 [[Full and faithful functors|faithful | + | These conditions ensure that ''S'' is a category in its own right: the collection of objects is ob(''S''), the collection of morphisms is hom(''S''), and the identities and composition are as in ''C''. There is an obvious [[Full and faithful functors|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'' | 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'' | ||

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==Embeddings== | ==Embeddings== | ||

+ | Given a subcategory ''S'' of ''C'' the inclusion [[functor]] ''I'' : ''S'' → ''C'' is both [[faithful functor|faithful]] and [[Injective function|injective]] on objects. It is [[full functor|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 (strictly).<ref>{{cite web|author=van Oosten|title=Basic category theory|url=http://www.staff.science.uu.nl/~ooste110/syllabi/catsmoeder.pdf}}</ref> | |

Other authors define a functor to be an '''embedding''' if it is | Other authors define a functor to be an '''embedding''' if it is | ||

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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. | 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 | + | 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 strictly injective on objects, the image of ''F'' is [[equivalence of categories|equivalent]] to ''B''. |

In some categories, one can also speak of morphisms of the category being [[embedding#Category theory|embedding]]s. | In some categories, one can also speak of morphisms of the category being [[embedding#Category theory|embedding]]s. | ||

==Types of subcategories== | ==Types of subcategories== | ||

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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''. A isomorphism-closed full subcategory is said to be '''strictly full'''. | 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''. 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<ref>{{cite book |last= Freyd|first= Peter|authorlink=Peter J. Freyd |year= 1991|month= |pages= | + | A subcategory of ''C'' is '''wide''' or '''lluf''' (a term first posed by P. Freyd<ref>{{cite book |last= Freyd|first= Peter|authorlink=Peter J. Freyd |year= 1991|month= |pages=95–104 |chapter= Algebraically complete categories|series=Lecture Notes in Mathematics |volume= 1488|publisher=Springer|title=Proceedings of the International Conference on Category Theory, Como, Italy (CT 1990)|doi=10.1007/BFb0084215}}</ref>) 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 | A '''Serre subcategory''' is a non-empty full subcategory ''S'' of an [[abelian category]] ''C'' such that for all short [[exact sequence]]s | ||

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:<math>0\to M'\to M\to M''\to 0</math> | :<math>0\to M'\to M\to M''\to 0</math> | ||

− | in ''C'', ''M'' belongs to ''S'' if and only if both <math>M'</math> and <math>M''</math> do. This notion arises from [[ | + | in ''C'', ''M'' belongs to ''S'' if and only if both <math>M'</math> and <math>M''</math> do. This notion arises from [[Localization of a category#Serre's C-theory|Serre's C-theory]]. |

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== See also == | == See also == | ||

*[[Reflective subcategory]] | *[[Reflective subcategory]] | ||

+ | *[[Exact category]], a full subcategory closed under extensions. | ||

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+ | ==References== | ||

+ | <references /> | ||

[[Category:Category theory]] | [[Category:Category theory]] | ||

− | + | [[Category:Hierarchy]] | |

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## Revision as of 12:36, 11 November 2013

{{#invoke:Hatnote|hatnote}}

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.

## 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: the collection of objects is ob(*S*), the collection of morphisms is hom(*S*), and the 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 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 faithful 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 (strictly).^{[1]}

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 an isomorphism of categories between *B* and *S*. If *F* is not strictly injective on objects, 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*. 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^{[2]}) 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 sequences

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
- Exact category, a full subcategory closed under extensions.

## References

- ↑ Template:Cite web
- ↑ {{#invoke:citation/CS1|citation |CitationClass=book }}