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==
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== Formal definition ==
 
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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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''.
 
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==
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== 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.
 
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.
  
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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''.
 
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.
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In some categories, one can also speak of morphisms of the category being [[embedding#Category theory|embeddings]].
  
==Types of subcategories==
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== 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'''.
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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''. An 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=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 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.
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== See also ==
 
== See also ==
 +
{{Wiktionary}}
 
*[[Reflective subcategory]]
 
*[[Reflective subcategory]]
 
*[[Exact category]], a full subcategory closed under extensions.
 
*[[Exact category]], a full subcategory closed under extensions.
  
==References==
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== References ==
 
<references />
 
<references />
  
 
[[Category:Category theory]]
 
[[Category:Category theory]]
 
[[Category:Hierarchy]]
 
[[Category:Hierarchy]]

Latest revision as of 18:09, 11 December 2014

{{#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 idX is in hom(S),
  • for every morphism f : XY 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 : SC, 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 : SC 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 : BC 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 : XY 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 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

Template:Sister

References

  1. Template:Cite web
  2. {{#invoke:citation/CS1|citation |CitationClass=book }}