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{{For|subcategories in wikipedia|WP:Subcategories}}
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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.
 
==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<sub>''X''</sub> 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 [[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''
:<math>\mathrm{Hom}_\mathcal{S}(X,Y)=\mathrm{Hom}_\mathcal{C}(X,Y).</math>
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 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
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 [[equivalence of categories|equivalent]] to ''B''.
 
In some categories, one can also speak of morphisms of the category being [[embedding#Category theory|embedding]]s.
 
==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<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
 
:<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 [[Localization of a category#Serre's C-theory|Serre's C-theory]].
 
== See also ==
*[[Reflective subcategory]]
*[[Exact category]], a full subcategory closed under extensions.
 
==References==
<references />
 
[[Category:Category theory]]
[[Category:Hierarchy]]

Latest revision as of 23:10, 21 April 2014

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