Editing Subcategory

{{Short description|Category whose objects and morphisms are inside a bigger category}}
{{For|subcategories on Wikipedia|Wikipedia:Subcategories}}

In [[mathematics]], specifically [[category theory]], a '''subcategory''' of a [[category (mathematics)|category]] ''C'' is a category ''S'' whose [[Object (category theory)|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: 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 [[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 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 set|finite sets]] forms a full subcategory of the [[category of sets]].
* The category whose objects are sets and whose morphisms are [[Bijection|bijections]] forms a non-full subcategory of the category of sets.
* The [[category of abelian groups]] forms a full subcategory of the [[category of groups]].
* The category of [[Ring (mathematics)|rings]] (whose morphisms are [[Unit (ring theory)|unit]]-preserving [[ring homomorphism]]s) forms a non-full subcategory of the category of [[Rng_(algebra)|rngs]].
* For a [[Field (mathematics)|field]] ''K'', the category of ''K''-[[vector space]]s forms a full subcategory of the category of (left or right) ''K''-[[Module (mathematics)|modules]].

== Embeddings ==
Given a subcategory ''S'' of ''C'', the inclusion functor {{math|''I'' : ''S'' → ''C''}} is both a faithful functor and [[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.<ref>{{cite web|author=Jaap 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.

With the definitions of the previous paragraph, for any (full) embedding ''F'' : ''B'' → ''C'' the [[Image (mathematics)|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 injective on objects then 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|embeddings]].

== Types of subcategories ==
A subcategory ''S'' of ''C'' is said to be '''[[isomorphism-closed subcategory|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'''.

{{anchor|Wide subcategory}}
A subcategory of ''C'' is '''wide''' or '''lluf''' (a term first posed by [[Peter Freyd]]<ref>{{cite book |last= Freyd|first= Peter|authorlink=Peter J. Freyd  |year= 1991|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|isbn= 978-3-540-54706-8}}</ref>) if it contains all the objects of ''C''.<ref>{{nlab|id=wide+subcategory|title=Wide subcategory}}</ref> 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]] ''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 ==
{{Wiktionary}}
*[[Reflective subcategory]]
*[[Exact category]], a full subcategory closed under extensions.

== References ==
<references />

{{Category theory}}

[[Category:Category theory]]
[[Category:Hierarchy]]