Editing Subcategory (section)

== 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]].