{{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 <math>\mathcal{C}</math> is a category <math>\mathcal{S}</math> whose objects are objects in <math>\mathcal{C}</math> and whose morphisms are morphisms in <math>\mathcal{C}</math> with the same identities and composition of morphisms. Intuitively, a subcategory of <math>\mathcal{C}</math> is a category obtained from <math>\mathcal{C}</math> by "removing" some of its objects and arrows.
== Formal definition == Let <math>\mathcal{C}</math> be a category. A '''subcategory''' <math>\mathcal{S}</math> of <math>\mathcal{C}</math> is given by *a subcollection of objects of <math>\mathcal{C}</math>, denoted <math>\operatorname{ob}(\mathcal{S})</math>, *a subcollection of morphisms of <math>\mathcal{C}</math>, denoted <math>\operatorname{mor}(\mathcal{S})</math>. such that *for every <math>X</math> in <math>\operatorname{ob}(\mathcal{S})</math>, the identity morphism id<sub><math>X</math></sub> is in <math>\operatorname{mor}(\mathcal{S})</math>, *for every morphism <math>f:X\to Y</math> in <math>\operatorname{mor}(\mathcal{S})</math>, both the source <math>X</math> and the target <math>Y</math> are in <math>\operatorname{ob}(\mathcal{S})</math>, *for every pair of morphisms <math>f</math> and <math>g</math> in <math>\operatorname{mor}(\mathcal{S})</math> the composite <math>f\circ g</math> is in <math>\operatorname{mor}(\mathcal{S})</math> whenever it is defined.
These conditions ensure that <math>\mathcal{S}</math> is a category in its own right: its collection of objects is <math>\operatorname{ob}(\mathcal{S})</math>, its collection of morphisms is <math>\operatorname{mor}(\mathcal{S})</math>, and its identities and composition are as in <math>\mathcal{C}</math>. There is an obvious faithful functor <math>I:\mathcal{S}\to\mathcal{C}</math>, called the '''inclusion functor''' which takes objects and morphisms to themselves.
Let <math>\mathcal{S}</math> be a subcategory of a category <math>\mathcal{C}</math>. We say that <math>\mathcal{S}</math> is a '''{{visible anchor|full subcategory}} of''' <math>\mathcal{C}</math> if for each pair of objects <math>X</math> and <math>Y</math> of <math>\mathcal{S}</math>, :<math>\mathrm{Hom}_\mathcal{S}(X,Y)=\mathrm{Hom}_\mathcal{C}(X,Y).</math> A full subcategory is one that includes ''all'' morphisms in <math>\mathcal{C}</math> between objects of <math>\mathcal{S}</math>. For any collection of objects <math>A</math> in <math>\mathcal{C}</math>, there is a unique full subcategory of <math>\mathcal{C}</math> whose objects are those in <math>A</math>.
== Examples == * The category of finite sets forms a full subcategory of the category of sets. * The category whose objects are sets and whose morphisms are 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 rings (whose morphisms are unit-preserving ring homomorphisms) forms a non-full subcategory of the category of rngs. * For a field <math>K</math>, the category of <math>K</math>-vector spaces forms a full subcategory of the category of (left or right) <math>K</math>-modules.
== Embeddings == Given a subcategory <math>\mathcal{S}</math> of <math>\mathcal{C}</math>, the inclusion functor <math>I:\mathcal{S}\to\mathcal{C}</math> is both a faithful functor and injective on objects. It is full if and only if <math>\mathcal{S}</math> 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, <math>F</math> is an embedding if it is injective on morphisms. A functor <math>F</math> 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 <math>F:\mathcal{B}\to\mathcal{C}</math> the image of <math>F</math> is a (full) subcategory <math>\mathcal{S}</math> of <math>\mathcal{C}</math>, and <math>F</math> induces an isomorphism of categories between <math>\mathcal{B}</math> and <math>\mathcal{S}</math>. If <math>F</math> is a full and faithful functor but not necessarily injective on objects, then the image of <math>F</math> is equivalent to <math>\mathcal{B}</math>.
In some categories, one can also speak of morphisms of the category being embeddings.
== Types of subcategories == A subcategory <math>\mathcal{S}</math> of <math>\mathcal{C}</math> is said to be '''isomorphism-closed''' or '''replete''' if every isomorphism <math>k:X\to Y</math> in <math>\mathcal{C}</math> such that <math>Y</math> is in <math>\mathcal{S}</math> also belongs to <math>\mathcal{S}</math>. An isomorphism-closed full subcategory is said to be '''strictly full'''.
{{anchor|Wide subcategory}} A subcategory of <math>\mathcal{C}</math> 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 <math>\mathcal{C}</math>.<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 <math>\mathcal{S}</math> of an abelian category <math>\mathcal{C}</math> such that for all short exact sequences
:<math>0\to M'\to M\to M''\to 0</math>
in <math>\mathcal{C}</math>, <math>M</math> belongs to <math>\mathcal{S}</math> if and only if both <math>M'</math> and <math>M''</math> do. This notion arises from 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