Subcategory

<h2 id="formal-definition">Formal definition</h2>
Let C be a category. A subcategory S of C is given by

<ul><li>a subcollection of objects of C, denoted ob(S),</li>
<li>a subcollection of morphisms of C, denoted hom(S).</li></ul>
such that

<ul><li>for every X in ob(S), the identity morphism idX is in hom(S),</li>
<li>for every morphism f : X → Y in hom(S), both the source X and the target Y are in ob(S),</li>
<li>for every pair of morphisms f and g in hom(S) the composite f o g is in hom(S) whenever it is defined.</li></ul>
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 <a href="/facts/Full_and_faithful_functors/bMM75v7e">faithful</a> <a href="/facts/Functor/l2u3rIBE">functor</a> 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,

H
            o
            m
          
          
            
              S
            
          
        
        (
        X
        ,
        Y
        )
        =
        
          
            H
            o
            m
          
          
            
              C
            
          
        
        (
        X
        ,
        Y
        )
        .
      
    
    {\displaystyle \mathrm {Hom} _{\mathcal {S}}(X,Y)=\mathrm {Hom} _{\mathcal {C}}(X,Y).}

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.

<h2 id="examples">Examples</h2>
<ul><li>The category of <a href="/facts/Finite_set/za1newlP">finite sets</a> forms a full subcategory of the <a href="/facts/Category_of_sets/LF5ykYl0">category of sets</a>.</li>
<li>The category whose objects are sets and whose morphisms are <a href="/facts/Bijection/ayC7YE3z">bijections</a> forms a non-full subcategory of the category of sets.</li>
<li>The <a href="/facts/Category_of_abelian_groups/jF3sXRFn">category of abelian groups</a> forms a full subcategory of the <a href="/facts/Category_of_groups/sxSd9Kbc">category of groups</a>.</li>
<li>The category of <a href="/facts/Ring_(mathematics)/JnCUXz63">rings</a> (whose morphisms are <a href="/facts/Unit_(ring_theory)/GJSBu8Y7">unit</a>-preserving <a href="/facts/Ring_homomorphism/UptcMpFk">ring homomorphisms</a>) forms a non-full subcategory of the category of <a href="/facts/Rng_(algebra)/SquNCXKe">rngs</a>.</li>
<li>For a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> K, the category of K-<a href="/facts/Vector_space/M1pxMLx2">vector spaces</a> forms a full subcategory of the category of (left or right) K-<a href="/facts/Module_(mathematics)/jP0LIhFL">modules</a>.</li></ul>
<h2 id="embeddings">Embeddings</h2>
Given a subcategory S of C, the inclusion functor I : S → C is both a faithful functor and <a href="/facts/Injective/5ES6tqf5">injective</a> on objects. It is <a href="/facts/Full_functor/bMM75v7e">full</a> if and only if S is a full subcategory.
Some authors define an embedding to be a <a href="/facts/Full_and_faithful_functor/bMM75v7e">full and faithful functor</a>. Such a functor is necessarily injective on objects up to <a href="/facts/Isomorphism/pj8KgkxU">isomorphism</a>. For instance, the <a href="/facts/Yoneda_embedding/j04vXd6x">Yoneda embedding</a> is an embedding in this sense.
Some authors define an embedding to be a full and faithful functor that is injective on objects.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>
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 <a href="/facts/Image_(mathematics)/KANefMAl">image</a> of F is a (full) subcategory S of C, and F induces an <a href="/facts/Isomorphism_of_categories/JxayqDcm">isomorphism of categories</a> between B and S. If F is not injective on objects then the image of F is <a href="/facts/Equivalence_of_categories/k6uzqAcy">equivalent</a> to B.
In some categories, one can also speak of morphisms of the category being <a href="/facts/Embedding/HKeVKc42">embeddings</a>.

<h2 id="types-of-subcategories">Types of subcategories</h2>
A subcategory S of C is said to be <a href="/facts/Isomorphism-closed_subcategory/QoWpVgeD">isomorphism-closed</a> 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 <a href="/facts/Peter_Freyd/jbQAncXD">Peter Freyd</a><a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>) if it contains all the objects of C.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> 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 <a href="/facts/Abelian_category/4WN473Yc">abelian category</a> C such that for all <a href="/facts/Short_exact_sequence/leUJnwJb">short exact sequences</a>

0
        →
        
          M
          ′
        
        →
        M
        →
        
          M
          ″
        
        →
        0
      
    
    {\displaystyle 0\to M'\to M\to M''\to 0}

in C, M belongs to S if and only if both 
 
 
 
 
 M
 ′
 
 
 
 {\displaystyle M'}
 
 and 
 
 
 
 
 M
 ″
 
 
 
 {\displaystyle M''}
 
 do. This notion arises from <a href="/facts/Localization_of_a_category/NJ36V2lu">Serre's C-theory</a>.

<h2 id="see-also">See also</h2>

Look up subcategory in Wiktionary, the free dictionary.

<ul><li><a href="/facts/Reflective_subcategory/vaF3ma2y">Reflective subcategory</a></li>
<li><a href="/facts/Exact_category/YS4a1UwU">Exact category</a>, a full subcategory closed under extensions.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Jaap van Oosten. "Basic category theory" (PDF). <a href="http://www.staff.science.uu.nl/~ooste110/syllabi/catsmoeder.pdf" target="_blank">http://www.staff.science.uu.nl/~ooste110/syllabi/catsmoeder.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Freyd, Peter (1991). "Algebraically complete categories". Proceedings of the International Conference on Category Theory, Como, Italy (CT 1990). Lecture Notes in Mathematics. Vol. 1488. Springer. pp. 95–104. doi:10.1007/BFb0084215. ISBN 978-3-540-54706-8. <a href="978-3-540-54706-8" target="_blank">978-3-540-54706-8</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Wide subcategory at the nLab <a href="https://ncatlab.org/nlab/show/wide+subcategory" target="_blank">https://ncatlab.org/nlab/show/wide+subcategory</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
</ol>

Subcategory open-in-new

Subcategory