η
        (
        X
        )
        :
        G
        (
        X
        )
        →
        H
        (
        X
        )
      
    
    {\displaystyle \eta (X):G(X)\to H(X)}
  
 is a natural transformation from the functor 
  
    
      
        G
      
    
    {\displaystyle G}
  
 to the functor 
  
    
      
        H
      
    
    {\displaystyle H}
  
, then the composition 
  
    
      
        η
        (
        X
        )
        μ
        (
        X
        )
        :
        F
        (
        X
        )
        →
        H
        (
        X
        )
      
    
    {\displaystyle \eta (X)\mu (X):F(X)\to H(X)}
  
 defines a natural transformation 
from 
  
    
      
        F
      
    
    {\displaystyle F}
  
 to 
  
    
      
        H
      
    
    {\displaystyle H}
  
. With this composition of natural transformations (known as vertical composition, see <a href="/facts/Natural_transformation/xwPumTHt">natural transformation</a>),

D
          
            C
          
        
      
    
    {\displaystyle D^{C}}
  
 satisfies the axioms of a category.
</p><p>In a completely analogous way, one can also consider the category of all <i>contravariant</i> functors from 
  
    
      
        C
      
    
    {\displaystyle C}
  
 to 
  
    
      
        D
      
    
    {\displaystyle D}
  
; we write this as Funct(
  
    
      
        
          C
          
            op
          
        
        ,
        D
      
    
    {\displaystyle C^{\text{op}},D}
  
).
</p><p>If 
  
    
      
        C
      
    
    {\displaystyle C}
  
 and 
  
    
      
        D
      
    
    {\displaystyle D}
  
 are both <a href="/facts/Preadditive_category/l308bN5A">preadditive categories</a> (i.e. their morphism sets are <a href="/facts/Abelian_group/FfWwmx4R">abelian groups</a> and the composition of morphisms is <a href="/facts/Bilinear_operator/LH1lHVeP">bilinear</a>), 
then we can consider the category of all <a href="/facts/Additive_functor/l308bN5A">additive functors</a> from 
  
    
      
        C
      
    
    {\displaystyle C}
  
 to 
  
    
      
        D
      
    
    {\displaystyle D}
  
, denoted by Add(
  
    
      
        C
      
    
    {\displaystyle C}
  
,
  
    
      
        D
      
    
    {\displaystyle D}
  
).
</p>
<h2 id="examples">Examples</h2>
<ul><li>If 
  
    
      
        I
      
    
    {\displaystyle I}
  
 is a small <a href="/facts/Discrete_category/Tk31I89n">discrete category</a> (i.e. its only morphisms are the identity morphisms), then a functor from 
  
    
      
        I
      
    
    {\displaystyle I}
  
 to 
  
    
      
        C
      
    
    {\displaystyle C}
  
 essentially consists of a family of objects of 
  
    
      
        C
      
    
    {\displaystyle C}
  
, indexed by 
  
    
      
        I
      
    
    {\displaystyle I}
  
; the functor category 
  
    
      
        
          C
          
            I
          
        
      
    
    {\displaystyle C^{I}}
  
 can be identified with the corresponding product category: its elements are families of objects in 
  
    
      
        C
      
    
    {\displaystyle C}
  
 and its morphisms are families of morphisms in 
  
    
      
        C
      
    
    {\displaystyle C}
  
.</li></ul>
<ul><li>An <a href="/facts/Arrow_category/wTa3tURp">arrow category</a> 
  
    
      
        
          
            
              C
            
          
          
            →
          
        
      
    
    {\displaystyle {\mathcal {C}}^{\rightarrow }}
  
 (whose objects are the morphisms of 
  
    
      
        
          
            C
          
        
      
    
    {\displaystyle {\mathcal {C}}}
  
, and whose morphisms are commuting squares in 
  
    
      
        
          
            C
          
        
      
    
    {\displaystyle {\mathcal {C}}}
  
) is just 
  
    
      
        
          
            
              C
            
          
          
            
              2
            
          
        
      
    
    {\displaystyle {\mathcal {C}}^{\mathbf {2} }}
  
, where 2 is the category with two objects and their identity morphisms as well as an arrow from one object to the other (but not another arrow back the other way).</li>
<li>A <a href="/facts/Graph_theory/VVPCd4TX">directed graph</a> consists of a set of arrows and a set of vertices, and two functions from the arrow set to the vertex set, specifying each arrow's start and end vertex. The category of all directed graphs is thus nothing but the functor category 
  
    
      
        
          
            
              Set
            
          
          
            C
          
        
      
    
    {\displaystyle {\textbf {Set}}^{C}}
  
, where 
  
    
      
        C
      
    
    {\displaystyle C}
  
 is the category with two objects connected by two parallel morphisms (source and target), and Set denotes the <a href="/facts/Category_of_sets/LF5ykYl0">category of sets</a>.</li>
<li>Any <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> 
  
    
      
        G
      
    
    {\displaystyle G}
  
 can be considered as a one-object category in which every morphism is invertible. The category of all <a href="/facts/Group_action_(mathematics)/Vh9VtUUw">
  
    
      
        G
      
    
    {\displaystyle G}
  
-sets</a> is the same as the functor category <a href="/facts/Category_of_sets/LF5ykYl0">Set</a>
  
    
      
        G
      
    
    {\displaystyle G}
  
. Natural transformations are <a href="/facts/Group_action_(mathematics)/Vh9VtUUw">
  
    
      
        G
      
    
    {\displaystyle G}
  
-maps</a>.</li>
<li>Similar to the previous example, the category of <i>K</i>-linear <a href="/facts/Group_representation/FsuuDagQ">representations</a> of the group 
  
    
      
        G
      
    
    {\displaystyle G}
  
 is the same as the functor category Vect<i>K</i>
  
    
      
        G
      
    
    {\displaystyle G}
  
 (where Vect<i>K</i> denotes the category of all <a href="/facts/Vector_space/M1pxMLx2">vector spaces</a> over the <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> <i>K</i>).</li>
<li>Any <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> 
  
    
      
        R
      
    
    {\displaystyle R}
  
 can be considered as a one-object preadditive category; the category of left <a href="/facts/Module_(mathematics)/jP0LIhFL">modules</a> over 
  
    
      
        R
      
    
    {\displaystyle R}
  
 is the same as the additive functor category Add(
  
    
      
        R
      
    
    {\displaystyle R}
  
,
  
    
      
        
          
            Ab
          
        
      
    
    {\displaystyle {\textbf {Ab}}}
  
) (where 
  
    
      
        
          
            Ab
          
        
      
    
    {\displaystyle {\textbf {Ab}}}
  
 denotes the <a href="/facts/Category_of_abelian_groups/jF3sXRFn">category of abelian groups</a>), and the category of right 
  
    
      
        R
      
    
    {\displaystyle R}
  
-modules is Add(
  
    
      
        
          R
          
            op
          
        
      
    
    {\displaystyle R^{\text{op}}}
  
,
  
    
      
        
          
            Ab
          
        
      
    
    {\displaystyle {\textbf {Ab}}}
  
). Because of this example, for any preadditive category 
  
    
      
        C
      
    
    {\displaystyle C}
  
, the category Add(
  
    
      
        C
      
    
    {\displaystyle C}
  
,
  
    
      
        
          
            Ab
          
        
      
    
    {\displaystyle {\textbf {Ab}}}
  
) is sometimes called the "category of left modules over 
  
    
      
        C
      
    
    {\displaystyle C}
  
" and Add(
  
    
      
        
          C
          
            op
          
        
      
    
    {\displaystyle C^{\text{op}}}
  
,
  
    
      
        
          
            Ab
          
        
      
    
    {\displaystyle {\textbf {Ab}}}
  
) is the "category of right modules over 
  
    
      
        C
      
    
    {\displaystyle C}
  
".</li>
<li>The category of <a href="/facts/Presheaf/MM3hcRw4">presheaves</a> on a topological space 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is a functor category: we turn the topological space into a category 
  
    
      
        C
      
    
    {\displaystyle C}
  
 having the open sets in 
  
    
      
        X
      
    
    {\displaystyle X}
  
 as objects and a single morphism from 
  
    
      
        U
      
    
    {\displaystyle U}
  
 to 
  
    
      
        V
      
    
    {\displaystyle V}
  
 if and only if 
  
    
      
        U
      
    
    {\displaystyle U}
  
 is contained in 
  
    
      
        V
      
    
    {\displaystyle V}
  
. The category of presheaves of sets (abelian groups, rings) on 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is then the same as the category of contravariant functors from 
  
    
      
        C
      
    
    {\displaystyle C}
  
 to 
  
    
      
        
          
            Set
          
        
      
    
    {\displaystyle {\textbf {Set}}}
  
 (or 
  
    
      
        
          
            Ab
          
        
      
    
    {\displaystyle {\textbf {Ab}}}
  
 or 
  
    
      
        
          
            Ring
          
        
      
    
    {\displaystyle {\textbf {Ring}}}
  
). Because of this example, the category Funct(
  
    
      
        
          C
          
            op
          
        
      
    
    {\displaystyle C^{\text{op}}}
  
, 
  
    
      
        
          
            Set
          
        
      
    
    {\displaystyle {\textbf {Set}}}
  
) is sometimes called the "<a href="/facts/Presheaf_(category_theory)/gWTNAFo3">category of presheaves</a> of sets on 
  
    
      
        C
      
    
    {\displaystyle C}
  
" even for general categories 
  
    
      
        C
      
    
    {\displaystyle C}
  
 not arising from a topological space. To define <a href="/facts/Sheaf_(mathematics)/MM3hcRw4">sheaves</a> on a general category 
  
    
      
        C
      
    
    {\displaystyle C}
  
, one needs more structure: a <a href="/facts/Grothendieck_topology/Przxx5sH">Grothendieck topology</a> on 
  
    
      
        C
      
    
    {\displaystyle C}
  
. (Some authors refer to categories that are  <a href="/facts/Equivalence_(category_theory)/k6uzqAcy">equivalent</a> to 
  
    
      
        
          
            
              Set
            
          
          
            C
          
        
      
    
    {\displaystyle {\textbf {Set}}^{C}}
  
 as <i><a href="/facts/Presheaf_(category_theory)/gWTNAFo3">presheaf</a> categories</i>.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>)</li></ul>
<h2 id="facts">Facts</h2>
<p>Most constructions that can be carried out in 
  
    
      
        D
      
    
    {\displaystyle D}
  
 can also be carried out in 
  
    
      
        
          D
          
            C
          
        
      
    
    {\displaystyle D^{C}}
  
 by performing them "componentwise", separately for 
each object in 
  
    
      
        C
      
    
    {\displaystyle C}
  
. For instance, if any two objects 
  
    
      
        X
      
    
    {\displaystyle X}
  
 and 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
 in 
  
    
      
        D
      
    
    {\displaystyle D}
  
 have a <a href="/facts/Product_(category_theory)/evFecz0t">product</a> 
  
    
      
        X
        ×
        Y
      
    
    {\displaystyle X\times Y}
  
, 
then any two functors 
  
    
      
        F
      
    
    {\displaystyle F}
  
 and 
  
    
      
        G
      
    
    {\displaystyle G}
  
 in 
  
    
      
        
          D
          
            C
          
        
      
    
    {\displaystyle D^{C}}
  
 have a product 
  
    
      
        F
        ×
        G
      
    
    {\displaystyle F\times G}
  
, defined by 
  
    
      
        (
        F
        ×
        G
        )
        (
        c
        )
        =
        F
        (
        c
        )
        ×
        G
        (
        c
        )
      
    
    {\displaystyle (F\times G)(c)=F(c)\times G(c)}
  
 
for every object 
  
    
      
        c
      
    
    {\displaystyle c}
  
 in 
  
    
      
        C
      
    
    {\displaystyle C}
  
. Similarly, if 
  
    
      
        
          η
          
            c
          
        
        :
        F
        (
        c
        )
        →
        G
        (
        c
        )
      
    
    {\displaystyle \eta _{c}:F(c)\to G(c)}
  
 is a natural transformation and each 
  
    
      
        
          η
          
            c
          
        
      
    
    {\displaystyle \eta _{c}}
  
 has a kernel 
  
    
      
        
          K
          
            c
          
        
      
    
    {\displaystyle K_{c}}
  
 in the category 
  
    
      
        D
      
    
    {\displaystyle D}
  
, 
then the kernel of 
  
    
      
        η
      
    
    {\displaystyle \eta }
  
 in the functor category 
  
    
      
        
          D
          
            C
          
        
      
    
    {\displaystyle D^{C}}
  
 is the functor 
  
    
      
        K
      
    
    {\displaystyle K}
  
 with 
  
    
      
        K
        (
        c
        )
        =
        
          K
          
            c
          
        
      
    
    {\displaystyle K(c)=K_{c}}
  
 for every object 
  
    
      
        c
      
    
    {\displaystyle c}
  
 in 
  
    
      
        C
      
    
    {\displaystyle C}
  
.
</p><p>As a consequence we have the general <a href="/facts/Rule_of_thumb/RopfGsJ0">rule of thumb</a> that the functor category 
  
    
      
        
          D
          
            C
          
        
      
    
    {\displaystyle D^{C}}
  
 shares most of the "nice" properties of 
  
    
      
        D
      
    
    {\displaystyle D}
  
:
</p>
<ul><li>if 
  
    
      
        D
      
    
    {\displaystyle D}
  
 is <a href="/facts/Complete_category/wpQzAbpB">complete</a> (or cocomplete), then so is 
  
    
      
        
          D
          
            C
          
        
      
    
    {\displaystyle D^{C}}
  
;</li>
<li>if 
  
    
      
        D
      
    
    {\displaystyle D}
  
 is an <a href="/facts/Abelian_category/4WN473Yc">abelian category</a>, then so is 
  
    
      
        
          D
          
            C
          
        
      
    
    {\displaystyle D^{C}}
  
;</li></ul>
<p>We also have:
</p>
<ul><li>if 
  
    
      
        C
      
    
    {\displaystyle C}
  
 is any small category, then the category 
  
    
      
        
          
            
              Set
            
          
          
            C
          
        
      
    
    {\displaystyle {\textbf {Set}}^{C}}
  
 of <a href="/facts/Presheaf_(category_theory)/gWTNAFo3">presheaves</a> is a <a href="/facts/Topos/zzttCIwC">topos</a>.</li></ul>
<p>So from the above examples, we can conclude right away that the categories of directed graphs, 
  
    
      
        G
      
    
    {\displaystyle G}
  
-sets and presheaves on a topological space are all complete and cocomplete topoi, and that the categories of representations of 
  
    
      
        G
      
    
    {\displaystyle G}
  
, modules over the ring 
  
    
      
        R
      
    
    {\displaystyle R}
  
, 
and presheaves of abelian groups on a topological space 
  
    
      
        X
      
    
    {\displaystyle X}
  
 are all abelian, complete and cocomplete.
</p><p>The embedding of the category 
  
    
      
        C
      
    
    {\displaystyle C}
  
 in a functor category that was mentioned earlier uses the <a href="/facts/Yoneda_lemma/j04vXd6x">Yoneda lemma</a> as its main tool. For every object 
  
    
      
        X
      
    
    {\displaystyle X}
  
 of 
  
    
      
        C
      
    
    {\displaystyle C}
  
, 
let 
  
    
      
        
          Hom
        
        (
        −
        ,
        X
        )
      
    
    {\displaystyle {\text{Hom}}(-,X)}
  
 be the contravariant <a href="/facts/Representable_functor/dILwN4oj">representable functor</a> from 
  
    
      
        C
      
    
    {\displaystyle C}
  
 to 
  
    
      
        
          
            Set
          
        
      
    
    {\displaystyle {\textbf {Set}}}
  
. The Yoneda lemma states that the assignment 
</p>

X
        ↦
        Hom
        ⁡
        (
        −
        ,
        X
        )
      
    
    {\displaystyle X\mapsto \operatorname {Hom} (-,X)}

<p>is a <a href="/facts/Full_embedding/k4lEeDDb">full embedding</a> of the category 
  
    
      
        C
      
    
    {\displaystyle C}
  
 into the category Funct(
  
    
      
        
          C
          
            op
          
        
      
    
    {\displaystyle C^{\text{op}}}
  
,
  
    
      
        
          
            Set
          
        
      
    
    {\displaystyle {\textbf {Set}}}
  
). So 
  
    
      
        C
      
    
    {\displaystyle C}
  
 naturally sits inside a topos.
</p><p>The same can be carried out for any preadditive category 
  
    
      
        C
      
    
    {\displaystyle C}
  
: Yoneda then yields a full embedding of 
  
    
      
        C
      
    
    {\displaystyle C}
  
 into the functor category Add(
  
    
      
        
          C
          
            op
          
        
      
    
    {\displaystyle C^{\text{op}}}
  
,
  
    
      
        
          
            Ab
          
        
      
    
    {\displaystyle {\textbf {Ab}}}
  
). 
So 
  
    
      
        C
      
    
    {\displaystyle C}
  
 naturally sits inside an abelian category.
</p><p>The intuition mentioned above (that constructions that can be carried out in 
  
    
      
        D
      
    
    {\displaystyle D}
  
 can be "lifted" to 
  
    
      
        
          D
          
            C
          
        
      
    
    {\displaystyle D^{C}}
  
) can be made precise in several ways; the most succinct formulation uses the language of <a href="/facts/Adjoint_functors/bJ1P7AN2">adjoint functors</a>. 
Every functor 
  
    
      
        F
        :
        D
        →
        E
      
    
    {\displaystyle F:D\to E}
  
 induces a functor 
  
    
      
        
          F
          
            C
          
        
        :
        
          D
          
            C
          
        
        →
        
          E
          
            C
          
        
      
    
    {\displaystyle F^{C}:D^{C}\to E^{C}}
  
 (by composition with 
  
    
      
        F
      
    
    {\displaystyle F}
  
). If 
  
    
      
        F
      
    
    {\displaystyle F}
  
 and 
  
    
      
        G
      
    
    {\displaystyle G}
  
 is a pair of adjoint functors, then 
  
    
      
        
          F
          
            C
          
        
      
    
    {\displaystyle F^{C}}
  
 and 
  
    
      
        
          G
          
            C
          
        
      
    
    {\displaystyle G^{C}}
  
 is also a pair of adjoint functors.
</p><p>The functor category 
  
    
      
        
          D
          
            C
          
        
      
    
    {\displaystyle D^{C}}
  
 has all the formal properties of an <a href="/facts/Exponential_object/4MYWRF9h">exponential object</a>; in particular the functors from 
  
    
      
        E
        ×
        C
        →
        D
      
    
    {\displaystyle E\times C\to D}
  
 
stand in a natural one-to-one correspondence with the functors from 
  
    
      
        E
      
    
    {\displaystyle E}
  
 to 
  
    
      
        
          D
          
            C
          
        
      
    
    {\displaystyle D^{C}}
  
. The category 
  
    
      
        
          
            Cat
          
        
      
    
    {\displaystyle {\textbf {Cat}}}
  
 of all small categories with functors as morphisms 
is therefore a <a href="/facts/Cartesian_closed_category/mPXAQps2">cartesian closed category</a>.
</p>
<h2 id="see-also">See also</h2>
<ul>
<li>Mathematics portal</li></ul>
<ul><li><a href="/facts/Diagram_(category_theory)/kGTlM945">Diagram (category theory)</a></li></ul>

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

<ol>
<li id="fn:1"><p>Tom Leinster (2004). Higher Operads, Higher Categories. Cambridge University Press. Bibcode:2004hohc.book.....L. Archived from the original on 2003-10-25. <a href="https://web.archive.org/web/20031025120434/http://www.maths.gla.ac.uk/~tl/book.html" target="_blank">https://web.archive.org/web/20031025120434/http://www.maths.gla.ac.uk/~tl/book.html</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Functor category open-in-new

Functor category