Closed category

<h2 id="definition">Definition</h2>
<p>A closed category can be defined as a category 
  
    
      
        
          
            C
          
        
      
    
    {\displaystyle {\mathcal {C}}}
  
 with a so-called <a href="/facts/Internal_Hom_functor/H76BLKCD">internal Hom functor</a>
</p>

[
          
            −
             
            −
          
          ]
        
        :
        
          
            
              C
            
          
          
            o
            p
          
        
        ×
        
          
            C
          
        
        →
        
          
            C
          
        
      
    
    {\displaystyle \left[-\ -\right]:{\mathcal {C}}^{op}\times {\mathcal {C}}\to {\mathcal {C}}}

<p>with left <a href="/facts/Yoneda_arrow/j04vXd6x">Yoneda arrows</a>
</p>

L
        :
        
          [
          
            B
             
            C
          
          ]
        
        →
        
          [
          
            
              [
              
                A
                 
                B
              
              ]
            
            
              [
              
                A
                 
                C
              
              ]
            
          
          ]
        
      
    
    {\displaystyle L:\left[B\ C\right]\to \left[\left[A\ B\right]\left[A\ C\right]\right]}

<p><a href="/facts/Natural_transformation/xwPumTHt">natural</a> in 
  
    
      
        B
      
    
    {\displaystyle B}
  
 and 
  
    
      
        C
      
    
    {\displaystyle C}
  
 and <a href="/facts/Dinatural_transformation/T3w3T7DI">dinatural</a> in 
  
    
      
        A
      
    
    {\displaystyle A}
  
, and a fixed object 
  
    
      
        I
      
    
    {\displaystyle I}
  
 of 
  
    
      
        
          
            C
          
        
      
    
    {\displaystyle {\mathcal {C}}}
  
 with a <a href="/facts/Natural_isomorphism/xwPumTHt">natural isomorphism</a>
</p>

i
          
            A
          
        
        :
        A
        ≅
        
          [
          
            I
             
            A
          
          ]
        
      
    
    {\displaystyle i_{A}:A\cong \left[I\ A\right]}

<p>and a <a href="/facts/Dinatural_transformation/T3w3T7DI">dinatural transformation</a>
</p>

j
          
            A
          
        
        :
        I
        →
        
          [
          
            A
             
            A
          
          ]
        
      
    
    {\displaystyle j_{A}:I\to \left[A\ A\right]}
  
,
<p>all satisfying certain coherence conditions.
</p>
<h2 id="examples">Examples</h2>
<ul><li><a href="/facts/Cartesian_closed_categories/mPXAQps2">Cartesian closed categories</a> are closed categories. In particular, any <a href="/facts/Topos/zzttCIwC">topos</a> is closed. The canonical example is the category of sets.</li>
<li><a href="/facts/Compact_closed_category/zrRPGzao">Compact closed categories</a> are closed categories. The canonical example is the category FdVect with <a href="/facts/Dimension_(vector_space)/z8m02A3W">finite-dimensional</a> <a href="/facts/Vector_space/M1pxMLx2">vector spaces</a> as objects and <a href="/facts/Linear_map/Q1PBKNVV">linear maps</a> as morphisms.</li>
<li>More generally, any <a href="/facts/Monoidal_closed_category/FBe3ecNJ">monoidal closed category</a> is a closed category. In this case, the object 
  
    
      
        I
      
    
    {\displaystyle I}
  
 is the <a href="/facts/Monoidal_category/F2l5oKMD">monoidal unit</a>.</li></ul>

<ul><li><a href="/facts/Samuel_Eilenberg/vs6aksKX">Eilenberg, S.</a>; <a href="/facts/Max_Kelly/PEUBaKX4">Kelly, G.M.</a> (2012) [1966]. <a href="https://books.google.com/books?id=rwX9CAAAQBAJ&pg=PA421">"Closed categories"</a>. <i>Proceedings of the Conference on Categorical Algebra. (La Jolla, 1965</i>. Springer. pp. 421–562. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-642-99902-4_22">10.1007/978-3-642-99902-4_22</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-642-99902-4.</li>
<li><a href="https://ncatlab.org/nlab/show/closed+category">Closed category</a> at the <a href="/facts/NLab/lpPvRmgo"><i>n</i>Lab</a></li></ul>

Closed category open-in-new

Closed category