Exponential object

<h2 id="definition">Definition</h2>
<p>Let 
  
    
      
        
          C
        
      
    
    {\displaystyle \mathbf {C} }
  
 be a category, let 
  
    
      
        Z
      
    
    {\displaystyle Z}
  
 and 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
 be <a href="/facts/Object_(category_theory)/7WzxUThf">objects</a> of 
  
    
      
        
          C
        
      
    
    {\displaystyle \mathbf {C} }
  
, and let 
  
    
      
        
          C
        
      
    
    {\displaystyle \mathbf {C} }
  
 have all <a href="/facts/Product_(category_theory)/evFecz0t">binary products</a> with 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
. An object 
  
    
      
        
          Z
          
            Y
          
        
      
    
    {\textstyle Z^{Y}}
  
 together with a <a href="/facts/Morphism/Kdpuyz3S">morphism</a> 
  
    
      
        
          e
          v
          a
          l
        
        :
        (
        
          Z
          
            Y
          
        
        ×
        Y
        )
        →
        Z
      
    
    {\textstyle \mathrm {eval} \colon (Z^{Y}\times Y)\to Z}
  
 is an <i>exponential object</i> if for any object 
  
    
      
        X
      
    
    {\displaystyle X}
  
 and morphism 
  
    
      
        g
        :
        X
        ×
        Y
        →
        Z
      
    
    {\textstyle g\colon X\times Y\to Z}
  
 there is a unique morphism 
  
    
      
        λ
        g
        :
        X
        →
        
          Z
          
            Y
          
        
      
    
    {\textstyle \lambda g\colon X\to Z^{Y}}
  
 (called the <i>transpose</i> of 
  
    
      
        g
      
    
    {\displaystyle g}
  
) such that the following diagram <a href="/facts/Commutative_diagram/bdG8xuer">commutes</a>:
</p>

<p>This assignment of a unique 
  
    
      
        λ
        g
      
    
    {\displaystyle \lambda g}
  
 to each 
  
    
      
        g
      
    
    {\displaystyle g}
  
 establishes an <a href="/facts/Isomorphism/pj8KgkxU">isomorphism</a> (<a href="/facts/Bijection/j4gTuTmW">bijection</a>) of <a href="/facts/Hom-set/Kdpuyz3S">hom-sets</a>, 
  
    
      
        
          H
          o
          m
        
        (
        X
        ×
        Y
        ,
        Z
        )
        ≅
        
          H
          o
          m
        
        (
        X
        ,
        
          Z
          
            Y
          
        
        )
        .
      
    
    {\textstyle \mathrm {Hom} (X\times Y,Z)\cong \mathrm {Hom} (X,Z^{Y}).}

</p><p>If 
  
    
      
        
          Z
          
            Y
          
        
      
    
    {\textstyle Z^{Y}}
  
exists for all objects 
  
    
      
        Z
        ,
        Y
      
    
    {\displaystyle Z,Y}
  
 in 
  
    
      
        
          C
        
      
    
    {\displaystyle \mathbf {C} }
  
, then the <a href="/facts/Functor/l2u3rIBE">functor</a> 
  
    
      
        (
        −
        
          )
          
            Y
          
        
        :
        
          C
        
        →
        
          C
        
      
    
    {\displaystyle (-)^{Y}\colon \mathbf {C} \to \mathbf {C} }
  
 defined on objects by 
  
    
      
        Z
        ↦
        
          Z
          
            Y
          
        
      
    
    {\displaystyle Z\mapsto Z^{Y}}
  
 and on arrows by 
  
    
      
        (
        f
        :
        X
        →
        Z
        )
        ↦
        (
        
          f
          
            Y
          
        
        :
        
          X
          
            Y
          
        
        →
        
          Z
          
            Y
          
        
        )
      
    
    {\displaystyle (f\colon X\to Z)\mapsto (f^{Y}\colon X^{Y}\to Z^{Y})}
  
, is a <a href="/facts/Right_adjoint/bJ1P7AN2">right adjoint</a> to the product functor 
  
    
      
        −
        ×
        Y
      
    
    {\displaystyle -\times Y}
  
. For this reason, the morphisms 
  
    
      
        λ
        g
      
    
    {\displaystyle \lambda g}
  
 and 
  
    
      
        g
      
    
    {\displaystyle g}
  
 are sometimes called <i>exponential adjoints</i> of one another.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p>
<h3>Equational definition</h3>
<p>Alternatively, the exponential object may be defined through equations:
</p>
<ul><li>Existence of 
  
    
      
        λ
        g
      
    
    {\displaystyle \lambda g}
  
 is guaranteed by existence of the operation 
  
    
      
        λ
        −
      
    
    {\displaystyle \lambda -}
  
.</li>
<li>Commutativity of the diagrams above is guaranteed by the equality 
  
    
      
        ∀
        g
        :
        X
        ×
        Y
        →
        Z
        ,
         
        
          e
          v
          a
          l
        
        ∘
        (
        λ
        g
        ×
        
          
            i
            d
          
          
            Y
          
        
        )
        =
        g
      
    
    {\displaystyle \forall g\colon X\times Y\to Z,\ \mathrm {eval} \circ (\lambda g\times \mathrm {id} _{Y})=g}
  
.</li>
<li>Uniqueness of 
  
    
      
        λ
        g
      
    
    {\displaystyle \lambda g}
  
 is guaranteed by the equality 
  
    
      
        ∀
        h
        :
        X
        →
        
          Z
          
            Y
          
        
        ,
         
        λ
        (
        
          e
          v
          a
          l
        
        ∘
        (
        h
        ×
        
          
            i
            d
          
          
            Y
          
        
        )
        )
        =
        h
      
    
    {\displaystyle \forall h\colon X\to Z^{Y},\ \lambda (\mathrm {eval} \circ (h\times \mathrm {id} _{Y}))=h}
  
.</li></ul>
<h3>Universal property</h3>
<p>The exponential 
  
    
      
        
          Z
          
            Y
          
        
      
    
    {\displaystyle Z^{Y}}
  
 is given by a <a href="/facts/Universal_morphism/5XJBYMNz">universal morphism</a> from the product functor 
  
    
      
        −
        ×
        Y
      
    
    {\displaystyle -\times Y}
  
 to the object 
  
    
      
        Z
      
    
    {\displaystyle Z}
  
. This universal morphism consists of an object 
  
    
      
        
          Z
          
            Y
          
        
      
    
    {\displaystyle Z^{Y}}
  
  and a morphism 
  
    
      
        
          e
          v
          a
          l
        
        :
        (
        
          Z
          
            Y
          
        
        ×
        Y
        )
        →
        Z
      
    
    {\textstyle \mathrm {eval} \colon (Z^{Y}\times Y)\to Z}
  
.
</p>
<h2 id="examples">Examples</h2>
<p>In the <a href="/facts/Category_of_sets/LF5ykYl0">category of sets</a>, an exponential object 
  
    
      
        
          Z
          
            Y
          
        
      
    
    {\displaystyle Z^{Y}}
  
 is the set of all <a href="/facts/Function_(mathematics)/CdReEKCh">functions</a> 
  
    
      
        Y
        →
        Z
      
    
    {\displaystyle Y\to Z}
  
.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> The map 
  
    
      
        
          e
          v
          a
          l
        
        :
        (
        
          Z
          
            Y
          
        
        ×
        Y
        )
        →
        Z
      
    
    {\displaystyle \mathrm {eval} \colon (Z^{Y}\times Y)\to Z}
  
 is just the <a href="/facts/Apply/2Emq4vwk">evaluation map</a>, which sends the pair 
  
    
      
        (
        f
        ,
        y
        )
      
    
    {\displaystyle (f,y)}
  
 to 
  
    
      
        f
        (
        y
        )
      
    
    {\displaystyle f(y)}
  
. For any map 
  
    
      
        g
        :
        X
        ×
        Y
        →
        Z
      
    
    {\displaystyle g\colon X\times Y\to Z}
  
 the map 
  
    
      
        λ
        g
        :
        X
        →
        
          Z
          
            Y
          
        
      
    
    {\displaystyle \lambda g\colon X\to Z^{Y}}
  
 is the <a href="/facts/Currying/AmD3PP5B">curried</a> form of 
  
    
      
        g
      
    
    {\displaystyle g}
  
:
</p>

λ
        g
        (
        x
        )
        (
        y
        )
        =
        g
        (
        x
        ,
        y
        )
        .
        
      
    
    {\displaystyle \lambda g(x)(y)=g(x,y).\,}

<p>A <a href="/facts/Heyting_algebra/Rrny2LRq">Heyting algebra</a> 
  
    
      
        H
      
    
    {\displaystyle H}
  
 is just a bounded <a href="/facts/Lattice_(order)/5ak6ufky">lattice</a> that has all exponential objects. Heyting implication, 
  
    
      
        Y
        ⇒
        Z
      
    
    {\displaystyle Y\Rightarrow Z}
  
, is an alternative notation for 
  
    
      
        
          Z
          
            Y
          
        
      
    
    {\displaystyle Z^{Y}}
  
. The above adjunction results translate to implication (
  
    
      
        ⇒:
        H
        ×
        H
        →
        H
      
    
    {\displaystyle \Rightarrow :H\times H\to H}
  
) being <a href="/facts/Right_adjoint/bJ1P7AN2">right adjoint</a> to <a href="/facts/Join_and_meet/cYcLhRIR">meet</a> (
  
    
      
        ∧
        :
        H
        ×
        H
        →
        H
      
    
    {\displaystyle \wedge :H\times H\to H}
  
). This adjunction can be written as 
  
    
      
        (
        −
        ∧
        Y
        )
        ⊣
        (
        Y
        ⇒
        −
        )
      
    
    {\displaystyle (-\wedge Y)\dashv (Y\Rightarrow -)}
  
, or more fully as:

(
        −
        ∧
        Y
        )
        :
        H
        
          
            
              
                
                  ⊤
                  ⟵
                
              
              
                ⟶
              
            
          
        
        H
        :
        (
        Y
        ⇒
        −
        )
      
    
    {\displaystyle (-\wedge Y):H{\stackrel {\longrightarrow }{\underset {\longleftarrow }{\top }}}H:(Y\Rightarrow -)}

</p><p>In the <a href="/facts/Category_of_topological_spaces/qH4SaEWa">category of topological spaces</a>, the exponential object 
  
    
      
        
          Z
          
            Y
          
        
      
    
    {\displaystyle Z^{Y}}
  
 exists provided that 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
 is a <a href="/facts/Locally_compact_space/hesalT7q">locally compact</a> <a href="/facts/Hausdorff_space/gbKIrusl">Hausdorff space</a>. In that case, the space 
  
    
      
        
          Z
          
            Y
          
        
      
    
    {\displaystyle Z^{Y}}
  
 is the set of all <a href="/facts/Continuous_function_(topology)/sKbl02pB">continuous functions</a> from 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
 to 
  
    
      
        Z
      
    
    {\displaystyle Z}
  
 together with the <a href="/facts/Compact-open_topology/qMGsW8gk">compact-open topology</a>. The evaluation map is the same as in the category of sets; it is continuous with the above topology.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> If 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
 is not locally compact Hausdorff, the exponential object may not exist (the space 
  
    
      
        
          Z
          
            Y
          
        
      
    
    {\displaystyle Z^{Y}}
  
 still exists, but it may fail to be an exponential object since the evaluation function need not be continuous). For this reason the category of topological spaces fails to be <a href="/facts/Cartesian_closed/mPXAQps2">cartesian closed</a>.
However, the category of locally compact topological spaces is not cartesian closed either, since 
  
    
      
        
          Z
          
            Y
          
        
      
    
    {\displaystyle Z^{Y}}
  
 need not be locally compact for locally compact spaces 
  
    
      
        Z
      
    
    {\displaystyle Z}
  
 and 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
. A cartesian closed category of spaces is, for example, given by the <a href="/facts/Subcategory/k4lEeDDb">full subcategory</a> spanned by the <a href="/facts/Compactly_generated_space/2A5umYGa">compactly generated</a> Hausdorff spaces.
</p><p>In <a href="/facts/Functional_programming_language/cdxqPEIM">functional programming languages</a>, the morphism 
  
    
      
        eval
      
    
    {\displaystyle \operatorname {eval} }
  
 is often <a href="/facts/Apply/2Emq4vwk"> called 
  
    
      
        apply
      
    
    {\displaystyle \operatorname {apply} }
  
</a>, and the syntax 
  
    
      
        λ
        g
      
    
    {\displaystyle \lambda g}
  
 is often <a href="/facts/Function_application/p4auE0QF">written 
  
    
      
        curry
        ⁡
        (
        g
        )
      
    
    {\displaystyle \operatorname {curry} (g)}
  
</a>. The morphism 
  
    
      
        eval
      
    
    {\displaystyle \operatorname {eval} }
  
 should not be confused with the <a href="/facts/Eval/lPVJnDNR">eval</a> function in some <a href="/facts/Programming_language/cWTYbgWa">programming languages</a>, which evaluates quoted expressions.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Closed_monoidal_category/FBe3ecNJ">Closed monoidal category</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li>Adámek, Jiří; <a href="/facts/Horst_Herrlich/p1Q5EPo7">Horst Herrlich</a>; George Strecker (2006) [1990]. <a href="http://katmat.math.uni-bremen.de/acc/"><i>Abstract and Concrete Categories (The Joy of Cats)</i></a>. John Wiley & Sons.</li>
<li><a href="/facts/Steve_Awodey/f5KVuGhv">Awodey, Steve</a> (2010). "Chapter 6: Exponentials". <i>Category theory</i>. Oxford New York: Oxford University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0199237180.</li>
<li>Mac Lane, Saunders (1998). "Chapter 4: Adjoints". <i><a href="/facts/Categories_for_the_working_mathematician/SLQDat9B">Categories for the working mathematician</a></i>. New York: Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0387984032.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="https://web.archive.org/web/20080916162345/http://www.j-paine.org/cgi-bin/webcats/webcats.php">Interactive Web page </a> which generates examples of exponential objects and other categorical constructions. Written by <a href="https://web.archive.org/web/20081223001815/http://www.j-paine.org/">Jocelyn Paine</a>.</li></ul>

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

<ol>
<li id="fn:1"><p>Exponential law for spaces at the nLab <a href="https://ncatlab.org/nlab/show/exponential+law+for+spaces" target="_blank">https://ncatlab.org/nlab/show/exponential+law+for+spaces</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Convenient category of topological spaces at the nLab <a href="https://ncatlab.org/nlab/show/convenient+category+of+topological+spaces" target="_blank">https://ncatlab.org/nlab/show/convenient+category+of+topological+spaces</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Goldblatt, Robert (1984). "Chapter 3: Arrows instead of epsilon". Topoi : the categorial analysis of logic. Studies in Logic and the Foundations of Mathematics #98 (Revised ed.). North-Holland. p. 72. ISBN 978-0-444-86711-7. <a href="978-0-444-86711-7" target="_blank">978-0-444-86711-7</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Mac Lane, Saunders (1978). "Chapter 4: Adjoints". Categories for the working mathematician. graduate texts in mathematics. Vol. 5 (2nd ed.). Springer-Verlag. p. 98. doi:10.1007/978-1-4757-4721-8_5. ISBN 978-0387984032. <a href="978-0387984032" target="_blank">978-0387984032</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Joseph J. Rotman, An Introduction to Algebraic Topology (1988) Springer-Verlag ISBN 0-387-96678-1 (See Chapter 11 for proof.) <a href="/wiki/Joseph_J._Rotman" target="_blank">/wiki/Joseph_J._Rotman</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
</ol>

Exponential object open-in-new

Exponential object