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A [[Category|category]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130030/c1300301.png" /> such that the following axioms are satisfied:
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A [[category]] $\mathfrak{C}$ such that the following axioms are satisfied:
  
A1) there exists a terminal object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130030/c1300302.png" />;
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A1) there exists a [[terminal object]] $\mathbf{1}$;
  
A2) for any pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130030/c1300303.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130030/c1300304.png" /> of objects of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130030/c1300305.png" /> there exist a product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130030/c1300306.png" /> and given projections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130030/c1300307.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130030/c1300308.png" />;
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A2) for any pair $A,B$ of objects of $\mathfrak{C}$ there exist a product $A \times B$ and given projections $\mathrm{pr}_1 : A \times B \rightarrow A$, $\mathrm{pr}_2 : A \times B \rightarrow B$;
  
A3) for any pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130030/c1300309.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130030/c13003010.png" /> of objects of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130030/c13003011.png" /> there exist an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130030/c13003012.png" /> and an evaluation arrow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130030/c13003013.png" /> such that for any arrow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130030/c13003014.png" /> there is a unique arrow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130030/c13003015.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130030/c13003016.png" />.
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A3) for any pair $A,B$ of objects of $\mathfrak{C}$ there exist an object $A^B$ and an evaluation arrow $\mathrm{ev} : A^B \times B \rightarrow A$ such that for any arrow $F : C \times A \rightarrow B$ there is a unique arrow $[f] : C \rightarrow A^B$ with $\mathrm{ev}\circ ([f]\times \mathrm{id}_A) = f$.
 
 
These conditions are equivalent to the following: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130030/c13003017.png" /> is a category with given products such that the functors
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130030/c13003018.png" /></td> </tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130030/c13003019.png" /></td> </tr></table>
 
  
 +
These conditions are equivalent to the following: $\mathfrak{C}$ is a category with given products such that the functors
 +
$$
 +
\mathfrak{C} \rightarrow \mathbf{1}\,\ \ c \mapsto 0\,;
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$$
 +
$$
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\mathfrak{C} \rightarrow \mathfrak{C} \times \mathfrak{C}\,\ \ c \mapsto \langle c,c \rangle \,;
 +
$$
 +
$$
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\mathfrak{C} \rightarrow \mathfrak{C}\,\ \ c \mapsto c \times b
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$$
 
have each a specified right-adjoint, written respectively as:
 
have each a specified right-adjoint, written respectively as:
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130030/c13003020.png" /></td> </tr></table>
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0 \mapsto t\,;
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130030/c13003021.png" /></td> </tr></table>
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$$
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\langle a,b \rangle \mapsto a \times b\,;
 +
$$
 +
$$
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c \mapsto  c^b \ .
 +
$$
  
 
Some examples of Cartesian-closed categories are:
 
Some examples of Cartesian-closed categories are:
  
E1) any Heyting algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130030/c13003022.png" />;
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E1) any [[Heyting algebra]] $\mathcal{H}$;
  
E2) the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130030/c13003023.png" /> for any [[Small category|small category]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130030/c13003024.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130030/c13003025.png" /> the category of (small) sets — in particular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130030/c13003026.png" /> itself;
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E2) the category $\mathsf{Set}^{\mathfrak{C}}$ for any [[small category]] $\mathfrak{C}$ with $\mathsf{Set}$ the category of (small) sets — in particular $\mathsf{Set}$ itself;
  
 
E3) the category of sheaves over a topological space, and more generally a (Grothendieck) topos;
 
E3) the category of sheaves over a topological space, and more generally a (Grothendieck) topos;
  
E4) any elementary [[Topos|topos]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130030/c13003027.png" />;
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E4) any elementary [[topos]] $\mathcal{E}$;
  
E5) the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130030/c13003028.png" /> of all (small) categories;
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E5) the category $\mathsf{Cat}$ of all (small) categories;
  
E6) the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130030/c13003029.png" /> of graphs and their homomorphisms;
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E6) the category $\mathsf{Graph}$ of graphs and their homomorphisms;
  
E7) the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130030/c13003030.png" />-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130030/c13003031.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130030/c13003032.png" />-CPOs.
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E7) the category $\omega$-$\mathsf{CPO}$ of $\omega$-CPOs.
  
 
These definitions can all be put into a purely equational form.
 
These definitions can all be put into a purely equational form.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Barr,  C. Wells,  "Category theory for computing science" , CRM  (1990)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Lambek,  P.J. Scott,  "Introduction to higher order categorical logic" , Cambridge Univ. Press  (1986)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. MacLane,  I. Moerdijk,  "Sheaves in geometry and logic" , Springer  (1992)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  S. MacLane,  "Categories for the working mathematician" , Springer  (1971)</TD></TR></table>
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<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Barr,  C. Wells,  "Category theory for computing science" , CRM  (1990)</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Lambek,  P.J. Scott,  "Introduction to higher order categorical logic" , Cambridge Univ. Press  (1986)</TD></TR>
 +
<TR><TD valign="top">[a3]</TD> <TD valign="top">  S. MacLane,  I. Moerdijk,  "Sheaves in geometry and logic" , Springer  (1992)</TD></TR>
 +
<TR><TD valign="top">[a4]</TD> <TD valign="top">  S. MacLane,  "Categories for the working mathematician" , Springer  (1971)</TD></TR>
 +
</table>
 +
 
 +
{{TEX|done}}

Latest revision as of 20:31, 27 December 2017

A category $\mathfrak{C}$ such that the following axioms are satisfied:

A1) there exists a terminal object $\mathbf{1}$;

A2) for any pair $A,B$ of objects of $\mathfrak{C}$ there exist a product $A \times B$ and given projections $\mathrm{pr}_1 : A \times B \rightarrow A$, $\mathrm{pr}_2 : A \times B \rightarrow B$;

A3) for any pair $A,B$ of objects of $\mathfrak{C}$ there exist an object $A^B$ and an evaluation arrow $\mathrm{ev} : A^B \times B \rightarrow A$ such that for any arrow $F : C \times A \rightarrow B$ there is a unique arrow $[f] : C \rightarrow A^B$ with $\mathrm{ev}\circ ([f]\times \mathrm{id}_A) = f$.

These conditions are equivalent to the following: $\mathfrak{C}$ is a category with given products such that the functors $$ \mathfrak{C} \rightarrow \mathbf{1}\,\ \ c \mapsto 0\,; $$ $$ \mathfrak{C} \rightarrow \mathfrak{C} \times \mathfrak{C}\,\ \ c \mapsto \langle c,c \rangle \,; $$ $$ \mathfrak{C} \rightarrow \mathfrak{C}\,\ \ c \mapsto c \times b $$ have each a specified right-adjoint, written respectively as: $$ 0 \mapsto t\,; $$ $$ \langle a,b \rangle \mapsto a \times b\,; $$ $$ c \mapsto c^b \ . $$

Some examples of Cartesian-closed categories are:

E1) any Heyting algebra $\mathcal{H}$;

E2) the category $\mathsf{Set}^{\mathfrak{C}}$ for any small category $\mathfrak{C}$ with $\mathsf{Set}$ the category of (small) sets — in particular $\mathsf{Set}$ itself;

E3) the category of sheaves over a topological space, and more generally a (Grothendieck) topos;

E4) any elementary topos $\mathcal{E}$;

E5) the category $\mathsf{Cat}$ of all (small) categories;

E6) the category $\mathsf{Graph}$ of graphs and their homomorphisms;

E7) the category $\omega$-$\mathsf{CPO}$ of $\omega$-CPOs.

These definitions can all be put into a purely equational form.

References

[a1] M. Barr, C. Wells, "Category theory for computing science" , CRM (1990)
[a2] J. Lambek, P.J. Scott, "Introduction to higher order categorical logic" , Cambridge Univ. Press (1986)
[a3] S. MacLane, I. Moerdijk, "Sheaves in geometry and logic" , Springer (1992)
[a4] S. MacLane, "Categories for the working mathematician" , Springer (1971)
How to Cite This Entry:
Cartesian-closed category. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cartesian-closed_category&oldid=14645
This article was adapted from an original article by M. Eytan (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article