View source for Cartesian-closed category

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:

A1) there exists a [[terminal object]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130030/c1300302.png" />;

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" />;

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" />.

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>

have each a specified right-adjoint, written respectively as:

<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>

<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>

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" />;

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;

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" />;

E5) the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130030/c13003028.png" /> 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;

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.

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

====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>