A category such that the following axioms are satisfied:
A1) there exists a terminal object ;
A2) for any pair , of objects of there exist a product and given projections , ;
A3) for any pair , of objects of there exist an object and an evaluation arrow such that for any arrow there is a unique arrow with .
These conditions are equivalent to the following: is a category with given products such that the functors
have each a specified right-adjoint, written respectively as:
Some examples of Cartesian-closed categories are:
E1) any Heyting algebra ;
E2) the category for any small category with the category of (small) sets — in particular itself;
E3) the category of sheaves over a topological space, and more generally a (Grothendieck) topos;
E4) any elementary topos ;
E5) the category of all (small) categories;
E6) the category of graphs and their homomorphisms;
E7) the category - of -CPOs.
These definitions can all be put into a purely equational form.
|[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)|
Cartesian-closed category. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cartesian-closed_category&oldid=14645