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The category of all directed graphs is cartesian closed; this is a functor category as explained under functor category.
In algebraic topology, cartesian closed categories are particularly easy to work with, and it is regrettable that neither the category of topological spaces with continuous maps nor the category of smooth manifolds with smooth maps is cartesian closed.
Certain cartesian closed categories, the topoi, have been proposed as a general setting for mathematics, instead of traditional set theory.
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