We consider two approaches to cartesian closed double categories generalizing two definitions which are equivalent for 1-categories, and give examples to show that the two differ in the double category case. One approach, previously considered in [N20], requires the lax functor (-) x Y on D to have a right adjoint (-)^Y, for every object Y, while the other supposes that the exponentials are given by a lax bifunctor D^op x D --> D also involving vertical (i.e., loose) morphisms of D. Examples include the double categories Cat, Pos, Top, Loc, and Quant, whose objects are small categories, posets, topological spaces, locales, and commutative quantales, respectively; as well as, the double categories Span(D) and Q-Rel, whose vertical morphisms are spans in a category D with pullback and relations valued in a locale Q, respectively.
Keywords: double categories, cartesian closed, spans/cospans, quantales, relations
2020 MSC: 18N10, 18D15, 18B10, 18F75, 54C35
Theory and Applications of Categories, Vol. 40, 2024, No. 3, pp 63-79.
Published 2024-03-26.
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