Exponentiability via Double Categories

For a small category B and a double category D, let LaxN (B,D) denote the category whose objects are vertical normal lax functors B //D and morphisms are horizontal lax transformations. It is well known that LaxN (B,Cat) ≃ Cat/B, where Cat is the double category of small categories, functors, and profunctors. In [19], we generalized this equivalence to certain double categories, in the case where B is a finite poset. In [23], Street showed that Y //B is exponentiable in Cat/B if and only if the corresponding normal lax functor B // Cat is a pseudo-functor. Using our generalized equivalence, we show that a morphism Y // B is exponentiable in D0/B if and only if the corresponding normal lax functor B // D is a pseudo-functor plus an additional condition that holds for allX //!B in Cat. Thus, we obtain a single theorem which yields characterizations of certain exponentiable morphisms of small categories, topological spaces, locales, and posets.