Exponentiable maps in the category Top of topological spaces are characterized by an easy ultrafilter-interpolation property, in generalization of a recent result by Pisani for spaces. From this characterization we deduce that perfect (= proper and separated) maps are exponentiable, generalizing the classical result for compact Hausdorff spaces. Furthermore, in generalization of the Whitehead-Michael characterization of locally compact Hausdorff spaces, we characterize exponentiable maps of Top between Hausdorff spaces as restrictions of perfect maps to open subspaces.

In connection with the so-called Hurwitz action of homeomorphisms in ramified covers we define a groupoid, which we call a ramification groupoid of the 2sphere, constructed as a certain path groupoid of the universal ramified cover of the 2-sphere with finitely many marked-points. Our approach to ramified covers is based on cosheaf spaces, which are closely related to Fox's complete spreads. A feature of a ramification groupoid is that it carries a certain order structure. The Artin group of braids of n strands has an order-invariant action in the ramification groupoid of the sphere with n + 1 marked-points. Left-invariant linear orderings of the braid group such as the Dehornoy ordering may be retrieved. Our work extends naturally to the braid group on countably many generators. In particular, we show that the underlying set of a free group on countably many generators (minus the identity element) can be linearly ordered in such a way that the classical Artin representation of a braid as an automorphism of the free group is an order-preserving action.

Two proofs of the exponentiability of perfect maps are presented and compared to two other recent approaches. One of the proofs is an elementary approach including a direct construction of the exponentials. The other, implicit in the literature, uses internal locales in the topos of set-valued sheaves on a topological space.

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 all X �!B in Cat. Thus, we obtain a single theorem which yields characterizations of certain exponentiable morphisms of small categories, topological spaces, locales, and posets. 1.