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The Convergence Approach to Exponentiable Maps
 352 MARIA MANUEL CLEMENTINO, DIRK HOFMANN AND WALTER
, 2000
"... Exponentiable maps in the category Top of topological spaces are characterized by an easy ultrafilterinterpolation 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 ..."
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Exponentiable maps in the category Top of topological spaces are characterized by an easy ultrafilterinterpolation 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 WhiteheadMichael characterization of locally compact Hausdorff spaces, we characterize exponentiable maps of Top between Hausdorff spaces as restrictions of perfect maps to open subspaces.
The Hurwitz Action and Braid Group Orderings
 Theory Appl. Categ
, 2001
"... In connection with the socalled 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 2sphere with finitely many markedpoints. Our approach to ra ..."
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In connection with the socalled 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 2sphere with finitely many markedpoints. 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 orderinvariant action in the ramification groupoid of the sphere with n + 1 markedpoints. Leftinvariant 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 orderpreserving action.
Exponentiability Of Perfect Maps: Four Approaches
, 2002
"... 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 setvalued sh ..."
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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 setvalued sheaves on a topological space.
EXPONENTIABILITY VIA DOUBLE CATEGORIES
"... 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 � ..."
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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 pseudofunctor. 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 pseudofunctor 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.