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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.
On Branched Covers In Topos Theory
 Theory and Applications of Categories
, 2000
"... We present some new findings concerning branched covers in topos theory. Our discussion involves a particular subtopos of a given topos that can be described as the smallest subtopos closed under small coproducts in the including topos. Our main result is a description of the covers of this subtopos ..."
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We present some new findings concerning branched covers in topos theory. Our discussion involves a particular subtopos of a given topos that can be described as the smallest subtopos closed under small coproducts in the including topos. Our main result is a description of the covers of this subtopos as a category of fractions of branched covers, in the sense of Fox [10], of the including topos. We also have some new results concerning the general theory of KZdoctrines, such as the closure under composition of discrete fibrations for a KZdoctrine, in the sense of Bunge and Funk [6].
FUNDAMENTAL PUSHOUT TOPOSES
"... Abstract. The author [2, 5] introduced and employed certain ‘fundamental pushout toposes ’ in the construction of the coverings fundamental groupoid of a locally connected topos. Our main purpose in this paper is to generalize this construction without the local connectedness assumption. In the spir ..."
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Abstract. The author [2, 5] introduced and employed certain ‘fundamental pushout toposes ’ in the construction of the coverings fundamental groupoid of a locally connected topos. Our main purpose in this paper is to generalize this construction without the local connectedness assumption. In the spirit of [16, 10, 8] we replace connected components by constructively complemented, or definable, monomorphisms [1]. Unlike the locally connected case, where the fundamental groupoid is localic prodiscrete and its classifying topos is a Galois topos, in the general case our version of the fundamental groupoid is a locally discrete progroupoid and there is no intrinsic Galois theory in the sense of [19]. We also discuss covering projections, locally trivial, and branched coverings without local connectedness by analogy with, but also necessarily departing from, the locally connected case [13, 11, 7]. Throughout, we work abstractly in a setting given axiomatically by a category V of locally discrete locales that has as examples the categories D of discrete locales, and Z of zerodimensional locales [9]. In this fashion we are led to give unified and often simpler proofs of old theorems in the locally connected case, as well as new ones without that assumption.
TOPOS THEORETIC ASPECTS OF SEMIGROUP ACTIONS
"... Abstract. We define the notion of a torsor for an inverse semigroup, which is based on semigroup actions, and prove that this is precisely the structure classified by the topos associated with an inverse semigroup. Unlike in the group case, not all settheoretic torsors are isomorphic: we shall give ..."
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Abstract. We define the notion of a torsor for an inverse semigroup, which is based on semigroup actions, and prove that this is precisely the structure classified by the topos associated with an inverse semigroup. Unlike in the group case, not all settheoretic torsors are isomorphic: we shall give a complete description of the category of torsors. We explain how a semigroup prehomomorphism gives rise to an adjunction between a restrictionsofscalars functor and a tensor product functor, which we relate to the theory of covering spaces and Eunitary semigroups. We also interpret for semigroups the Lawvereproduct of a sheaf and distribution, and finally, we indicate how the theory might be extended to general semigroups, by defining a notion of torsor and a classifying topos for those. 1.