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.

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 KZ-doctrines, such as the closure under composition of discrete fibrations for a KZ-doctrine, in the sense of Bunge and Funk [6].

We continue the investigation of the extension into the topos realm of the concepts introduced by R.H. Fox [10] and E. Michael [22] in connection with topological singular coverings. In particular, we construct an analogue of the Michael completion of a spread and compare it with the analogue of the Fox completion obtained earlier by the first two named authors [4]. Two ingredients are present in our analysis of geometric morphisms ': F ! E between toposes bounded over a base topos S. The first is the nature of the domain of ', which need only be assumed to be a "definable dominance" over S, a condition that is trivially satisfied if S is a Boolean topos. The Heyting algebras arising from the object S of truth values in the base topos play a special role in that they classify the de nable monomorhisms in those toposes. The geometric morphisms F ! F 0 over E which preserve these Heyting algebras (and that are not typically complete) are said to be strongly pure. The second is the nature of ' itself, which is assumed to be some kind of a spread. Applied to a spread, the (strongly pure, weakly entire) factorization obtained here gives what we call the "Michael completion" of the given spread. Whereas the Fox complete spreads over a topos E correspond to the S-valued Lawvere distributions on E [21] and relate to the distribution algebras [7], the Michael complete spreads seem to correspond to some sort of "S-additive measures" on E whose analysis we do not pursue here. We close the paper with several other open questions and directions for future work.

this paper a systematic study is made of various notions of "proper map" in the context of toposes. Modulo some separation conditions, a proper map Y ! X of spaces is generally understood to be a continuous function which preserves compactness of subspaces under inverse image, and which therefore in particular has compact fibers. In this spirit, a first definition of proper map between toposes was put forward by Johnstone in []. There, a map of toposes f : F ! E was called proper if f

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 zero-dimensional 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.