We consider processes consisting of a category of states varying over a control category as prescribed by a unique factorisation lifting functor. After a brief analysis of the structure of general processes in this setting, we restrict attention to linearly-controlled ones. To this end, we introduce and study a notion of path-linearisable category in which any two paths of morphisms with equal composites can be linearised (or interleaved) in a canonical fashion. Our main result is that categories of linearly-controlled processes (viz., processes controlled by path-linearisable categories) are sheaf models. Introduction This work is an investigation into the mathematical structure of processes. The processes to be considered embody a notion of state space varying according to a control. This we formalise as a category of states (and their inter-relations) Xequipped with a control functor X C f . There are different ways in which the control category C may be required to control t...

ABSTRACT. Inthis paper, we consider those morphisms p: P − → B of posets for which the induced geometric morphism of presheaf toposes is exponentiable in the category of Grothendieck toposes. In particular, we show that a necessary condition is that the induced map p ↓ : P ↓ − → B ↓ is exponentiable in the category of topological spaces, where P ↓ is the space whose points are elements of P and open sets are downward closed subsets of P. Along the way, we show that p ↓ : P ↓ − → B ↓ is exponentiable if and only if p: P − → B is exponentiable in the category of posets and satisfies an additional compactness condition. The criteria for exponentiability of morphisms of posets is related to (but weaker than) the factorization-lifting property for exponentiability of morphisms in the

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.

ABSTRACT. Some basic features of the simultaneous inclusion of discrete fibrations and discrete opfibrations in categories over a base category X are considered. In particular, we illustrate the formulas (↓P)x = ten(x/X, P) ; (P↓)x = hom(X/x, P)

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.

Abstract. We illustrate the formula (↓p)x = Γ!(x/p), which gives the reflection ↓p of a category p: P → X over X in discrete fibrations. One of its proofs is based on a “complement operator ” which takes a discrete fibration A to the functor ¬A, right adjoint to Γ!(A × −) : Cat/X → Set and valued in discrete opfibrations. Some consequences and applications are presented. 1.

In this paper we introduce the notion of an extensive 2-category, to be thought of as a "2-category of generalized spaces". We consider an extensive 2-category K equipped with a binary-product-preserving pseudofunctor C : K CAT, which we think of as specifying the "coverings" of our generalized spaces. We prove, in this context, a van Kampen theorem which generalizes and refines one of Brown and Janelidze. The local properties required in this theorem are stated in terms of morphisms of effective descent for the pseudofunctor C . We specialize the general van Kampen theorem to the 2-category Top S of toposes bounded over an elementary topos S , and to its full sub 2-category LTop S determined by the locally connected toposes, after showing both of these 2-categories to be extensive. We then consider three particular notions of coverings on toposes corresponding respectively to local homeomorphisms, covering projections, and unramified morphisms; in each case we deduce a suitable version of a van Kampen theorem in terms of coverings and, under further hypotheses, also one in terms of fundamental groupoids. An application is also given to knot groupoids and branched coverings. Along the way

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.