In this paper we introduce a model for a wide class of computational systems, whose behaviour can be described by certain rewriting rules. We gathered our inspiration both from the world of term rewriting, in particular from the rewriting logic framework [Mes92], and of concurrency theory: among the others, the structured operational semantics [Plo81], the context systems [LX90] and the structured transition systems [CM92] approaches. Our model recollects many properties of these sources: first, it provides a compositional way to describe both the states and the sequences of transitions performed by a given system, stressing their distributed nature. Second, a suitable notion of typed proof allows to take into account also those formalisms relying on the notions of synchronization and side-effects to determine the actual behaviour of a system. Finally, an equivalence relation over sequences of transitions is defined, equipping the system under analysis with a concurrent semantics, ...

We explore the combinatorial properties of the branching areas of execution paths in higher dimensional automata. Mathematically, this means that we investigate the combinatorics of the negative corner (or branching) homology of a globular #-category and the combinatorics of a new homology theory called the reduced branching homology. The latter is the homology of the quotient of the branching complex by the sub-complex generated by its thin elements. Conjecturally it coincides with the non reduced theory for higher dimensional automata, that is #-categories freely generated by precubical sets. As application, we calculate the branching homology of some #-categories and we give some invariance results for the reduced branching homology. We only treat the branching side. The merging side, that is the case of merging areas of execution paths is similar and can be easily deduced from the branching side.

Abstract. As an example of the categorical apparatus of pseudo algebras over 2-theories, we show that pseudo algebras over the 2-theory of categories can be viewed as pseudo double categories with folding or as appropriate 2-functors into bicategories. Foldings are equivalent to connection pairs, and also to thin structures if the vertical and horizontal morphisms coincide. In a sense, the squares of a double category with folding are determined in a functorial way by the 2-cells of the horizontal 2-category. As a special case, strict 2-algebras with one object and everything invertible are crossed modules under a group.

This paper is the second in a series exploring the properties of a functor which assigns a homotopy double groupoid with connections to a Hausdorff space. We show that this functor satisfies a version of the van Kampen theorem, and so is a suitable tool for nonabelian, 2-dimensional, local-to-global problems. The methods are analogous to those developed by Brown and Higgins for similar theorems for other higher homotopy groupoids. An integral part of the proof is a detailed discussion of commutative cubes in a double category with connections, and a proof of the key result that any composition of commutative cubes is commutative. These results have recently been generalised to all dimensions by Philip Higgins.

The relationships between thin elements, commutative shells and connections in cubical ω-categories are explored by a method which does not involve the use of pasting theory or nerves of ω-categories (both of which were previously needed for this purpose; see [2], Section 9). It is shown that composites of commutative shells are commutative and that thin structures are equivalent to appropriate sets of connections; this work extends to all dimensions the results proved in dimensions 2 and 3 in [7, 6].

We give a geometric description of the free double category generated by a double reflexive graph. Its cells are homotopy classes of colourings of certain rectangular complexes in the plane. A number of examples illustrate the wide variety of combinatorial properties of the plane this touches. 1 Introduction Double categories were introduced in 1963 by Ehresmann [3]. Since then, considerable work has been done, much of it in the context of homotopy theory (see XXX). Of course, a 2-category is a special kind of double category [8] so that all work done on 2-categories (and bicategories) is also saying something about double categories. In particular, the pasting schemes of [4] and the computads of [9] bear a close relationship to our work. They can be thought of in various ways, each with its own advantages. The concise definition is that they are category objects in Cat; however, this phrase slightly obscures the symmetry between the "category object" and "Cat". It can also be u...

Abstract. Two constructions of paths in double categories are studied, providing algebraic versions of the homotopy groupoid of a space. Universal properties of these constructions are presented. The first is seen as the codomain of the universal oplax morphism of double categories and the second, which is a quotient of the first, gives the universal normal oplax morphism. Normality forces an equivalence relation on cells, a special case of which was seen before in the free adjoint construction. These constructions are the object part of 2-comonads which are shown to be oplax idempotent. The coalgebras for these comonads turn out to be Leinster’s fc-multicategories, with representable identities in the second case.

Abstract. This is a survey on the use of some internal higher categorical structures in algebraic topology and homotopy theory. After providing a general view of the area and its applications, we concentrate on the algebraic modelling of connected (n + 1)-types through cat n-groups. 1.

Abstract. The relationships between thin elements, commutative shells and connec-tions in cubical!-categories are explored by a method which does not involve the useof pasting theory or nerves of!-categories (both of which were previously needed forthis purpose; see [2], Section 9). It is shown that composites of commutative shells are commutative and that thin structures are equivalent to appropriate sets of connections;this work extends to all dimensions the results proved in dimensions 2 and 3 in [7, 6]. Introduction Thin structures in simplicial sets were introduced by Dakin in [8] and were applied to cubical sets in [3, 4, 5]. In the cubical case a thin structure is equivalent to an!-groupoid structure.

Abstract. In this paper we obtain several model structures on DblCat, the category of small double categories. Our model structures have three sources. We first transfer across a categorificationnerve adjunction. Secondly, we view double categories as internal categories in Cat and take as our weak equivalences various internal equivalences defined via Grothendieck topologies. Thirdly, DblCat inherits a model structure as a category of algebras over a 2-monad. Some of these model structures coincide and the different points of view give us further results about cofibrant replacements and cofibrant objects. As part of this program we give explicit descriptions and discuss properties of free double categories, quotient double categories, colimits of double categories, and several nerves