The strict globular omega-categories formalize the execution paths of a parallel automaton and the homotopies between them. One associates to such (and any) omega-category C three homology theories. The first one is called the globular homology. It contains the oriented loops of C. The two other ones are called the negative (resp. positive) corner homology. They contain in a certain manner the branching areas of execution paths or negative corners (resp. the merging areas of execution paths or positive corners) of C. Two natural linear maps called the negative (resp. the positive) Hurewicz morphism from the globular homology to the negative (resp. positive) corner homology are constructed. We explain the reason why these constructions allow the reinterpretation of some geometric problems coming from computer science.

A local po-space is a gluing of topological spaces which are equipped with a closed partial ordering representing the time flow. They are used as a formalization of higher dimensional automata (see for instance [6]) which model concurrent systems in computer science. It is known [11] that there are two distinct notions of deformation of higher dimensional automata, “spatial” and “temporal”, leaving invariant computer scientific properties like presence or absence of deadlocks. Unfortunately, the formalization of these notions is still unknown in the general case of local po-spaces. We introduce here a particular kind of local po-space, the “globular CW-complexes”, for which we formalize these notions of deformations and which are sufficient to formalize

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.

This paper is a survey of the new notions and results scattered in [13], [11] and [12]. Starting from a formalization of higher dimensional automata (HDA) by strict globular !-categories, the construction of a diagram of simplicial sets over the three-object small category gl ! + is exposed. Some of the properties discovered so far on the corresponding simplicial homology theories are explained, in particular their links with geometric problems coming from concurrency theory in computer science.

This work is a contribution to a recent field, Directed Algebraic Topology. Categories which appear as fundamental categories of ‘directed structures’, e.g. ordered topological spaces, have to be studied up to appropriate notions of directed homotopy equivalence, which are more general than ordinary equivalence of categories. Here we introduce past and future equivalences of categories—sort of symmetric versions of an adjunction—and use them and their combinations to get ‘directed models ’ of a category; in the simplest case, these are the join of the least full reflective and the least full coreflective subcategory.

this report belongs to topology. Its motivations are, however, derived from investigations of concurrent processes. For a general discussion of concurrent processes giving rise to topological spaces, the reader is referred for instance to Pratt [7] or [8]; or to Fajstrup, Goubault, Raussen [1]; or to Goubault [6]. In the most general terms, the mathematical structures modeling cooperation between processes are endowed with a partial order and with a topology. The partial order relates events to other events they may be causes of, which captures causality and time succession. Every component of a system of processes may have its own time and these private times have only to be in tune when inter-process communication is taking place. Because of this, the order does not have to be total. The topology in a model provides a notion of closeness between particular execution trajectories. For instance, when a process is slightly slowed down without changing the order in which it communicates with other processes (cooperates or competes for shared resources), the resulting execution is close to the original one. On the other hand, if two executions differ in the way the communication takes place, they are far from each other. In this sense, communications create "holes" in the spaces modeling concurrency. Topology makes a convenient language to discuss this. A standard topological toolbox containing homotopies and homologies is not readily applicable for a classification of the holes, since it disregards the time- or causality-induced ordering which is paramount in computer science applications. Researchers have, therefore, attempted to adapt these concepts. Some discussion and comparison to earlier work may be found in Sec. 5. This report is a step towards finding useful invar...

This thesis is concerned with formal semantics and models for concurrent computational systems, that is, systems consisting of a number of parallel computing sequential systems, interacting with each other and the environment. A formal semantics gives meaning to computational systems by describing their behaviour in a mathematical model. For concurrent systems the interesting aspect of their computation is often how they interact with the environment during a computation and not in which state they terminate, indeed they may not be intended to terminate at all. For this reason they are often referred to as reactive systems, to distinguish them from traditional calculational systems, as e.g. a program calculating your income tax, for which the interesting behaviour is the answer it gives when (or if) it terminates, in other words the (possibly partial) function it computes between input and output. Church's thesis tells us that regardless of whether we choose the lambda calculus, Turing machines, or almost any modern programming language such as C or Java to describe calculational systems, we are able to describe exactly the same class of functions. However, there is no agreement on observable behaviour for concurrent reactive systems, and consequently there is no correspondent to Church's thesis. A result of this fact is that an overwhelming number of di#erent and often competing notions of observable behaviours, primitive operations, languages and mathematical models for describing their semantics, have been proposed in the litterature on concurrency.

Abstract. Many categories have been used to model concurrency. Using any of these, the challenge is to reduce a given model to a smaller representation which nevertheless preserves the relevant computer-scientific information. That is, one wants to replace a given model with a simpler model with the same directed homotopy-type. Unfortunately, the obvious definition of directed homotopy equivalence is too coarse. This paper introduces the notion of context to refine this definition. 1.

Abstract. Locally partial-ordered spaces (local po-spaces) have been used to model concurrent systems. We provide equivalences for these spaces by

This report gives a formal topological semantics to inductively defined concurrent systems and investigates the properties of such systems. We allow loops and infinitely running computations, which is new in the topological investigations of concurrency. In this more general setting, we prove the equivalent to the result from [2] that deadlocks and unsafe points can be found using a finite number of deloopings. 1 Introduction The idea of using geometric methods for concurrency is not new. The geometric viewpoint referred to in the title goes back to Dijkstra [1], who introduces higher dimensional geometric objects, progress graphs, and abstracts a process to be a series of actions locking and releasing a set of resources, which may then be shared with other processes thus giving rise to coordination problems. This idea has later been refined or independently reinvented by a number of authors. For an overview see, for instance, [3]. Concurrent systems, as most things in computer scienc...