In order to model concurrent parallel computing, I study topological spaces with a distinguished set of paths, called directed paths. The topological space models the state space of the system, and the directed paths model the execution paths. To reduce to the essentially different executions, we reduce to homotopy classes of directed paths. To reduce the size of the state space we apply future retracts and past retracts. We also prove some theorems for doing these constructions in a piece-by-piece fashion.
More technically, since the directed paths are generally not reversible, the directed homotopy classes of directed paths do not assemble into a groupoid, and there is no direct analog of the fundamental group. However, they do assemble into a category, called the fundamental category. I define models of the fundamental category, such as the fundamental bipartite graph, and minimal extremal models which are shown to generalize the fundamental group. In addition, I prove van Kampen theorems for subcategories, retracts, and models of the fundamental category.
For a simply connected topological space X there is a differential graded Lie algebra, called the Quillen model, which determines the rational homotopy-type of X. Its homology is isomorphic to the rational homotopy groups of the loop space on X, called the homotopy Lie algebra.
In this paper I show that for spaces with finite (rational) LS category, these can be assumed to satisfy a condition I call separated, which is useful for calculations of the homotopy Lie algebra. This separated condition implies the free condition introduced in "Free and semi-inert cell attachments".The separated Lie models give a nice characterization of the rational homotopy Lie algebra in the case where the "top" differential creates at least two new homology classes: the radical is contained in previous homology and the rational homotopy Lie algebra contains a free Lie algebra on two generators. So it satisfies the Avramov-Felix conjecture.
Any rational space constructed using a sequence of cell attachments of length N is equivalent to a space constructed using a sequence of free cell attachments of length N+1.
This is shown by proving a similar result for differential graded Lie algebras (dgLs). As a results, one obtains a method for calculating the homotopy Lie algebra, and the homology of certain dgLs.
A random balanced sample (RBS) is a multivariate distribution with $n$ components $X_k$, each uniformly distributed on $[-1,1]$, such that the sum of these components is precisely 0. The corresponding vectors $\vec X$ lie in an $(n-1)$--dimensional polytope $M(n)$.
We present new methods for the construction of such RBS via densities over $M(n)$ and these apply for arbitrary $n$. While simple densities had been known previously for small values of $n$ ($2,3,4$), for larger $n$ the known distributions with large support were fractal distributions (with fractal dimension asymptotic to $n$ as $n\to\infty$).
Applications of RBS distributions include sampling with antithetic coupling to reduce variance, isolation of nonlinearities, and goodness of fit testing.
We also show that the previously known densities (for $n\leq4$) are in fact the only solutions in a natural and very large class of potential RBS densities. It follows that the new methods lead in another direction entirely.
Local po-spaces are topological spaces together with a local partial-order. Maps between these spaces are continuous maps which respect the order. Some computer scientists use local po-spaces to model concurrent systems and would like a good theory of equivalences for these spaces.
In this paper we construct a homotopy theory for these spaces. Spaces which are trivial or nearly trivial in the classical undirected case can be highly non-trivial when homotopies are forced to respect directions.
Our aim is to construct a (Quillen) model category for local po-spaces as a framework for a directed homotopy theory. This is technically difficult because local po-spaces are not closed under colimits, which is a necessary condition in a model category.
Our method is inspired by the construction of Voevodsky's A1-homotopy theory and Dugger's universal homotopy theories. We pass to the simplicial presheaf category and apply Jardine's model structure. We analyze the weak equivalences in this model category. Finally one uses the right context (see "Context for models of concurrency") and localizes with respect to a class of equivalences - for example, the dihomotopy equivalences. Further work will analyze the weak equivalences in this model category.
In this paper I examine many simple examples of partially-ordered spaces which model concurrent systems. It would be useful to be able to replace a given model with a simpler one. What one wants is a good notion of a directed homology equivalence. However, I show that the usual definition of directed homotopy equivalence is too coarse.
To solve this problem I introduce the notion of context. More precisely, I show that dihomotopy equivalences are best defined in the category of po-spaces under a po-space A where the choice of A depends on the 'pastings' that one would like to consider.
In this paper I give some new results on the cell attachment problem, which was perhaps first studied by J.H.C. Whitehead around 1940:
If one attaches one or more cells to a topological space, what is the effect on the homology of the loop space, and on the homotopy-type?
I introduce the free and semi-inert conditions under which I determine the loop space homology as a module and as an algebra respectively. Under a further condition I determine the homotopy-type of the space.