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.

Introduction Recently it was shown by Kapranov [4] that the combinatorics of the permutohedra and associahedra can be combined to give a `hybrid' family of polytopes, the permutoassociahedra. In this short note we put forward a slightly different point of view: the associahedra can themselves be seen as retracts of the permutohedra. We construct a natural cellular quotient map from the permutohedron P n to the associahedron K n+1 . In dimension 3 we also give K 5 as the convex hull of a particular subset of the usual vertices of P 4 . 1. The quotient map We begin by recalling the definitions of the permutohedra and the associahedra. See [4] and the references there for more details. The permutohedron [5, 8] (or zilchgon [2], or parallelohedron [1]) P n is the convex hull of the<F13.5

this paper we try to lay some of the foundations of such a theory of categories `up to homotopy' or more exactly `up to coherent homotopies'. The method we use is based on earlier work on:

We show that the nerve of a braided monoidal category carries a natural action of a simplicial E2-operad and is thus up to group completion a double loop space. Shifting up dimension twice associates to each braided monoidal category a 1-reduced lax 3-category whose nerve realizes an explicit double delooping whenever all cells are invertible. We deduce that lax 3-groupoids are algebraic models for homotopy 3-types. Introduction The concept of braiding as a refinement of symmetry is the starting point of a rich interplay between geometry (knot theory) and algebra (representation theory). The underlying structure of a braided monoidal category reveals an interest of its own in that it encompasses two at first sight different geometrical objects : double loop spaces and homotopy 3-types. The link to double loop spaces was pointed out by J. Stasheff [38] and made precise by Z. Fiedorowicz [15], who proves that double loop spaces may be characterized (up to group completion) as algebras o...

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Abstract. We survey the generalized Baues problem of Billera and Sturmfels. The problem is one of discrete geometry and topology, and asks about the topology of the set of subdivisions of a certain kind of a convex polytope. Along with a discussion of most of the known results, we survey the motivation for the problem and its relation to triangulations, zonotopal tilings, monotone paths in linear programming, oriented matroid Grassmannians, singularities, and homotopy theory. Included are several open questions and problems. 1.

... categories. The associated cellular nerve of an o-category extends the well-known simplicial nerve of a small category. Cellular sets (like simplicial sets) carry a closed model structure in Quillen’s sense with weak equivalences induced by a geometric realisation functor. More generally, there exists a dense subcategory YA of the category of A-algebras for each o-operad A in Batanin’s sense. Whenever A is contractible, the resulting homotopy category of A-algebras (i.e. weak o-categories) is

this paper is the extension of this result to the case of twisted coefficients given by

Abstract. In the paper the notion of a truncating twisting function from a simplicial set to a cubical set and the corresponding notion of twisted Cartesian product of these sets are introduced. The latter becomes a cubical set. This construction together with the theory of twisted tensor products for homotopy G-algebras allows to obtain multiplicative models for fibrations. 1.

The classical “computation ” methods in Algebraic Topology most often work by means of highly infinite objects and in fact are not constructive. Typical examples are shown to describe the nature of the problem. The Rubio-Sergeraert solution for Constructive Algebraic Topology is recalled. This is not only a theoretical solution: the concrete computer program Kenzo has been written down which precisely follows this method. This program has been used in various cases, opening new research subjects and producing in several cases significant results unreachable by hand. In particular the Kenzo program can compute the first homotopy groups of a simply connected arbitrary simplicial set.