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27
Homotopy Invariants of Higher Dimensional Categories and Concurrency in Computer Science
, 1999
"... The strict globular omegacategories formalize the execution paths of a parallel automaton and the homotopies between them. One associates to such (and any) omegacategory C three homology theories. The first one is called the globular homology. It contains the oriented loops of C. The two other one ..."
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Cited by 49 (10 self)
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The strict globular omegacategories formalize the execution paths of a parallel automaton and the homotopies between them. One associates to such (and any) omegacategory 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.
Combinatorics Of Branchings In Higher Dimensional Automata
 Theory Appl. Categ
, 2001
"... 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 ca ..."
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Cited by 35 (9 self)
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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 subcomplex 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.
Triangulations of cyclic polytopes and higher Bruhat orders
 MATHEMATIKA
, 1997
"... Recently EDELMAN & REINER suggested two poset structures S 1 (n;d) and S 2 (n;d) on the set of all triangulations of the cyclic dpolytope C(n;d) with n vertices. Both posets are generalizations of the wellstudied Tamari lattice. While S 2 (n;d) is bounded by definition, the same is not obvious f ..."
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Cited by 26 (5 self)
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Recently EDELMAN & REINER suggested two poset structures S 1 (n;d) and S 2 (n;d) on the set of all triangulations of the cyclic dpolytope C(n;d) with n vertices. Both posets are generalizations of the wellstudied Tamari lattice. While S 2 (n;d) is bounded by definition, the same is not obvious for S 1 (n;d). In the paper by EDELMAN & REINER the bounds of S 2 (n;d) were also confirmed for S 1 (n;d) whenever d 5, leaving the general case as a conjecture. In this paper their conjecture is answered in the affirmative for all d, using several new functorial constructions. Moreover, a structure theorem is presented, stating that the elements of S 1 (n;d + 1) are in onetoone correspondence to certain equivalence classes of maximal chains in S 1 (n;d). By similar methods it is proved that all triangulations of cyclic polytopes are shellable. In order to clarify the connection between S 1 (n;d) and the higher Bruhat order B(n \Gamma 2;d \Gamma 1) of MANIN & SCHECHTMAN, we define an orderpreserving map from B(n \Gamma 2;d \Gamma 1) to S 1 (n;d), thereby concretizing a result by KAPRANOV & VOEVODSKY in the theory of ordered ncategories.
From Concurrency to Algebraic Topology
, 2000
"... 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 threeobject small category gl ! + is exposed. Some of ..."
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Cited by 24 (8 self)
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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 threeobject 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.
About the Globular Homology of Higher Dimensional Automata
, 2000
"... We introduce a new simplicial nerve of higher dimensional automata whose homology groups yield a new definition of the globular homology. With this new definition, the drawbacks noticed with the construction of [Gau99] disappear. Moreover the important morphisms which associate to every globe its co ..."
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Cited by 23 (8 self)
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We introduce a new simplicial nerve of higher dimensional automata whose homology groups yield a new definition of the globular homology. With this new definition, the drawbacks noticed with the construction of [Gau99] disappear. Moreover the important morphisms which associate to every globe its corresponding branching area and merging area of execution paths become morphisms of simplicial sets.
Pasting Schemes for the Monoidal Biclosed Structure on
, 1995
"... Using the theory of pasting presentations, developed in chapter 2, I give a detailed description of the tensor product on !categories, which extends Gray's tensor product on 2categories and which is closely related to BrownHiggins's tensor product on !groupoids. Immediate consequences are a gen ..."
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Cited by 18 (0 self)
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Using the theory of pasting presentations, developed in chapter 2, I give a detailed description of the tensor product on !categories, which extends Gray's tensor product on 2categories and which is closely related to BrownHiggins's tensor product on !groupoids. Immediate consequences are a general and uniform definition of higher dimensional lax natural transformations, and a nice and transparent description of the corresponding internal homs. Further consequences will be in the development of a theory for weak ncategories, since both tensor products and lax structures are crucial in this. Contents 1 Introduction 3 2 Cubes and cubical sets 5 2.1 Cubes combinatorially : : : : : : : : : : : : : : : : : : : : : : : : 5 2.2 A model category for cubes : : : : : : : : : : : : : : : : : : : : : 6 2.3 Generating the model category for cubes : : : : : : : : : : : : : : 7 2.4 Cubical sets : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.5 Duality : : : : : : : : : : : : : ...
The generalized Baues problem
, 1998
"... 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 motivat ..."
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Cited by 17 (0 self)
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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.
Pasting In Multiple Categories
 Theory Appl. Categ
, 1998
"... . In the literature there are several kinds of concrete and abstract cell complexes representing composition in ncategories, !categories or 1categories, and the slightly more general partial !categories. Some examples are parity complexes, pasting schemes and directed complexes. In this paper we ..."
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Cited by 11 (2 self)
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. In the literature there are several kinds of concrete and abstract cell complexes representing composition in ncategories, !categories or 1categories, and the slightly more general partial !categories. Some examples are parity complexes, pasting schemes and directed complexes. In this paper we give an axiomatic treatment: that is to say, we study the class of `!complexes' which consists of all complexes representing partial !categories. We show that !complexes can be given geometric structures and that in most important examples they become wellbehaved CW complexes; we characterise !complexes by conditions on their cells; we show that a product of !complexes is again an !complex; and we describe some products in detail. 1. Introduction In this paper we consider pasting diagrams representing compositions in multiple categories. To be specific, the multiple categories concerned are ncategories and their infinitedimensional analogues, which are called !categories or 1cat...
Fiber Polytopes For The Projections Between Cyclic Polytopes
, 1997
"... The cyclic polytope C(n; d) is the convex hull of any n points on the moment curve f(t; t 2 ; : : : ; t d ) : t 2 Rg in R d . For d 0 ? d, we consider the fiber polytope (in the sense of Billera and Sturmfels) associated to the natural projection of cyclic polytopes : C(n; d 0 ) ! C(n; ..."
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Cited by 9 (9 self)
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The cyclic polytope C(n; d) is the convex hull of any n points on the moment curve f(t; t 2 ; : : : ; t d ) : t 2 Rg in R d . For d 0 ? d, we consider the fiber polytope (in the sense of Billera and Sturmfels) associated to the natural projection of cyclic polytopes : C(n; d 0 ) ! C(n; d) which "forgets" the last d 0 \Gamma d coordinates. It is known that this fiber polytope has face lattice indexed by the coherent polytopal subdivisions of C(n; d) which are induced by the map . Our main result characterizes the triples (n; d; d 0 ) for which the fiber polytope is canonical in either of the following two senses: ffl all polytopal subdivisions induced by are coherent, ffl the structure of the fiber polytope does not depend upon the choice of points on the moment curve. We also discuss a new instance with a positive answer to the Generalized Baues Problem, namely that of a projection : P ! Q where Q has only regular subdivisions and P has two more vertices than its...
An Australian conspectus of higher categories

, 2004
"... Much Australian work on categories is part of, or relevant to, the development of higher categories and their theory. In this note, I hope to describe some of the origins and achievements of our efforts that they might perchance serve as a guide to the development of aspects of higherdimensional wo ..."
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Cited by 6 (0 self)
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Much Australian work on categories is part of, or relevant to, the development of higher categories and their theory. In this note, I hope to describe some of the origins and achievements of our efforts that they might perchance serve as a guide to the development of aspects of higherdimensional work. I trust that the somewhat autobiographical style will add interest rather than be a distraction. For so long I have felt rather apologetic when describing how categories might be helpful to other mathematicians; I have often felt even worse when mentioning enriched and higher categories to category theorists. This is not to say that I have doubted the value of our work, rather that I have felt slowed down by the continual pressure to defend it. At last, at this meeting, I feel justified in speaking freely amongst motivated researchers who know the need for the subject is well established. Australian Category Theory has its roots in homology theory: more precisely, in the treatment of the cohomology ring and the Künneth formulas in the book by Hilton and Wylie [HW]. The first edition of the book had a mistake concerning the cohomology ring of a product. The Künneth formulas arise from splittings of the natural short exact sequences