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27
A Suspension Lemma For Bounded Posets
, 1997
"... Let P and Q be bounded posets. In this note, a lemma is introduced that provides a set of sufficient conditions for the proper part of P being homotopy equivalent to the suspension of the proper part of Q. An application of this lemma is a unified proof of the sphericity of the higher Bruhat orders ..."
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Let P and Q be bounded posets. In this note, a lemma is introduced that provides a set of sufficient conditions for the proper part of P being homotopy equivalent to the suspension of the proper part of Q. An application of this lemma is a unified proof of the sphericity of the higher Bruhat orders under both inclusion order (which is a known result by ZIEGLER) and single step inclusion order (which was not known so far).
Presentations of OmegaCategories By Directed Complexes
, 1997
"... The theory of directed complexes is extended from free !categories to arbitrary ! categories by defining presentations in which the generators are atoms and the relations are equations between molecules. Our main result relates these presentations to the more standard algebraic presentations; we ..."
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Cited by 2 (2 self)
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The theory of directed complexes is extended from free !categories to arbitrary ! categories by defining presentations in which the generators are atoms and the relations are equations between molecules. Our main result relates these presentations to the more standard algebraic presentations; we also show that every !category has a presentation by directed complexes. The approach is similar to that used by Crans for pasting presentations. 1991 Mathematics Subject Classification: 18D05. 1 Introduction There are at present three combinatorial structures for constructing !categories: pasting schemes, defined in 1988 by Johnson [8], parity complexes, introduced in 1991 by Street [16, 17] and directed complexes, given by Steiner in 1993 [15]. These structures each consist of cells which have collections of lower dimensional cells as domain and codomain; see for example Definition 2.2 below. They also have `local' conditions on the cells, ensuring that a cell together with its boundin...
Homotopy Invariants of Multiple Categories and Concurrency in Computer Science
, 1999
"... this paper is to build several homology theories which are strongly related with the geometric properties of these paths. Concurrent machines can be formalized using cubical complexes (definition (2.1.1)) : the main idea is that a ncube represents the simultaneous execution of n 1transitions (Prat ..."
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this paper is to build several homology theories which are strongly related with the geometric properties of these paths. Concurrent machines can be formalized using cubical complexes (definition (2.1.1)) : the main idea is that a ncube represents the simultaneous execution of n 1transitions (Pratt, 1991). For example, take the following automaton :
Pasting Presentations for OmegaCategories
, 1995
"... The pasting theorem showed that pasting schemes are useful in studying free !categories. It was thought that their inflexibility with respect to composition and identities prohibited wider use. This is not the case: there is a way of dealing with identities which makes it possible to describe !ca ..."
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The pasting theorem showed that pasting schemes are useful in studying free !categories. It was thought that their inflexibility with respect to composition and identities prohibited wider use. This is not the case: there is a way of dealing with identities which makes it possible to describe !categories in terms of generating pasting schemes and relations between generated pastings, i.e., with pasting presentations. In this chapter I develop the necessary machinery for this. The main result, that the !category generated by a pasting presentation is universal with respect to respectable families of realizations, is a generalization of the pasting theorem. Contents 1 Introduction 3 2 Pasting schemes according to Johnson 4 2.1 Graded sets : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.2 !categories : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 2.3 Pasting schemes : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 2.4 The pasting theorem : : ...
Zamolodchikov Systems
, 2000
"... 7.2> 1 R f;B 0 (BT303) AB A BA A AB 0 B 0 A R A;g R A;g 0 (BT402) AB B A 0 B BA B BA 0 R f 0 ;B R f;B (BT204) AB A(g 0 #g) BA AB 0 B 0 A AB 00 B 00 A R A;g 0 #g R A;g R A;g 0 (BT32302) AB (f 0 #f)B A 00 B A 0 B BA BA 00 BA 0 R f 0 #f;B R f 0 ;B R f;B (BT20323) ..."
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7.2> 1 R f;B 0 (BT303) AB A BA A AB 0 B 0 A R A;g R A;g 0 (BT402) AB B A 0 B BA B BA 0 R f 0 ;B R f;B (BT204) AB A(g 0 #g) BA AB 0 B 0 A AB 00 B 00 A R A;g 0 #g R A;g R A;g 0 (BT32302) AB (f 0 #f)B A 00 B A 0 B BA BA 00 BA 0 R f 0 #f;B R f 0 ;B R f;B (BT20323) 2 ABC RA;BC RA;B C BCA BAC BR A;C (BT21202)
Extension Spaces of Oriented Matriods
, 1991
"... We study the space of all extensions of a real hyperplane arrangement by a new pseudohyperplane, and, more generally, of an oriented matroid by a new element. The question whether this space has the homotopy type of a sphere is a special case of the "Generalized Baues Problem" of Billera, Kaprano ..."
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We study the space of all extensions of a real hyperplane arrangement by a new pseudohyperplane, and, more generally, of an oriented matroid by a new element. The question whether this space has the homotopy type of a sphere is a special case of the "Generalized Baues Problem" of Billera, Kapranov & Sturmfels, via the BohneDress Theorem on zonotopal tilings. We prove that the extension space is spherical for the class of strongly euclidean oriented matroids. This class includes the alternating matroids and all oriented matroids of rank at most 3 or of corank at most 2. In general it is not even known whether the extension space is connected. We show that the subspace of realizable extensions is always connected but not necessarily spherical.
1 OMEGACATEGORIES AND CHAIN COMPLEXES
, 2004
"... There are several ways to construct omegacategories from combinatorial objects such as pasting schemes or parity complexes. We make these constructions into a functor on a category of chain complexes with additional structure, which we call augmented directed complexes. This functor from augmented ..."
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There are several ways to construct omegacategories from combinatorial objects such as pasting schemes or parity complexes. We make these constructions into a functor on a category of chain complexes with additional structure, which we call augmented directed complexes. This functor from augmented directed complexes to omegacategories has a left adjoint, and the adjunction restricts to an equivalence on a category of augmented directed complexes with good bases. The omegacategories equivalent to augmented directed complexes with good bases include the omegacategories associated to globes, simplexes and cubes; thus the morphisms between these omegacategories are determined by morphisms between chain complexes. It follows that the entire theory of omegacategories can be expressed in terms of chain complexes; in particular we describe the biclosed monoidal structure on omegacategories and calculate some internal homomorphism objects. 1.
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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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.
MAPS BETWEEN HIGHER BRUHAT ORDERS AND HIGHER STASHEFFTAMARI POSETS
, 2002
"... Abstract. We make explicit a description in terms of convex geometry of the higher Bruhat orders B(n, d) sketched by Kapranov and Voevodsky. We give an analogous description of the higher StasheffTamari poset S1(n, d). We show that the map f sketched by Kapranov and Voevodsky from B(n, d) to S1([0, ..."
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Abstract. We make explicit a description in terms of convex geometry of the higher Bruhat orders B(n, d) sketched by Kapranov and Voevodsky. We give an analogous description of the higher StasheffTamari poset S1(n, d). We show that the map f sketched by Kapranov and Voevodsky from B(n, d) to S1([0, n + 1], d + 1) coincides with the map constructed by Rambau, and is a surjection for d ≤ 2. We also give geometric descriptions of lk0 ◦f and lk {0,n+1} ◦f. We construct a map analogous to f from S1(n, d) to B(n − 1, d), and show that it is always a poset embedding. We also give an explicit criterion to determine if an element of B(n −1, d) is in the image of this map. 1.
Combinatorics of branchings in higher dimensional automata
, 1999
"... We explore the combinatorial properties of the branching areas of paths in higher dimensional automata. Mathematically, this means that we investigate the combinatorics of the negative corner homology of a globular ωcategory and the combinatorics of a new homology theory called the reduced negative ..."
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We explore the combinatorial properties of the branching areas of paths in higher dimensional automata. Mathematically, this means that we investigate the combinatorics of the negative corner homology of a globular ωcategory and the combinatorics of a new homology theory called the reduced negative corner homology. This latter is the homology of the quotient of the corner complex by the subcomplex generated by its thin elements. As application, we calculate the corner homology of some ωcategories and we give some invariance results for the reduced corner homology. We only treat the negative side. The positive side, that is to say the case of merging areas of paths is similar and can be easily deduced from the negative side.