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Directed combinatorial homology and noncommutative tori (The breaking of symmetries in algebraic topology)
"... This is a brief study of the homology of cubical sets, with two main purposes. First, this combinatorial structure is viewed as representing directed spaces, breaking the intrinsic symmetries of topological spaces. Cubical sets have a directed homology, consisting of preordered abelian groups where ..."
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This is a brief study of the homology of cubical sets, with two main purposes. First, this combinatorial structure is viewed as representing directed spaces, breaking the intrinsic symmetries of topological spaces. Cubical sets have a directed homology, consisting of preordered abelian groups where the positive cone comes from the structural cubes. But cubical sets can also express topological facts missed by ordinary topology. This happens, for instance, in the study of group actions or foliations, where a topologicallytrivial quotient (the orbit set or the set of leaves) can be enriched with a natural cubical structure whose directed cohomology agrees with Connes ' analysis in noncommutative geometry. Thus, cubical sets can provide a sort of 'noncommutative topology', without the metric information of C*algebras.
Higher cospans and weak cubical categories (Cospans in Algebraic Topology
 I), Theory Appl. Categ
"... form a cubical set with compositions x +i y in all directions, which are computed using pushouts and behave ‘categorically ’ in a weak sense, up to suitable comparisons. Actually, we work with a ‘symmetric cubical structure’, which includes the transposition symmetries, because this allows for a str ..."
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Cited by 5 (2 self)
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form a cubical set with compositions x +i y in all directions, which are computed using pushouts and behave ‘categorically ’ in a weak sense, up to suitable comparisons. Actually, we work with a ‘symmetric cubical structure’, which includes the transposition symmetries, because this allows for a strong simplification of the coherence conditions. These notions will be used in subsequent papers to study topological cospans and their use in Algebraic Topology, from tangles to cobordisms of manifolds. We also introduce the more general notion of a multiple category, where to start witharrows belong to different sorts, varying in a countable family, and symmetries must be dropped. The present examples seem to show that the symmetric cubical case is better suited for topological applications.
Combinatorics of labelling in higher dimensional automata
"... Abstract. The main idea for interpreting concurrent processes as labelled precubical sets is that a given set of n actions running concurrently must be assembled to a labelled ncube, in exactly one way. The main ingredient is the nonfunctorial construction called labelled directed coskeleton. It i ..."
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Abstract. The main idea for interpreting concurrent processes as labelled precubical sets is that a given set of n actions running concurrently must be assembled to a labelled ncube, in exactly one way. The main ingredient is the nonfunctorial construction called labelled directed coskeleton. It is defined as a subobject of the labelled coskeleton, the latter coinciding in the unlabelled case with the right adjoint to the truncation functor. This nonfunctorial construction is necessary since the labelled coskeleton functor of the category of labelled precubical sets does not fulfil the above requirement. We prove in this paper that it is possible to force the labelled coskeleton functor to be wellbehaved by working with labelled transverse symmetric precubical sets. Moreover, we prove that this solution is the only one. A transverse symmetric precubical set is a precubical set equipped with symmetry maps and with a new kind of degeneracy map called transverse degeneracy. Finally, we also prove that the two settings are equivalent from a directed algebraic topological viewpoint. To illustrate, a new semantics of CCS, equivalent to the old one, is given. Contents
Higher homotopy operations and cohomology
"... Abstract. We explain how higher homotopy operations, defined topologically, may be identified under mild assumptions with (the last of) the DwyerKanSmith cohomological obstructions to rectifying homotopycommutative diagrams. ..."
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Abstract. We explain how higher homotopy operations, defined topologically, may be identified under mild assumptions with (the last of) the DwyerKanSmith cohomological obstructions to rectifying homotopycommutative diagrams.
Formal Relationships Between Geometrical and Classical Models for Concurrency
"... Abstract. A wide variety of models for concurrent programs has been proposed during the past decades, each one focusing on various aspects of computations: trace equivalence, causality between events, conflicts and schedules due to resource accesses, etc. More recently, models with a geometrical fla ..."
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Abstract. A wide variety of models for concurrent programs has been proposed during the past decades, each one focusing on various aspects of computations: trace equivalence, causality between events, conflicts and schedules due to resource accesses, etc. More recently, models with a geometrical flavor have been introduced, based on the notion of cubical set. These models are very rich and expressive since they can represent commutation between any number of events, thus generalizing the principle of true concurrency. While they seem to be very promising – because they make possible the use of techniques from algebraic topology in order to study concurrent computations – they have not yet been precisely related to the previous models, and the purpose of this paper is to fill this gap. In particular, we describe an adjunction between Petri nets and cubical sets which extends the previously known adjunction between Petri nets and asynchronous transition systems by Nielsen and Winskel. 1 1
unknown title
, 2009
"... Moore hyperrectangles on a space form a strict cubical omegacategory with connections ..."
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Moore hyperrectangles on a space form a strict cubical omegacategory with connections
HOMOLOGY FOR HIGHERRANK GRAPHS AND TWISTED C ∗ALGEBRAS
"... Abstract. We introduce a homology theory for kgraphs and explore its fundamental properties. We establish connections with algebraic topology by showing that the homology of a kgraph coincides with the homology of its topological realisation as described by Kaliszewski et al. We exhibit combinator ..."
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Abstract. We introduce a homology theory for kgraphs and explore its fundamental properties. We establish connections with algebraic topology by showing that the homology of a kgraph coincides with the homology of its topological realisation as described by Kaliszewski et al. We exhibit combinatorial versions of a number of standard topological constructions, and show that they are compatible, from a homological point of view, with their topological counterparts. We show how to twist the C ∗algebra of a kgraph by a Tvalued 2cocycle and demonstrate that examples include all noncommutative tori. In the appendices, we construct a cubical set ˜ Q(Λ) from a kgraph Λ and demonstrate that the homology and topological realisation of Λ coincide with those of ˜ Q(Λ) as defined by Grandis. 1.
Journal of Homotopy and Related Structures, vol.??(??),????, pp.1–36 CUBICAL COSPANS AND HIGHER COBORDISMS (COSPANS IN ALGEBRAIC TOPOLOGY, III)
, 806
"... After two papers on weak cubical categories and collarable cospans, respectively, we put things together and construct a weak cubical category of cubical collared cospans of topological ..."
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After two papers on weak cubical categories and collarable cospans, respectively, we put things together and construct a weak cubical category of cubical collared cospans of topological