## Directed combinatorial homology and noncommutative tori (The breaking of symmetries in algebraic topology)

Citations: | 12 - 7 self |

### BibTeX

@MISC{Grandis_directedcombinatorial,

author = {Marco Grandis},

title = { Directed combinatorial homology and noncommutative tori (The breaking of symmetries in algebraic topology)},

year = {}

}

### OpenURL

### Abstract

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 topologically-trivial 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.

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Citation Context ...to the set of leaves of an irrational Kronecker foliation of the 2-torus, has been interpreted as a ‘noncommutative space’, the irrational rotation C ∗ -algebra Aϑ, also called a noncommutative torus =-=[4, 5, 11, 1]-=-. As proved in [10, 11], Work supported by MIUR Research Projects. 2000 Mathematics Subject Classification: 55U10, 81R60, 55Nxx. Key words and phrases: Cubical sets, noncommutative C*-algebras, combin... |

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Citation Context ...to the set of leaves of an irrational Kronecker foliation of the 2-torus, has been interpreted as a ‘noncommutative space’, the irrational rotation C ∗ -algebra Aϑ, also called a noncommutative torus =-=[4, 5, 11, 1]-=-. As proved in [10, 11], Work supported by MIUR Research Projects. 2000 Mathematics Subject Classification: 55U10, 81R60, 55Nxx. Key words and phrases: Cubical sets, noncommutative C*-algebras, combin... |

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Citation Context ...to the set of leaves of an irrational Kronecker foliation of the 2-torus, has been interpreted as a ‘noncommutative space’, the irrational rotation C ∗ -algebra Aϑ, also called a noncommutative torus =-=[4, 5, 11, 1]-=-. As proved in [10, 11], Work supported by MIUR Research Projects. 2000 Mathematics Subject Classification: 55U10, 81R60, 55Nxx. Key words and phrases: Cubical sets, noncommutative C*-algebras, combin... |

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Citation Context ...ght be more convenient). We shall use a generalised, non-symmetric notion of metric space adequate for directed algebraic topology (cf. [6]), and natural within the theory of enriched categories (cf. =-=[9]-=-).Thus, a directed metric space or d-metric space, isaset X equipped with a d-metric δ: X ×X → [0, ∞], satisfying the axioms δ(x, x) =0, δ(x, y)+δ(y, z) ≥ δ(x, z). (13) (If the value ∞ is forbidden, ... |

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Citation Context ...equipped with a d-metric δ: X ×X → [0, ∞], satisfying the axioms δ(x, x) =0, δ(x, y)+δ(y, z) ≥ δ(x, z). (13) (If the value ∞ is forbidden, such a function is usually called a quasi-pseudo-metric, cf. =-=[8]-=-.) A symmetric d-metric (satisfying δ(x, y) =δ(y, x)) will be called here a (generalised) metric; itisthe same as an écart in Bourbaki [2]. dMtr will denote the category of such d-metric spaces, with ... |

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Citation Context ... such a function is usually called a quasi-pseudo-metric, cf. [8].) A symmetric d-metric (satisfying δ(x, y) =δ(y, x)) will be called here a (generalised) metric; itisthe same as an écart in Bourbaki =-=[2]-=-. dMtr will denote the category of such d-metric spaces, with d-contractions f: X → Y (δ(x, x ′) ≥ δ(f(x),f(x ′))). Limits and colimits exist and are calculated as in Set; products have the l∞ d-metri... |

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Citation Context ...pings, gives back the contracting ones B1(Lip∞(Y,Z)) = NSet(Y,Z), (24) as it happens in the well-known case of Banach spaces, in the interplay between bounded linear maps and linear contractions (cf. =-=[12]-=-). 2.3. Normed cubical sets. We have already defined such objects and their category, NCub (1.1). Recall that normal cubical sets have norm 1 on all non-degenerate entries (1.2). As for normed sets (1... |

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