2,091 research outputs found

    Projective Systems of Noncommutative Lattices as a Pregeometric Substratum

    Full text link
    We present an approximation to topological spaces by {\it noncommutative} lattices. This approximation has a deep physical flavour based on the impossibility to fully localize particles in any position measurement. The original space being approximated is recovered out of a projective limit.Comment: 30 pages, Latex. To appear in `Quantum Groups and Fundamental Physical Applications', ISI Guccia, Palermo, December 1997, D. Kastler and M. Rosso Eds., (Nova Science Publishers, USA

    Noncommutative Lattices and Their Continuum Limits

    Get PDF
    We consider finite approximations of a topological space MM by noncommutative lattices of points. These lattices are structure spaces of noncommutative C∗C^*-algebras which in turn approximate the algebra \cc(M) of continuous functions on MM. We show how to recover the space MM and the algebra \cc(M) from a projective system of noncommutative lattices and an inductive system of noncommutative C∗C^*-algebras, respectively.Comment: 22 pages, 8 Figures included in the LaTeX Source New version, minor modifications (typos corrected) and a correction in the list of author

    Lim colim versus colim lim. II: Derived limits over a pospace

    Full text link
    \v{C}ech cohomology Hn(X)H^n(X) of a separable metrizable space XX is defined in terms of cohomology of its nerves (or ANR neighborhoods) PβP_\beta whereas Steenrod-Sitnikov homology Hn(X)H_n(X) is defined in terms of homology of compact subsets Kα⊂XK_\alpha\subset X. We show that one can also go vice versa: in a sense, Hn(X)H^n(X) can be reconstructed from Hn(Kα)H^n(K_\alpha), and if XX is finite dimensional, Hn(X)H_n(X) can be reconstructed from Hn(Pβ)H_n(P_\beta). The reconstruction is via a Bousfield-Kan/Araki-Yoshimura type spectral sequence, except that the derived limits have to be "corrected" so as to take into account a natural topology on the indexing set. The corrected derived limits coincide with the usual ones when the topology is discrete, and in general are applied not to an inverse system but to a "partially ordered sheaf". The "correction" of the derived limit functors in turn involves constructing a "correct" (metrizable) topology on the order complex ∣P∣|P| of a partially ordered metrizable space PP (such as the hyperspace K(X)K(X) of nonempty compact subsets of XX with the Hausdorff metric). It turns out that three natural approaches (by using the space of measurable functions, the space of probability measures, or the usual embedding K(X)→C(X;R)K(X)\to C(X;\mathbb R)) all lead to the same topology on ∣P∣|P|.Comment: 30 page

    The homotopy theory of Khovanov homology

    Get PDF
    We show that the unnormalised Khovanov homology of an oriented link can be identified with the derived functors of the inverse limit. This leads to a homotopy theoretic interpretation of Khovanov homology.Comment: 36 pages; reformatted and some minor change
    • …
    corecore