2,091 research outputs found
Projective Systems of Noncommutative Lattices as a Pregeometric Substratum
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
We consider finite approximations of a topological space by
noncommutative lattices of points. These lattices are structure spaces of
noncommutative -algebras which in turn approximate the algebra \cc(M) of
continuous functions on . We show how to recover the space and the
algebra \cc(M) from a projective system of noncommutative lattices and an
inductive system of noncommutative -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
\v{C}ech cohomology of a separable metrizable space is defined
in terms of cohomology of its nerves (or ANR neighborhoods) whereas
Steenrod-Sitnikov homology is defined in terms of homology of compact
subsets .
We show that one can also go vice versa: in a sense, can be
reconstructed from , and if is finite dimensional,
can be reconstructed from .
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 of a partially
ordered metrizable space (such as the hyperspace of nonempty compact
subsets of 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 ) all lead
to the same topology on .Comment: 30 page
The homotopy theory of Khovanov homology
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
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