9,174 research outputs found
Duality and canonical extensions for stably compact spaces
We construct a canonical extension for strong proximity lattices in order to
give an algebraic, point-free description of a finitary duality for stably
compact spaces. In this setting not only morphisms, but also objects may have
distinct pi- and sigma-extensions.Comment: 29 pages, 1 figur
Some notes on Esakia spaces
Under Stone/Priestley duality for distributive lattices, Esakia spaces
correspond to Heyting algebras which leads to the well-known dual equivalence
between the category of Esakia spaces and morphisms on one side and the
category of Heyting algebras and Heyting morphisms on the other. Based on the
technique of idempotent split completion, we give a simple proof of a more
general result involving certain relations rather then functions as morphisms.
We also extend the notion of Esakia space to all stably locally compact spaces
and show that these spaces define the idempotent split completion of compact
Hausdorff spaces. Finally, we exhibit connections with split algebras for
related monads
Weakly complex homogeneous spaces
We complete our recent classification [GMS11] of compact inner symmetric
spaces with weakly complex tangent bundle by filling up a case which was left open, and
extend this classification to the larger category of compact homogeneous spaces with positive
Euler characteristic. We show that a simply connected compact equal rank homogeneous
space has weakly complex tangent bundle if and only if it is a product of compact equal
rank homogeneous spaces which either carry an invariant almost complex structure (and are
classified by Hermann [H56]), or have stably trivial tangent bundle (and are classified by
Singhof and Wemmer [SW86]), or belong to an explicit list of weakly complex spaces which
have neither stably trivial tangent bundle, nor carry invariant almost complex structures
Representable (T,V)-categories
Working in the framework of -categories, for a symmetric monoidal closed category V and a (not necessarily cartesian) monad , we present a common account to the study of ordered compact Hausdorff spaces and stably compact spaces on one side and monoidal categories and representable multicategories on the other one. In this setting we introduce the notion of dual for -categories.Working in the framework of (T, V)-categories, for a symmetric monoidal closed
category V and a (not necessarily cartesian) monad T, we present a common account to the
study of ordered compact Hausdorff spaces and stably compact spaces on one side and monoidal
categories and representable multicategories on the other one. In this setting we introduce the
notion of dual for (T, V)-categories
Priestley-Stone Duality for Subbases of Stably Locally Compact Spaces
We extend the classic Priestley-Stone duality to a Wallman-like duality for
subbases of general stably locally compact spaces. As a corollary, we show that
any locally compact T_0 space X has a unique minimal 'stabilisation', i.e. a
stably locally compact space containing X as a patch-dense subspace, which is
moreover functorial with respect to proper maps
Stably dualizable groups
We extend the duality theory for topological groups from the classical theory
for compact Lie groups, via the topological study by J. R. Klein [Kl01] and the
p-complete study for p-compact groups by T. Bauer [Ba04], to a general duality
theory for stably dualizable groups in the E-local stable homotopy category,
for any spectrum E. The principal new examples occur in the K(n)-local
category, where the Eilenberg-Mac Lane spaces G = K(Z/p, q) are stably
dualizable and nontrivial for 0 <= q <= n.
We show how to associate to each E-locally stably dualizable group G a stably
defined representation sphere S^{adG}, called the dualizing spectrum, which is
dualizable and invertible in the E-local category. Each stably dualizable group
is Atiyah-Poincare self-dual in the E-local category, up to a shift by S^{adG}.
There are dimension-shifting norm- and transfer maps for spectra with G-action,
again with a shift given by S^{adG}. The stably dualizable group G also admits
a kind of framed bordism class [G] in pi_*(L_E S), in degree dim_E(G) =
[S^{adG}] of the Pic_E-graded homotopy groups of the E-localized sphere
spectrum.Comment: Final version, to appear in the Memoirs of the A.M.
Equivalence and stable isomorphism of groupoids, and diagonal-preserving stable isomorphisms of graph C*-algebras and Leavitt path algebras
We prove that ample groupoids with sigma-compact unit spaces are equivalent
if and only if they are stably isomorphic in an appropriate sense, and relate
this to Matui's notion of Kakutani equivalence. We use this result to show that
diagonal-preserving stable isomorphisms of graph C*-algebras or Leavitt path
algebras give rise to isomorphisms of the groupoids of the associated
stabilised graphs. We deduce that the Leavitt path algebras and
are not stably *-isomorphic.Comment: 12 pages. Minor corrections. This is the version that will be
publishe
Topological Orthoalgebras
We define topological orthoalgebras (TOAs) and study their properties. While
every topological orthomodular lattice is a TOA, the lattice of projections of
a Hilbert space is an example of a lattice-ordered TOA that is not a toplogical
lattice. On the other hand, we show that every compact Boolean TOA is a
topological Boolean algebra. We also show that a compact TOA in which 0 is an
isolated point is atomic and of finite height. We identify and study a
particularly tractable class of TOAs, which we call {\em stably ordered}: those
in which the upper-set generated by an open set is open. This includes all
topological OMLs, and also the projection lattices of Hilbert spaces. Finally,
we obtain a topological version of the Foulis-Randall representation theory for
stably ordered TOAsComment: 16 pp, LaTex. Minor changes and corrections in sections 1; more
substantial corrections in section
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