2,662 research outputs found
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
Tensor products and regularity properties of Cuntz semigroups
The Cuntz semigroup of a C*-algebra is an important invariant in the
structure and classification theory of C*-algebras. It captures more
information than K-theory but is often more delicate to handle. We
systematically study the lattice and category theoretic aspects of Cuntz
semigroups.
Given a C*-algebra , its (concrete) Cuntz semigroup is an object
in the category of (abstract) Cuntz semigroups, as introduced by Coward,
Elliott and Ivanescu. To clarify the distinction between concrete and abstract
Cuntz semigroups, we will call the latter -semigroups.
We establish the existence of tensor products in the category and study
the basic properties of this construction. We show that is a symmetric,
monoidal category and relate with for
certain classes of C*-algebras.
As a main tool for our approach we introduce the category of
pre-completed Cuntz semigroups. We show that is a full, reflective
subcategory of . One can then easily deduce properties of from
respective properties of , e.g. the existence of tensor products and
inductive limits. The advantage is that constructions in are much easier
since the objects are purely algebraic.
We also develop a theory of -semirings and their semimodules. The Cuntz
semigroup of a strongly self-absorbing C*-algebra has a natural product giving
it the structure of a -semiring. We give explicit characterizations of
-semimodules over such -semirings. For instance, we show that a
-semigroup tensorially absorbs the -semiring of the Jiang-Su
algebra if and only if is almost unperforated and almost divisible, thus
establishing a semigroup version of the Toms-Winter conjecture.Comment: 195 pages; revised version; several proofs streamlined; some results
corrected, in particular added 5.2.3-5.2.
Topology from enrichment: the curious case of partial metrics
For any small quantaloid \Q, there is a new quantaloid \D(\Q) of
diagonals in \Q. If \Q is divisible then so is \D(\Q) (and vice versa),
and then it is particularly interesting to compare categories enriched in \Q
with categories enriched in \D(\Q). Taking Lawvere's quantale of extended
positive real numbers as base quantale, \Q-categories are generalised metric
spaces, and \D(\Q)-categories are generalised partial metric spaces, i.e.\
metric spaces in which self-distance need not be zero and with a suitably
modified triangular inequality. We show how every small quantaloid-enriched
category has a canonical closure operator on its set of objects: this makes for
a functor from quantaloid-enriched categories to closure spaces. Under mild
necessary-and-sufficient conditions on the base quantaloid, this functor lands
in the category of topological spaces; and an involutive quantaloid is
Cauchy-bilateral (a property discovered earlier in the context of distributive
laws) if and only if the closure on any enriched category is identical to the
closure on its symmetrisation. As this now applies to metric spaces and partial
metric spaces alike, we demonstrate how these general categorical constructions
produce the "correct" definitions of convergence and Cauchyness of sequences in
generalised partial metric spaces. Finally we describe the Cauchy-completion,
the Hausdorff contruction and exponentiability of a partial metric space, again
by application of general quantaloid-enriched category theory.Comment: Apart from some minor corrections, this second version contains a
revised section on Cauchy sequences in a partial metric spac
- …