44 research outputs found
Towards "dynamic domains": totally continuous cocomplete Q-categories
It is common practice in both theoretical computer science and theoretical
physics to describe the (static) logic of a system by means of a complete
lattice. When formalizing the dynamics of such a system, the updates of that
system organize themselves quite naturally in a quantale, or more generally, a
quantaloid. In fact, we are lead to consider cocomplete quantaloid-enriched
categories as fundamental mathematical structure for a dynamic logic common to
both computer science and physics. Here we explain the theory of totally
continuous cocomplete categories as generalization of the well-known theory of
totally continuous suplattices. That is to say, we undertake some first steps
towards a theory of "dynamic domains''.Comment: 29 pages; contains a more elaborate introduction, corrects some
typos, and has a sexier title than the previously posted version, but the
mathematics are essentially the sam
Q-modules are Q-suplattices
It is well known that the internal suplattices in the topos of sheaves on a
locale are precisely the modules on that locale. Using enriched category theory
and a lemma on KZ doctrines we prove (the generalization of) this fact in the
case of ordered sheaves on a small quantaloid. Comparing module-equivalence
with sheaf-equivalence for quantaloids and using the notion of centre of a
quantaloid, we refine a result of F. Borceux and E. Vitale.Comment: 12 page
Categorical structures enriched in a quantaloid: regular presheaves, regular semicategories
We study presheaves on semicategories enriched in a quantaloid: this gives
rise to the notion of regular presheaf. A semicategory is regular when its
representable presheaves are regular, and its regular presheaves then
constitute an essential (co)localization of the category of all of its
presheaves. The notion of regular semidistributor allows to establish the
Morita equivalence of regular semicategories. Continuous orders and Omega-sets
provide examples.Comment: 21 page
'Hausdorff distance' via conical cocompletion
In the context of quantaloid-enriched categories, we explain how each
saturated class of weights defines, and is defined by, an essentially unique
full sub-KZ-doctrine of the free cocompletion KZ-doctrine. The KZ-doctrines
which arise as full sub-KZ-doctrines of the free cocompletion, are
characterised by two simple "fully faithfulness" conditions. Conical weights
form a saturated class, and the corresponding KZ-doctrine is precisely (the
generalisation to quantaloid-enriched categories of) the Hausdorff doctrine of
[Akhvlediani et al., 2009].Comment: Minor change
Categorical structures enriched in a quantaloid: orders and ideals over a base quantaloid
Applying (enriched) categorical structures we define the notion of ordered
sheaf on a quantaloid Q, which we call `Q-order'. This requires a theory of
semicategories enriched in the quantaloid Q, that admit a suitable Cauchy
completion. There is a quantaloid Idl(Q) of Q-orders and ideal relations, and a
locally ordered category Ord(Q) of Q-orders and monotone maps; actually,
Ord(Q)=Map(Idl(Q)). In particular is Ord(Omega), with Omega a locale, the
category of ordered objects in the topos of sheaves on Omega. In general
Q-orders can equivalently be described as Cauchy complete categories enriched
in the split-idempotent completion of Q. Applied to a locale Omega this
generalizes and unifies previous treatments of (ordered) sheaves on Omega in
terms of Omega-enriched structures.Comment: 21 page
Modules on involutive quantales: canonical Hilbert structure, applications to sheaf theory
We explain the precise relationship between two module-theoretic descriptions
of sheaves on an involutive quantale, namely the description via so-called
Hilbert structures on modules and that via so-called principally generated
modules. For a principally generated module satisfying a suitable symmetry
condition we observe the existence of a canonical Hilbert structure. We prove
that, when working over a modular quantal frame, a module bears a Hilbert
structure if and only if it is principally generated and symmetric, in which
case its Hilbert structure is necessarily the canonical one. We indicate
applications to sheaves on locales, on quantal frames and even on sites.Comment: 21 pages, revised version accepted for publicatio
State transitions as morphisms for complete lattices
We enlarge the hom-sets of categories of complete lattices by introducing
`state transitions' as generalized morphisms. The obtained category will then
be compared with a functorial quantaloidal enrichment and a contextual
quantaloidal enrichment that uses a specific concretization in the category of
sets and partially defined maps ().Comment: 9 page
Grothendieck quantaloids for allegories of enriched categories
For any small involutive quantaloid Q we define, in terms of symmetric
quantaloid-enriched categories, an involutive quantaloid Rel(Q) of Q-sheaves
and relations, and a category Sh(Q) of Q-sheaves and functions; the latter is
equivalent to the category of symmetric maps in the former. We prove that
Rel(Q) is the category of relations in a topos if and only if Q is a modular,
locally localic and weakly semi-simple quantaloid; in this case we call Q a
Grothendieck quantaloid. It follows that Sh(Q) is a Grothendieck topos whenever
Q is a Grothendieck quantaloid. Any locale L is a Grothendieck quantale, and
Sh(L) is the topos of sheaves on L. Any small quantaloid of closed cribles is a
Grothendieck quantaloid, and if Q is the quantaloid of closed cribles in a
Grothendieck site (C,J) then Sh(Q) is equivalent to the topos Sh(C,J). Any
inverse quantal frame is a Grothendieck quantale, and if O(G) is the inverse
quantal frame naturally associated with an \'etale groupoid G then Sh(O(G)) is
the classifying topos of G.Comment: 28 pages, final versio
Operational resolutions and state transitions in a categorical setting
We define a category with as objects operational resolutions and with as
morphisms - not necessarily deterministic - state transitions. We study
connections with closure spaces and join-complete lattices and sketch physical
applications related to evolution and compoundness. An appendix with
preliminaries on quantaloids is included.Comment: 21 page