343 research outputs found
Convolution, Separation and Concurrency
A notion of convolution is presented in the context of formal power series
together with lifting constructions characterising algebras of such series,
which usually are quantales. A number of examples underpin the universality of
these constructions, the most prominent ones being separation logics, where
convolution is separating conjunction in an assertion quantale; interval
logics, where convolution is the chop operation; and stream interval functions,
where convolution is used for analysing the trajectories of dynamical or
real-time systems. A Hoare logic is constructed in a generic fashion on the
power series quantale, which applies to each of these examples. In many cases,
commutative notions of convolution have natural interpretations as concurrency
operations.Comment: 39 page
Quantales of open groupoids
It is well known that inverse semigroups are closely related to \'etale
groupoids. In particular, it has recently been shown that there is a
(non-functorial) equivalence between localic \'etale groupoids, on one hand,
and complete and infinitely distributive inverse semigroups (abstract complete
pseudogroups), on the other. This correspondence is mediated by a class of
quantales, known as inverse quantal frames, that are obtained from the inverse
semigroups by a simple join completion that yields an equivalence of
categories. Hence, we can regard abstract complete pseudogroups as being
essentially ``the same'' as inverse quantal frames, and in this paper we
exploit this fact in order to find a suitable replacement for inverse
semigroups in the context of open groupoids that are not necessarily \'etale.
The interest of such a generalization lies in the importance and ubiquity of
open groupoids in areas such as operator algebras, differential geometry and
topos theory, and we achieve it by means of a class of quantales, called open
quantal frames, which generalize inverse quantal frames and whose properties we
study in detail. The resulting correspondence between quantales and open
groupoids is not a straightforward generalization of the previous results
concerning \'etale groupoids, and it depends heavily on the existence of
inverse semigroups of local bisections of the quantales involved.Comment: 55 page
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