25 research outputs found
The Involutive Quantaloid of Completely Distributive Lattices
Let L be a complete lattice and let Q(L) be the unital quantale of join-continuous endo-functions of L. We prove the following result: Q(L) is an involutive (that is, non-commutative cyclic ⋆-autonomous) quantale if and only if L is a completely distributive lattice. If this is the case, then the dual tensor operation corresponds, via Raney's transforms, to composition in the (dual) quantale of meet-continuous endo-functions of L. Let sLatt be the category of sup-lattices and join-continuous functions and let cdLatt be the full subcategory of sLatt whose objects are the completely distributive lattices. We argue that (i) cdLatt is itself an involutive quantaloid, and therefore it is the largest full-subcategory of sLatt with this property; (ii) cdLatt is closed under the monoidal operations of sLatt and, consequently, if Q(L) is involutive, then Q(L) is completely distributive as well
Unitless Frobenius quantales
It is often stated that Frobenius quantales are necessarily unital. By taking
negation as a primitive operation, we can define Frobenius quantales that may
not have a unit. We develop the elementary theory of these structures and show,
in particular, how to define nuclei whose quotients are Frobenius quantales.
This yields a phase semantics and a representation theorem via phase quantales.
Important examples of these structures arise from Raney's notion of tight
Galois connection: tight endomaps of a complete lattice always form a Girard
quantale which is unital if and only if the lattice is completely distributive.
We give a characterisation and an enumeration of tight endomaps of the diamond
lattices Mn and exemplify the Frobenius structure on these maps. By means of
phase semantics, we exhibit analogous examples built up from trace class
operators on an infinite dimensional Hilbert space. Finally, we argue that
units cannot be properly added to Frobenius quantales: every possible extention
to a unital quantale fails to preserve negations
Lineales
The first aim of this note is to describe an algebraic structure,
more primitive than lattices and quantales, which corresponds to the
intuitionistic flavour of Linear Logic we prefer. This part of the note
is a total trivialisation of ideas from category theory and we play with
a toy-structure a not distant cousin of a toy-language.
The second goal of the note is to show a generic categorical construction,
which builds models for Linear Logic, similar to categorical models GC of
[deP1990], but more general. The ultimate aim is to relate different categorical models of linear logic
An Internal Language for Categories Enriched over Generalised Metric Spaces
Programs with a continuous state space or that interact with physical processes often require notions of equivalence going beyond the standard binary setting in which equivalence either holds or does not hold. In this paper we explore the idea of equivalence taking values in a quantale ?, which covers the cases of (in)equations and (ultra)metric equations among others.
Our main result is the introduction of a ?-equational deductive system for linear ?-calculus together with a proof that it is sound and complete (in fact, an internal language) for a class of enriched autonomous categories. In the case of inequations, we get an internal language for autonomous categories enriched over partial orders. In the case of (ultra)metric equations, we get an internal language for autonomous categories enriched over (ultra)metric spaces.
We use our results to obtain examples of inequational and metric equational systems for higher-order programs that contain real-time and probabilistic behaviour