124 research outputs found
Dualities and dual pairs in Heyting algebras
We extract the abstract core of finite homomorphism dualities using the
techniques of Heyting algebras and (combinatorial) categories.Comment: 17 pages; v2: minor correction
Dualities and Dual Pairs in Heyting Algebras
We extract the abstract core of finite homomorphism dualities using the techniques of Heyting algebras and (combinatorial) categorie
Variable sets over an algebra of lifetimes: a contribution of lattice theory to the study of computational topology
A topos theoretic generalisation of the category of sets allows for modelling
spaces which vary according to time intervals. Persistent homology, or more
generally, persistence is a central tool in topological data analysis, which
examines the structure of data through topology. The basic techniques have been
extended in several different directions, permuting the encoding of topological
features by so called barcodes or equivalently persistence diagrams. The set of
points of all such diagrams determines a complete Heyting algebra that can
explain aspects of the relations between persistent bars through the algebraic
properties of its underlying lattice structure. In this paper, we investigate
the topos of sheaves over such algebra, as well as discuss its construction and
potential for a generalised simplicial homology over it. In particular we are
interested in establishing a topos theoretic unifying theory for the various
flavours of persistent homology that have emerged so far, providing a global
perspective over the algebraic foundations of applied and computational
topology.Comment: 20 pages, 12 figures, AAA88 Conference proceedings at Demonstratio
Mathematica. The new version has restructured arguments, clearer intuition is
provided, and several typos correcte
Stone-Type Dualities for Separation Logics
Stone-type duality theorems, which relate algebraic and
relational/topological models, are important tools in logic because -- in
addition to elegant abstraction -- they strengthen soundness and completeness
to a categorical equivalence, yielding a framework through which both algebraic
and topological methods can be brought to bear on a logic. We give a systematic
treatment of Stone-type duality for the structures that interpret bunched
logics, starting with the weakest systems, recovering the familiar BI and
Boolean BI (BBI), and extending to both classical and intuitionistic Separation
Logic. We demonstrate the uniformity and modularity of this analysis by
additionally capturing the bunched logics obtained by extending BI and BBI with
modalities and multiplicative connectives corresponding to disjunction,
negation and falsum. This includes the logic of separating modalities (LSM), De
Morgan BI (DMBI), Classical BI (CBI), and the sub-classical family of logics
extending Bi-intuitionistic (B)BI (Bi(B)BI). We additionally obtain as
corollaries soundness and completeness theorems for the specific Kripke-style
models of these logics as presented in the literature: for DMBI, the
sub-classical logics extending BiBI and a new bunched logic, Concurrent Kleene
BI (connecting our work to Concurrent Separation Logic), this is the first time
soundness and completeness theorems have been proved. We thus obtain a
comprehensive semantic account of the multiplicative variants of all standard
propositional connectives in the bunched logic setting. This approach
synthesises a variety of techniques from modal, substructural and categorical
logic and contextualizes the "resource semantics" interpretation underpinning
Separation Logic amongst them
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