14 research outputs found
Generalizations of Hedbergâs Theorem
Streicherâs axiom K or the uniqueness of identity proofs (UIP) is the statement that every identity type has at most one inhabitant. The groupoid interpretation by Hofmann and Streicher shows that this is not provable for an arbitrary type, but a theorem by Hedberg gives a sufficient condition. Types satisfying UIP are known as h-sets. The main contributions of this paper are: (i) Natural, more general sufficient conditions for a type being an h-set. (ii) A new characterization of h-sets. (iii) A new result inspired by these characterizations, with some applications. All the proofs have been formalized in Agda
Continuous Truth II: Reflections
Abstract. In the late 1960s, Dana Scott first showed how the Stone-Tarski topological interpretation of Heytingâs calculus could be extended to model intuitionistic analysis; in particular Brouwerâs continuity prin-ciple. In the early â80s we and others outlined a general treatment of non-constructive objects, using sheaf modelsâconstructions from topos theoryâto model not only Brouwerâs non-classical conclusions, but also his creation of ânew mathematical entitiesâ. These categorical models are intimately related to, but more general than Scottâs topological model. The primary goal of this paper is to consider the question of iterated extensions. Can we derive new insights by repeating the second act? In Continuous Truth I, presented at Logic Colloquium â82 in Florence, we showed that general principles of continuity, local choice and local com-pactness hold in the gros topos of sheaves over the category of separable locales equipped with the open cover topology. We touched on the question of iteration. Here we develop a more gen-eral analysis of iterated categorical extensions, that leads to a reflection schema for statements of predicative analysis. We also take the opportunity to revisit some aspects of both Continuous Truth I and Formal Spaces (Fourman & Grayson 1982), and correct two long-standing errors therein
A topos for algebraic quantum theory
The aim of this paper is to relate algebraic quantum mechanics to topos
theory, so as to construct new foundations for quantum logic and quantum
spaces. Motivated by Bohr's idea that the empirical content of quantum physics
is accessible only through classical physics, we show how a C*-algebra of
observables A induces a topos T(A) in which the amalgamation of all of its
commutative subalgebras comprises a single commutative C*-algebra. According to
the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter
has an internal spectrum S(A) in T(A), which in our approach plays the role of
a quantum phase space of the system. Thus we associate a locale (which is the
topos-theoretical notion of a space and which intrinsically carries the
intuitionistic logical structure of a Heyting algebra) to a C*-algebra (which
is the noncommutative notion of a space). In this setting, states on A become
probability measures (more precisely, valuations) on S(A), and self-adjoint
elements of A define continuous functions (more precisely, locale maps) from
S(A) to Scott's interval domain. Noting that open subsets of S(A) correspond to
propositions about the system, the pairing map that assigns a (generalized)
truth value to a state and a proposition assumes an extremely simple
categorical form. Formulated in this way, the quantum theory defined by A is
essentially turned into a classical theory, internal to the topos T(A).Comment: 52 pages, final version, to appear in Communications in Mathematical
Physic
ParadisEO-MOEO: A Software Framework for Evolutionary Multi-Objective Optimization
This chapter presents ParadisEO-MOEO, a white-box object-oriented software framework dedicated to the flexible design of metaheuristics for multi-objective optimization. This paradigm-free software proposes a unified view for major evolutionary multi-objective metaheuristics. It embeds some features and techniques for multi-objective resolution and aims to provide a set of classes allowing to ease and speed up the development of computationally efficient programs. It is based on a clear conceptual distinction between the solution methods and the problems they are intended to solve. This separation confers a maximum design and code reuse. This general-purpose framework provides a broad range of fitness assignment strategies, the most common diversity preservation mechanisms, some elitistrelated features as well as statistical tools. Furthermore, a number of state-of-the-art search methods, including NSGA-II, SPEA2 and IBEA, have been implemented in a user-friendly way, based on the fine-grained ParadisEO-MOEO components
Whole-genome sequencing reveals host factors underlying critical COVID-19
Critical COVID-19 is caused by immune-mediated inflammatory lung injury. Host genetic variation influences the development of illness requiring critical care1 or hospitalization2,3,4 after infection with SARS-CoV-2. The GenOMICC (Genetics of Mortality in Critical Care) study enables the comparison of genomes from individuals who are critically ill with those of population controls to find underlying disease mechanisms. Here we use whole-genome sequencing in 7,491 critically ill individuals compared with 48,400 controls to discover and replicate 23 independent variants that significantly predispose to critical COVID-19. We identify 16 new independent associations, including variants within genes that are involved in interferon signalling (IL10RB and PLSCR1), leucocyte differentiation (BCL11A) and blood-type antigen secretor status (FUT2). Using transcriptome-wide association and colocalization to infer the effect of gene expression on disease severity, we find evidence that implicates multiple genesâincluding reduced expression of a membrane flippase (ATP11A), and increased expression of a mucin (MUC1)âin critical disease. Mendelian randomization provides evidence in support of causal roles for myeloid cell adhesion molecules (SELE, ICAM5 and CD209) and the coagulation factor F8, all of which are potentially druggable targets. Our results are broadly consistent with a multi-component model of COVID-19 pathophysiology, in which at least two distinct mechanisms can predispose to life-threatening disease: failure to control viral replication; or an enhanced tendency towards pulmonary inflammation and intravascular coagulation. We show that comparison between cases of critical illness and population controls is highly efficient for the detection of therapeutically relevant mechanisms of disease
The âworld's simplest axiom of choiceâ fails
We use topos-theoretic methods to show that intuitionistic set theory with countable or dependent choice does not imply that every family, all of whose elements are doubletons and which has at most one element, has a choice function.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46645/1/229_2005_Article_BF01170929.pd