We study some particular examples of quantaloids and corresponding morphisms, originating from primitive physical reasonings on the lattices of properties of physical systems. AMS classification code: 03G10, 18D20, 81P10. Key words: Complete Lattice, Galois Adjoint, Physical System, Quantaloid. 1 Introduction The starting point for our research program is the fact, already observed in Eilenberg and Mac Lane's seminal paper [8], that preordered sets may be considered as small thin categories. One can then not only reformulate a large part of the theory of order structures in categorical terms, but also apply general categorical techniques to specific order theoretic problems. In particular, the notion of an adjunction reduces to that of a residuation [15] 4.5, whereas the notion of a monad reduces to that of a closure operator [15] 6.1-2. Now the above categorical notions have direct physical interpretations in the context of axiomatic quantum theory, the order relation in the prop...

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We present a detailed synthetic overview of the utilisation of categorical techniques in the study of order...

The development ofoperationalquantum logicpointsoutthatclassicalbooleanstructuresaretoo rigidtodescribe theactualand potentialpropertiesofquantum systems.Operationalquantum logic bearsupon basicaxiomswhich aremotivatedby empiricalfactsand assuch supportsthedynamicshiftfromclassicaltonon-classicallogic resultinginto a dynamicsoflogic.On theotherhand,a dynamic extensionofoperationalquantum logicallows ustoexpressdynamic 1 reasoninginthesensethatwecancapturehow actualpropertiespropagate, includingtheirtemporalcausalstructure.Itisinthissense thatpassingfromstaticoperationalquantum logictodynamicoperationalquantum logicresultsina truelogicofdynamicsthatprovides a unifledlogicaldescriptionofsystemswhich evolve orwhich aresubmittedtomeasurements. Whilefocusingon thequantalesemantics fordynamicoperationalquantum logic,we cananalyzethepointsof difierencewiththeexistingquantalesemanticsfor(non)-commutative linearlogic.Linearlogicisheretobeconceivedasa resource-sensitive logiccapableofdealingwi...

Instances of non-commutativity are pervasive in human behavior. In this paper, we suggest that psychological properties such as attitudes, values, preferences and beliefs may be suitably described in terms of the mathematical formalism of quantum mechanics. We expose the foundations of nonclassical measurement theory building on a simple notion of orthospace and ortholattice (logic). Two axioms are formulated and the characteristic state-property duality is derived. A last axiom concerned with the impact of measurements on the state takes us with a leap toward the Hilbert space model of Quantum Mechanics. An application to behavioral sciences is proposed. First, we suggest an interpretation of the axioms and basic properties for human behavior. Then we explore an application to decision theory in an example of preference reversal. We conclude by formulating basic ingredients of a theory of actualized preferences based in non-classical measurement theory.

Abstract. State property systems were created on the basis of physical intuition in order to describe a mathematical model for physical systems. A state property system consists of a triple: a set of states, a complete lattice of properties and a specified function linking the other two components. The definition of morphisms between such objects was inspired by the physical idea of a subsystem. In this paper we give an isomorphic description of the category of state property systems, thus introducing a category SP, which is concrete over Set. This isomorphic description enables us to investigate further categorical properties of SP. It turns out that the category SP is not amnestic. In our main theorem we prove that the amnestic modification of SP is the construct Cls of closure spaces and continuous maps. Moreover we observe that the categorical product and coproduct in Cls find, through SP, application in physics.

The term quantum logic has different connotations for different people, having been considered as everything from a metaphysical attack on classical reasoning to an exercise in abstract algebra. Our aim here is to give a uniform presentation of what we call operational quantum logic, highlighting both its concrete physical origins and its purely mathematical structure. To orient readers new to this subject, we shall recount some of the historical development of quantum logic, attempting to show how the physical and mathematical sides of the subject have influenced and enriched one another. 1 This paper is a slightly modified version of the introductory chapter to the volume B. Coecke,

We present a detailed synthetic overview of the utilisation of categorical techniques in the study of order structures together with their applications in operational quantum theory. First, after reviewing the notion of residuation and its implementation at the level of quantaloids we consider some standard universal constructions and the extension of adjunctions to weak morphisms. Second, we present the categorical formulation of closure operators and introduce a hierarchy of contextual enrichments of the quantaloid of complete join lattices. Third, we briefly survey physical state-property duality and the categorical analysis of derived notions such as causal assignment and the propagation of properties. 1.

The term quantum logic has different connotations for different people, having been considered as everything from a metaphysical attack on classical reasoning to an exercise in abstract algebra. Our aim here is to give a uniform presentation of what we call operational quantum logic, highlighting both its concrete physical origins and its purely mathematical structure. To orient readers new to this subject, we shall recount some of the historical development of quantum logic, attempting to show how the physical and mathematical sides of the subject have influenced and enriched one another. 1 This paper is a slightly modified version of the introductory chapter to the volume B. Coecke,