The physical "continuum" is being modeled as a complex of events interconnected by the relation of extension and forming an abstract partially ordered structure. Operational physical procedures for discerning observable events assume their existence and validity locally, by coordinatizing the informational content of those observable events in terms of real-valued local observables. The localization process is effectuated in terms of topological covering systems on the events "continuum", that

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... o not presuppose an underlying structure of points on the real line, and moreover, respect only the fundamental relation of extension between events. In this sense, the physical "continuum" is represented by means of a fibered topos-theoretic structure, modeled as a sheaf of algebras of continuous real-valued functions. Finally, the dynamics of information propagation, formulated in terms of continuous real-valued observables, is described in the context of an appropriate sheaf-cohomological framework corresponding to the localization process. 0 The semantics of the physical "continuum" in the standard interpretation of physical systems theories is associated with the codomain of valuation of physical attributes (Butterfield and Isham (2000) ). Usually the notion of "continuum" is tied with the attribute of position, serving as the range of values characterizing this particular attribution. The model adopted to represent these values is the real line R and its powers, specified as a set theoretical structure of points that are independent and possess the property of infinite distinguishability with absolute precision. The adoption of the set-theoretic real line model is usually justified on the basis of operational arguments. Physical attributes are associated with the conception of observables, that is, physical quantities which, in principle, can be measured. Furthermore, physical systems theories stipulate that quantities admissible as measured results must be real numbers, since, it is accepted that the resort to real numbers has the advantage of making our empirical access secure. Thus, the crucial assumption underlying the employment of the real line and its powers for the modeling of the physical "continuum" is that real number representability constitutes our form of observation. In this context, the geometrization of classical field theories as fiber bundles of some kind over a background spacetime points manifold, that is locally a power of the real line, is necessitated by the requirement of conferring numerical identity to the corresponding field events, conceived as being localized on charts of the spacetime manifold, being homeomorphic to a power of the real line. Closely related to the conceptualization of these geometrical models is the issue of localization in the physical "continuum". Operational procedures accompanying the development of physical systems theories can be understood as providing the means of probing the physical "continuum", via appropriate processes of localization in the "continuum", referring to localized events in terms of real-valued observable quantities. 3 Thus, physical observation presupposes, at the fundamental level, the development of localization processes in the "continuum" that accomplish the task of discerning observable events from it, and subsequently, assigning an individuality to them. It is important to notice however, that ascribing individuality to an event that has been observed by means of a localization scheme is not always equivalent to conferring a numerical identity to it, by means of a real value corresponding to a physical attribute. This is exactly the crucial assumption underlying the almost undisputed employment of the set theoretical model of the real line and its powers as models of the physical "continuum". The consequences of this common assumption, overlooked mainly because of the successful integration of the techniques of real analysis and classical differential geometry of smooth manifolds, in the argumentation and predictive power of physical theories, has posed enormous technical and interpretational problems related mainly with the appearance of singularities. In this work we will attempt a transition in the semantics of the events "continuum" from a set-theoretic to a sheaf-theoretic one. The transition will be effectuated by using the syntax and technical machinery provided by category and topos theory ). Conceptually, the proposed semantic transition is implemented and necessitated by the introduction of the following basic requirements admitting a sound physical basis of reasoning: Requirement I: The primary conception of the physical "continuum" constitutes an inexhaustible complex of overlapping and non-overlapping events. The consideration of the notion of event as a primary concept immediately poses the following question: How are events being related to each other? If continuity is to be ascribed in the relations among events, then the 4 fundamental relation is extension. The relata in the relation of extension are the events, such that each event is part of a larger whole and each event encompasses smaller events. Extension is also inextricably tied with the assumption of divisibility of events signifying a part-whole type of relation. In this sense, the physical "continuum" should constitute a representation of events ontology respecting the fundamental relation of extension. A natural working hypothesis in this sense, would be the modeling of the physical "continuum" by a partial order of events. Requirement II: The notion of a "continuum" of events should not be necessarily based on the existence of an underlying structure of points on the real line. This equivalently means that localization processes for the individuation of events from the physical "continuum" should not depend on the existence of points. In this sense, ascribing individuality to an event that has been observed by means of a localization scheme should not be tautosemous to conferring a numerical identity to it, by means of a real value corresponding to a physical attribute, but only a limiting case of the localization process. In order to construct a sheaf-theoretic model of the physical "continuum" based on the above physical requirements, and thus, accomplish the announced semantic transition, we further assume that, the localization process is being effectuated operationally in terms of suitable topological covering systems, which, do not presuppose an underlying structure of points on the real line, and moreover, respect only the fundamental relation of extension between events. In this sense, it will become apparent that the physical "continuum" can be precisely represented by means of a generalized fibered structure, such that, the partial order of events fibers over a base category of varying reference loci, corresponding to the open sets of a topological measurement space, ordered by inclusion. Moreover, we will explicitly show 5 that, this fibered structure is being modeled as a sheaf of algebras of continuous real-valued functions, corresponding to observables. Finally, we are going to demonstrate that a sheaf-theoretic fibered construct of the physical continuum, as briefly described above, permits the formulation of the dynamical aspects of information propagation, in terms of a purely algebraic cohomological framework. Generally speaking, the concept of sheaf expresses essentially gluing conditions, or equivalently, it formalizes the requirements needed for collating local observable information into global ones. The notion of local is characterized mathematically by means of a topological covering system, which, is the referent of topological closure conditions on the collection of covers, instantiating a localization process in the "continuum". It is important to emphasize that, the transition from locally defined observable information into global ones, elucidated by the concept of sheaf, takes place via a globally compatible family of localized information elements over a topological covering system of the "continuum". For a general mathematical and philosophical discussion of sheaves, variable sets, and related structures, the interested reader should consult (Lawvere (1975), Zafiris (2005)). Technical expositions of sheaf theory, being of particular interest in relation to the focus of the present work on topological localization processes, are provided by (Mac Lane and Moerdijk (1992), Bell (1986), Mallios (2004), and Bredon (1997)). Various applications of sheaf-theoretic fibered structures, based on the development of suitable localization schemes referring to the modeling and interpretation of quantum and complex systems, have been communicated, both conceptually and technically by the author, in the literature (The behavior of physical systems is adequately described by the collection of all observed data determined by the functioning of measurement devices in suitably specified experimental environments. Observables are precisely associated with physical quantities that, in principle, can be measured. The mathematical formalization of this procedure relies on the idea of expressing the observables by functions corresponding to measuring devices. Moreover, the usual underlying assumption on the basis of physical theories postulates that our global form of observation is represented by real-valued coefficients, and subsequently, global observables are modeled by real-valued functions corresponding to measuring devices. At a further stage of development of this notion, two fundamental requirements are being postulated on the structure of observables: Postulate I: The first postulate specifies the algebraic nature of the set of all observables, by assuming the structure of a commutative unital algebra A over the real numbers. The basic intuition behind this requirement is related with the fact that we can legitimately associate to any commutative algebra with unit a geometric object, called the spectrum of the algebra, such that the elements of the algebra, viz. the observables, can be considered as functions on the spectrum. The implemented principle is that the geometric structure of a measurement space can be completely recovered from the commutative algebra of observables defined on it. From a mathematical perspective, this principle has been well demonstrated in a variety of different contexts, known as Stone-Gel'fand duality in a functional analytic setting, or Grothendieck duality in an algebraic geometrical setting. Thus, to any commutative unital algebra of observables over the real numbers R, we can associate a measurement space, namely its real spectrum, such that, each element of the algebra becomes a real-valued function on sented equivalently, as derivations, are representable as left A-modules of n-forms Ω n (A) in the category of left A-modules M (A) . We emphasize that the intelligibility of the algebraic modeling framework of dynamics, giving rise to variable geometric spectra, is based on the conception that infinitesimal variations in the observables of A, are caused by interactions, meaning that they are being effectuated by the presence of a physical field. Thus, it is necessary to establish a purely algebraic representation of the notion of a physical field, as the causal agent of local interactions, and moreover, explain the functional role it assumes for the interpretation of the theory. The key idea for this purpose amounts vectors of the left A-module E can be expressed by means of the following comparison morphism of left A-modules: Equivalently, the irreducible amount of information incorporated in the comparison morphism, can be now expressed as a connection on E, viz., as an R-linear morphism of A-modules: such that, the following Leibniz type constraint is satisfied: Consequently, after having expressed the process of scalars extension in functorial algebraic terms, we can identify the functor of infinitesimal scalars extension, due to interactions, with the functional dependence induced by a physical field causing it. Thus, a local causal agent of a variable interaction geometry, viz., a physical field acting locally and causing infinitesimal variations of local observables, can be faithfully represented by means of a pair (E, ∇ E ), consisting of a left A-module E and a connection ∇ E on E. We conclude, by emphasizing that, the functorial modeling of the universal mechanism of encoding physical interactions, by means of causal agents, as above, namely, physical fields effectuating infinitesimal scalars extension, is covariant with the algebra-theoretic specification of the structure of observables. Equivalently stated, the only actual requirement for the intelligibility of functoriality of interactions, by means of physical fields, rests on the algebra-theoretic specification of what we characterize structures of observables.