Leray's Vision

Leray's Vision and Semantic Unfolding of the Notion of a Topological Space: Sheaves as Models of the Compatible Passage from the Local to the Global and the Articulation of Points

The concept of a sheaf expresses essentially gluing conditions, namely the way by which local information, organized in terms of the structure of groups, or modules, or algebras, or even sets, can be semantically unfolded from the local to the global, thus collated compatibly into global ones over a partially order covering system, like the one defined by the open covers of a topological space. In general, a sheaf may be thought of as a continuously variable algebraic information structure, whose continuous variation, expressed in terms of its sections, is considered over specified local covers of an implicit global topological space, such that sectional information can be amalgamated together via the compatible passage from the local to the global.

Unfolding through Spectral Becoming

One of the most interesting aspects of the theory of sheaves in relation to the semantic unfolding of the notion of a topological space is that a sheaf can be viewed as capturing the model of a space in its spectral becoming.

Given that a topological space, for instance a manifold, is physically utilized to represent the event or state structure of a physical theory corresponding to the evaluation of observables, its associated sheaf model pertains to its actual taking place, i.e. to its actual constitution by amalgamating compatibly local observables into global ones.

This is because the underlying notion of a global topological space, referring to its structure of points, both standard and singular, is initially only implicitly assumed, and then, step-by-step induced indirectly by completion via the germs of the sheaf, culminating in extracting the invariant information pertaining to these germs after their integration.

Sheaf Ontophainetic Platform of Local Germs

According to Leray, the sheaf model of a space does not have a local structure defined by its points, assumed as absolute and pre-existing, but a local structure which is intrinsically continuous with respect to the totality of its covers partially ordered through inclusion.

A closed partially ordered family of intersecting local covers, capable of unravelling gradually the invariant information that germs of a complete presheaf bear at a point, acquires the semantics of a temporal order pertaining to the apeiron aspect of reality.

The algebraic structure of germs at a point subsumes in this manner the role of an ontophainetic platform delineating spectrally the time-space of the present at that point. The sheaf completion property at a point is tantamount to the integration of these germs, though of as expressing differentially the local or infinitesimal variations around this point.

Eventually, this is precisely the unfolding process that characterizes not only the underlying point itself sheaf-theoretically, but also its genetic constitution and variation in statu-nascendi in relation to the considered temporal order.

This is because the global characterization of a point, for example of a physical state or an event, is the final result, that is, the trace of a continuous unfolding process of genetic constitution in terms of granular elements, i.e. the germs of sections around that point bearing the resolution capacity of their local ontophainetic cover-horizon incorporating or integrating all their compatible hereditary subcovers within the pertinent temporal order.

Sieves of Localization and Traces

The idea of considering hereditary covers can be implemented through refinement of the resolution grain of information, which is topologically qualified as localization taking place via the spectral process of sieving.

A sieve on a local cover can be thought of as consisting of spectral horizons distributed across different nested layers, such that every partial information which is compatible with respect to one of these horizons passes through it but not otherwise.

The consideration of covering sieves as hereditary covers used for localizing, and thus sharpening, information with respect to intersecting local covers at deeper ordered depths, gives rise to localization systems of global observable algebras, which induce a semantic transition of events from the set-theoretic to the sheaf-theoretic level.

In this manner, events are just the traces of the process of semantically unfolding observable information via sheaf-theoretic localization taking place in statu-nascendi.

Amalgamation of the Local to the Global and the Ontophainetic Display Space of a Sheaf

The sheaf model of a space takes place and is constituted in three steps, after the specification of a covering sieve:

The first step organizes compatibly the local cover-infiltrated information of sections, such that the latter can be compatibly restricted from the global to the local. This process produces a structure that is only a presheaf. Notice that conceptually the implicitly assumed global, from the presheaf-theoretic viewpoint, refers to the global in its potentiality to germinate in the context of some temporal order, and eventually produce facts.

The second step involves the process of functional localization of the sections of the presheaf. This is necessary because the sections of the presheaf are not a priori functional elements. This is accomplished by means of the topological realization of the presheaf of sections as a spectral partition, that is, an ontophainetic display space constituted by disjoint stalks of germs over each point of the underlying base space.

The third step involves the completion of the presheaf, or equivalently the completion of its associated display space, such that the germs belonging to each stalk can be evaluated and produce a fact, temporalizing in this way the pertinent point of the stalk, by disclosing genetically the temporal order they descent from.

In this way, it is realized a sheaf, incorporating all the necessary and sufficient conditions for the bidirectional compatibility of information under restriction or reduction from the global to the local, and inversely under unique extension or induction from the local to the global.

The latter sheaf-theoretic conceptualization of the global refers now to this term in its actuality to explicate globally points in terms of events following the germination of sections with respect to the temporal order they participate and descent from.