Grothendieck's Vision II

Grothendiecks's Vision and Semantic Unfolding in the Conception of Geometric Form: Group Sheaves as Locally Variable Coefficient Systems for Cohomology

The far-reaching applicability of Grassmann's semantic unfolding of the field of lineal geometric analysis can be adequately appreciated only if it can be lifted appropriately from the level of vector spaces to the level of locally free modules defined over a smooth manifold or, more generally, over an arbitrary topological space, called vector sheaves.

The essential aspect of this generalization marking the powerful unifying combination of methods and ideas from the fields of geometry, topology, analysis, and homological algebra, is the notion of localization of a mathematical form, which reaches its greatest heights with the invention of sheaf theory and its subsequent application in the machinery of cohomology theory.

Conceptually, the invariant transference of Grassmann's lineal extension framework from the level of vectors to the level of locally free modules necessitates the consideration of locally definable and variable mathematical entities.

The Horizon of an Implicit Point

Grothendieck's notion of a cover is thought of as an observation horizon of a point, represented as an opening in a covering sieve that infiltrates observable information compatibly at different resolution layers by refinement until its temporal completion, via which the essence of a point as a bearer of some globally irreducible, quantifiable, and invariant information may be unfolded cohomologically.

The defining requirements of these covers are the following: Covers are transitive meaning that covers of covers are also covers, covers are stable under pullback operations implementing in this manner the stability of the notion of a cover under change of base, and isomorphisms are qualified as covers.

Cohomological Unfolding

The notion of a coboundary operator is the encoding of a multiplicity of point-bounding temporal orders in infinitesimal terms, subject to the coboundary law, which establishes the nilpotency of the square of this infinitesimal operator.

In this context, the conception of Cech cohomology is based on an articulation of the notion of a temporal order, which is independent of the concept of a local linearly extended geometric magnitude, on which the notion of a de Rham coboundary operator is based on in terms of differential forms.

The amazing fact that these two different types of cohomological unfolding can be unified is based on Grothendieck's vision that a locally variable sheaf of coefficients is actually the natural argument of all cohomological theories in this context.

In other words, the natural argument of a cohomology theory is not just a space, as it was initially thought of, but a space together with an observable sheaf of coefficients, such that the space constitutes the observed spectrum of the sheaf employed cohomologically for this purpose.

The global nature of points of this spectrum is typically determined, as it is actually expected, by a constant sheaf of coefficients, i.e. a sheaf of locally constant sections valued in the integers, or the reals, or the complex numbers. In this sense, a commutative observable sheaf of coefficients plays locally the equivalent role of a measurement procedure, or apparatus in physical terminology, that is capable of capturing some global invariant feature only by cohomological means.

Resolution of Points via Encoding/Decoding Inductive/Projective Systems

The basic characteristic of these open coverings is that they are partially ordered by inclusion. Note that this is just a partial order and not a total order of open covers, meaning that it is capable of subsuming a variety of potential local directed total orders or chains.

The crucial idea is to force an inductive system out of these open covers that is capable of resolving points in terms of a corresponding dual projective system of abelian groups of locally defined function elements or sections of a sheaf over these covers at varying resolution horizons. Note that a section is conceived extensively as a whole over its domain or locus of definition.

In the initial conception of this theory these function elements have been considered as constants, for instance constant real-valued functions, i.e. the constant coefficient system of the real numbers. In this sense, an extension process of function elements is initiated by gluing them together via joining together open covers, which has to be compatible with the inverse process of restriction of these elements to smaller open covers.

For this purpose, it is required that for any pair of them there exists an infimum expressed by their intersection, or meet, or more generally pull-back operation, such that their overlap is totally contractible. Clearly, for every sub-collection of these covers there also exists a supremum with respect to their partial order relation. Implicitly also there is considered a minimal cover to serve as the inductive limit of all pairwise intersections of all covers forming a chain.

Novel Notion of a Directed Temporal Order

The crucial issue is that, in the case of a topological covering sieve, there is required a novel notion of a directed temporal order to be applied for distinguishing singular points, which is of a different quality from the one conceptualized in terms of boundary chains.

It is intuitive enough to think that a singular point, since it cannot be bounded by infinitesimal flows meeting at this point, it can be only amenable to a process of circulating around it, where all these circulations are not equivalent, but they are in principle distinguishable in terms of different global attributes.

Dually thinking, the type of global chains needed for this purpose, called cycles, are not boundaries, and most significant, they encapsulate a novel type of temporal order that it is qualitatively different from the former one.

The type of temporal order encapsulated by cycles is characterized in terms of the periods of the locally-generated process of circulating around it in terms of a cocycle, to be thought of as a global integration procedure, and not in terms of instants.

Grothendieck's Phraseology

Consider the set formed by all sheaves over a (given) topological space or, if you like, the formidable arsenal of all the "rulers'' that can be used in taking measurements on it. We will treat this "ensemble'', or "arsenal'' as one equipped with a structure that may be considered "self-evident'', one that crops up "in front of one's nose'': that is to say, a Categorical structure ... It functions as a kind of "superstructure of measurement'', called the "Category of Sheaves'' (over the given space), which henceforth shall be taken to incorporate all that is most essential about that space. This is in all respects a lawful procedure, (in terms of "mathematical common sense'') because it turns out that one can "reconstitute'' in all respects, the topological space by means of the associated "category of sheaves'' (or "arsenal'' of measuring instruments).