Duality, manifolds, structures
Creators: name: Kondratiev, G. V
Publisher: name: Université d'Ottawa / University of Ottawa
Year: 2007
Subjects: subject: Mathematics.

description: Concrete duality is a key tool in the development of modern algebraic geometry. It gives general directions, vistas of the subject, as well as concrete machinery (local and global techniques). The sort of higher order duality proposed in this thesis reflects still more of the close relationship of the associated algebraic and geometric categories. By higher order here, we mean duality with respect to n-categorical structure. It is significant that all famous dualities (by Stone, Pontryagin, Gelfand-Naimark, Grothendieck) can be obtained in a similar categorical fashion. This was proved by H.-E. Porst and W. Tholen [P-Th]. Once the structure of concrete duality was clarified, people were encouraged to discover new dualities (e.g., see [Luk]). What is the practical sense of that? On a basic level, duality gives concrete "functional representations" of objects via "functions" on their duals. More precisely, we have a duality between geometric and algebraic categories, so that each object is presented by a functional space on its dual. For example, in this sense, the solution spaces of algebraic or differential equations become visible. On a more abstract level it brings sense and often ease of calculation to categorical constructions via their duals (for example, complicated algebraic colimits often become clear via geometric limits). In this work a concept of higher order concrete duality is introduced and a criterion of existence of natural such dualities is given. One of the aims was to strictly prove that the extension of Gelfand-Naimark duality over 2-cells (homotopy classes of homotopies) is a 2-duality in a proper categorical sense. For that, a notion of infinity category is introduced in the first chapter and developed to cover such usual categorical topics as representability, the Yoneda lemma, and adjunctions for this infinite-dimensional environment. We include a discussion of the distinction between weak (up to equivalence) versus strict (up to isomorphism). For pointed objects of an infinity category, homotopy-like groups are introduced (which coincide in special cases with the usual ones). The advantage of these objects is that they are defined more categorically (i.e. internally) and so have more chance to be preserved by functors. For example, the usual homotopy groups are very rarely preserved by functors. In the second chapter a concept of category of manifolds over a Grothendieck site is introduced. It is shown that usual manifolds, fibre bundles, and foliations fit into this scheme. The construction of such categories makes use of stacks. The third and final chapter contains examples of concrete duality of first and second order. Some of them are new at least in their categorical formulation (Vinogradov duality, duality for differential equations, Gelfand-Naimark 2-duality, Pontryagin-Lukacs duality). They provide a framework in which each concrete theory has a duality which we successfully develop. The main results of this work are presented in two papers in the electronic (non refereed) website xxx.lanl.gov.