Homotopical algebraic geometry I: topos theory

This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts. In this first part we investigate a notion of higher topos.For this, we use S-categories (i.e. simplicially enriched categories) as models for certain kind of ∞-categories, and we develop the notions of S-topologies, S-sites and stacks over them. We prove in particular, that for an S-category T endowed with an S-topology, there exists a model category of stacks over T, generalizing the model category structure on simplicial presheaves over a Grothendieck site of Joyal and Jardine. We also prove some analogs of the relations between topologies and localizing subcategories of the categories of presheaves, by proving that there exists a one-to-one correspondence between S-topologies on an S-category T, and certain left exact Bousfield localizations of the model category of pre-stacks on T. Based on the above results, we study the notion of model topos introduced by Rezk, and we relate it to our model categories of stacks over S-sites.In the second part of the paper, we present a parallel theory where S-categories, S-topologies and S-sites are replaced by model categories, model topologies and model sites. We prove that a canonical way to pass from the theory of stacks over model sites to the theory of stacks over S-sites is provided by the simplicial localization construction of Dwyer and Kan. As an example of application, we propose a definition of étale K-theory of ring spectra, extending the étale K-theory of commutative rings.     

Principal functions,index theory,geometric measure theory and function algebras

The aim of this paper is twofold: first to provide evidence of a nice relationship between index theory and operator algebras within the framework of geometric measure theory by exhibiting basic examples involving one dimensional singular integral operators; second to expose certain connections that exist involving the principal function associated to an operator having trace class self-commutator and the theory of function algebras.     

Cyclic sieving,rotation, and geometric representation theory

We study rotation of invariant vectors in tensor products of minuscule representations. We define a combinatorial notion of rotation of minuscule Littelmann paths. Using affine Grassmannians, we show that this rotation action is realized geometrically as rotation of components of the Satake fiber. As a consequence, we have a basis for invariant spaces, which is permuted by rotation (up to global sign). Finally, we diagonalize the rotation operator by showing that its eigenspaces are given by intersection homology of quiver varieties. As a consequence, we generalize Rhoades’ work on the cyclic sieving phenomenon.     

Morse theory and Lyusternik-Shnirelman theory in geometric control theory

Questions, related to the application of the ideas of global analysis to optimal control problems, are considered. A theory of Lyusternik-Shnirelman type is constructed for Hilbert manifolds with singularities, the so-called transversally convex subsets. Conditions for the nondegeneracy of the critical points (the extremal controls) are established in the optimal control problem, related to a smooth control system of constant rank, and a formula for their Morse index is given.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 39, pp. 41–117, 1991.     

The fundamental progroupoid of a general topos

It is well known that the category of covering projections (that is, locally constant objects) of a locally connected topos is equivalent to the classifying topos of a strict progroupoid (or, equivalently, a localic prodiscrete groupoid), the fundamental progroupoid, and that this progroupoid represents first degree cohomology. In this paper we generalize these results to an arbitrary topos. The fundamental progroupoid is now a localic progroupoid, and cannot be replaced by a localic groupoid. The classifying topos is no longer a Galois topos. Not all locally constant objects can be considered as covering projections. The key contribution of this paper is a novel definition of covering projection for a general topos, which coincides with the usual definition when the topos is locally connected. The results in this paper were presented in a talk at the Category Theory Conference, Vancouver, July 2004.     

The etale homotopy theory of a geometric fibration

A geometric fibration, f:XY, is a smooth map of schemes which locally on Y admits a smooth, relative compactification. The etale homotopy type of the geometric fibre when completed away from the residue characteristics of Y, , is shown to be weakly homotopy equivalent to the completion of the Hurewicz fibre of the etale homotopy type of f,F(f _et ^r )^{^}. This implies a homotopy sequence for f. A key topological fact is verified in the appendix: for any pointed Hurewicz fibre triple FEB, the action of _l(F) on the homotopy type of various covering spaces of F extends to an action of _l(E).Partially supported by the N.S.F., I.H.E.S., and University of Warwick.     

Categorical abstract algebraic logic: The largest theory system included in a theory family

In this note, it is shown that, given a π ‐institution ? = 〈 Sign , SEN, C 〉, with N a category of natural transformations on SEN, every theory family T of ? includes a unique largest theory system of ?. satisfies the important property that its N ‐Leibniz congruence system always includes that of T . As a consequence, it is shown, on the one hand, that the relation Ω^N ( ) = Ω^N (T ) characterizes N ‐protoalgebraicity inside the class of N ‐prealgebraic π ‐institutions and, on the other, that all N ‐Leibniz theory families associated with theory families of a protoalgebraic π ‐institution ? are in fact N ‐Leibniz theory systems. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Basic concepts for a theory of evaluation: The aggregative operator

Starting with an explication of the “aggregative”-concept and deducing a general structure which satisfies a number of minimal requirements (properties of clustering) the main features of a new mathematical theory — called “theory of evaluation” — are developed. The theory sheds new light on such well-known concepts as membership, conjunction and disjunction and seems to be a very promising tool to handle representation problems as they grow from the fields of theory of fuzzy set, and its many applications, of human decision making and of multicriteria analysis.     

Eigenvalues,geometric expanders,sorting in rounds,and ramsey theory

Expanding graphs are relevant to theoretical computer science in several ways. Here we show that the points versus hyperplanes incidence graphs of finite geometries form highly (nonlinear) expanding graphs with essentially the smallest possible number of edges. The expansion properties of the graphs are proved using the eigenvalues of their adjacency matrices. These graphs enable us to improve previous results on a parallel sorting problem that arises in structural modeling, by describing an explicit algorithm to sortn elements ink time units using parallel processors, where, e.g., α₂=7/4, α₃=8/5, α₄=26/17 and α₅=22/15. Our approach also yields several applications to Ramsey Theory and other extremal problems in combinatorics.     

Stirling numbers and a geometric,structure from voting theory

In the spatial theory of voting, m candidates are each represented by a point in a p-dimensional Euclidean “attribute” space. The hyperplanes bisecting the line segments joining pairs of these points divide the space into regions, and each region corresponds to a definite ranking of the distances to the candidates. This paper discusses the combinatorics of such a structure and shows that the Stirling numbers have a geometrical significance.     

Notes on logic and set theory

Book Reviews

Notes on logic and set theoryP. T. Johnstone: (Cambridge Mathematical Text-books), Cambridge University Press, 1987, paperback, 110pp., £6.95/$12.95