This paper is an expanded version of notes for a set of lectures given at the Isaac Newton Institute for Mathematical Sciences during a NATO ASI Workshop entitled “Homotopy Theory of Geometric Categories” on September 23 and 24, 2002. This workshop was part of a program entitled New Contexts in Stable Homotopy Theory that was held at the Institute during the fall of 2002. The intent for the lec...

Sheaf semantics is developed within a constructive and predicative framework, Martin-Lof’s type theory. We prove strong completeness of many sorted, first order intuitionistic logic with respect to this semantics, by using sites of provably functional relations. Mathematics Subject Classification: 03B20, 03C90, 18F10, 18F25.

Let X be the big etale site of schemes over k = C. If S is a scheme of finite type over k, let X/S denote the big etale site of schemes over S. The goal of this paper is to introduce a full subcategory of the category of sheaves of groups on X/S, which we will call the category of presentable group sheaves (§2), with the following properties. 1. The category of presentable group sheaves contain...

In this article we consider sheaf quotients of affine superschemes by finite supergroups that act on them freely. More precisely, if a finite supergroup G acts on an affine superscheme X freely, then the quotient K-sheaf ̃ X/G is again an affine superscheme Y , where K[Y ] ≃ K[X ]. Besides, K[X ] is a finitely presented projective K[X ]-module.

The papers introduces a new complex of differential forms which provides a fine resolution for the sheaf of regular functions in two quaternionic variables and the sheaf of monogenic functions in two vector variables. The paper announces some applications of this complex to the construction of sheaves of quaternionic and Clifford hyperfunctions as equivalence classes of such differential forms.

In this chapter we explain constructive methods for computing the cohomol-ogy of a sheaf on a projective variety. We also give a construction for the Beilinson monad, a tool for studying the sheaf from partial knowledge of its cohomology. Finally, we give some examples illustrating the use of the Beilinson monad.

Algebraic structures such as the lattices of attractors, repellers, and Morse representations provide a computable description global dynamics. In this paper, sheaf-theoretic approach to their continuation is developed. The algebraic are cast into categorical framework study systematically simultaneously. Sheaves built from abstract formulation, which track data systems vary. Sheaf cohomology c...

The method of intersection spaces associates rational Poincaré complexes to singular stratified spaces. For a conifold transition, the resulting cohomology theory yields the correct count of all present massless 3-branes in type IIB string theory, while intersection cohomology yields the correct count of massless 2-branes in type IIA theory. For complex projective hypersurfaces with an isolated...

We determine the (arithmetic) local monodromy at 0 and at ∞ of the Kloosterman sheaf using local Fourier transformations and Laumon’s stationary phase principle. We then calculate -factors for symmetric products of the Kloosterman sheaf. Using Laumon’s product formula, we get functional equations of L-functions for these symmetric products and prove a conjecture of Evans on signs of constants o...