Grothendieck Topology Research Articles

Cover semantics for quantified lax logic

Lax modalities occur in intuitionistic logics concerned with hardware verification, the computational lambda calculus and access control in secure systems. They also encapsulate the logic of Lawvere–Tierney–Grothendieck topologies on topoi. This article provides a complete semantics for quantified lax logic by combining the Beth–Kripke–Joyal cover semantics for first-order intuitionistic logic with the classical relational semantics for a ‘diamond’ modality. The main technique used is the lifting of a multiplicative closure operator (nucleus) from a Heyting algebra to its MacNeille completion, and the representation of an arbitrary locale as the lattice of ‘propositions’ of a suitable cover system. In addition, the theory is worked out for certain constructive versions of the classical logics K and S4. An alternative completeness proof is given for (non-modal) first-order intuitionistic logic itself with respect to the cover semantics, using a simple and explicit Henkin-style construction of a characteristic model whose points are principal theories rather than prime-saturated ones. The article provides further evidence that there is more to intuitionistic modal logic than the generalization of properties of boxes and diamonds from Boolean modal logic.

The Weil-étale topology for number rings

There should be a Grothendieck topology for an arithmetic scheme X such that the Euler characteristic of the cohomology groups of the constant sheaf Z with compact support at infinity gives, up to sign, the leading term of the zeta-function of X at s = 0. We construct a topology (the Weil-etale topology) for the ring of integers in a number field whose cohomology groups Hl (Z) determine such an Euler characterstic if we restrict to / < 3.

Grothendieck topologies on Chu spaces

We consider the Grothendieck topologies on low semi-lattices, defined by one family, and the corresponding sheaf cohomology. This is a basis to define and study the left and right cohomologies and the left and right dimensions of the Chu spaces. The construction of Chu spaces allows to characterize a large class of quantities, for example, the dimension of a Noether space or the Krull dimension of a ring, the Lebesgue-type dimensions, as well as to compare them with the cohomology dimensions of the corresponding Chu spaces. We prove existence of spectral sequences of the morphisms of the Chu spaces.

Model structures on the category of small double categories

In this paper we obtain several model structures on {\bf DblCat}, the category of small double categories. Our model structures have three sources. We first transfer across a categorification-nerve adjunction. Secondly, we view double categories as internal categories in {\bf Cat} and take as our weak equivalences various internal equivalences defined via Grothendieck topologies. Thirdly, {\bf DblCat} inherits a model structure as a category of algebras over a 2-monad. Some of these model structures coincide and the different points of view give us further results about cofibrant replacements and cofibrant objects. As part of this program we give explicit descriptions and discuss properties of free double categories, quotient double categories, colimits of double categories, several nerves, and horizontal categorification.

Quantum Observables Algebras and Abstract Differential Geometry: The Topos-Theoretic Dynamics of Diagrams of Commutative Algebraic Localizations

We construct a sheaf-theoretic representation of quantum observables algebras over a base category equipped with a Grothendieck topology, consisting of epimorphic families of commutative observables algebras, playing the role of local arithmetics in measurement situations. This construction makes possible the adaptation of the methodology of Abstract Differential Geometry (ADG), a la Mallios, in a topos-theoretic environment, and hence, the extension of the “mechanism of differentials” in the quantum regime. The process of gluing information, within diagrams of commutative algebraic localizations, generates dynamics, involving the transition from the classical to the quantum regime, formulated cohomologically in terms of a functorial quantum connection, and subsequently, detected via the associated curvature of that connection.

Sheaf tools for computation

The paper presents first how we can compute sheaf tools for accomplishing our goal, i.e. an authentication method by sheaves. We are using the most general notion of a sheaf, as a functor on a site, which is a category equipped with a Grothendieck topology. Moreover, we have introduced a notion in sheaf theory, family for matching, in order to simplify the sheaf definition in a computation class of sites called well structured sites. This has given us the possibility to construct an optimal validation algorithm of sheaves. Also, we present an algorithm to generate sheaves. By handling the concepts of Goguen, the behaviour of a system is computed in the internal language of a (pre)sheaf topos. Non-interference by Goguen is also computed in the internal language of a presheaf topos by using predicates.

A Relation Between Closure Operators on a Small Category and Its Category of Presheaves

In this paper we show that each factorization structure \({\mathcal {M}}\) on a small category \({\mathcal {X}}\), satisfying certain conditions, yields a presheaf \({{\boldsymbol{M}}}\) on \({\mathcal {X}}\) and a morphism of presheaves \({\mathbf{m}}:\Omega \xrightarrow{.}{\mathbf{M}}\). We then give connections, and set up one to one correspondences, between subclasses of the following classes: (a) closure operators on \({\mathcal {X}}\) (b) subobjects of \({\boldsymbol{M}}\) (c) morphisms from \({\boldsymbol{M}}\) to \({\boldsymbol{\Omega}}\) (d) weak Lawvere–Tierney topologies (e) weak Grothendieck topologies (f) closure operators on \(Sets^{{\mathcal {X}}^{op}}\).

The Canonical Topology on a Meet-Semilattice

Considering the lattice of properties of a physical system, it has been argued elsewhere that-to build a calculus of propositions having a well-behaved notion of disjunction (and implication)-one should consider a very particular frame completion of this lattice. We show that the pertinent frame completion is obtained as sheafification of the presheaves on the given meet-semilattice with respect to its canonical Grothendieck topology, an explicit description of which is easily given. Our conclusion is that there is an intrinsic categorical quality to the notion of "disjunction" in the context of property lattices of physical systems.

A note on global pth powers of rigid analytic functions

Assume that K is a perfect field of characteristic p > 0 that is complete with respect to an ultrametric valuation, and let X be a rigid analytic variety over K. Suppose that X is smooth and connected with respect to its Grothendieck topology. Let f be a (global) function on X the differential of which vanishes locally at some point of X; then f is the pth power of a (global) function.

The Weil-étale topology on schemes over finite fields

We introduce an essentially new Grothendieck topology, the Weil-étale topology, on schemes over finite fields. The cohomology groups associated with this topology should behave better than the standard étale cohomology groups. In particular there is a very natural definition of an Euler characteristic and a plausible conjecture relating the Euler characteristic of Z to the value of the zeta-function at s = 0. This conjecture is proved in certain cases.

Quantum Event Structures from the Perspective of Grothendieck Topoi

We develop a categorical scheme of interpretation of quantum event structures from the viewpoint of Grothendieck topoi. The construction is based on the existence of an adjunctive correspondence between Boolean presheaves of event algebras and Quantum event algebras, which we construct explicitly. We show that the established adjunction can be transformed to a categorical equivalence if the base category of Boolean event algebras, defining variation, is endowed with a suitable Grothendieck topology of covering systems. The scheme leads to a sheaf theoretical representation of Quantum structure in terms of variation taking place over epimorphic families of Boolean reference frames.

Morita theory for Hopf algebroids and presheaves of groupoids

Comodules over Hopf algebroids are of central importance in algebraic topology. It is well known that a Hopf algebroid is the same thing as a presheaf of groupoids on Aff , the opposite category of commutative rings. We show in this paper that a comodule is the same thing as a quasi-coherent sheaf over this presheaf of groupoids. We prove the general theorem that internal equivalences of presheaves of groupoids with respect to a Grothendieck topology T on Aff give rise to equivalences of categories of sheaves in that topology. We then show using faithfully flat descent that an internal equivalence in the flat topology gives rise to an equivalence of categories of quasi-coherent sheaves. The corresponding statement for Hopf algebroids is that weakly equivalent Hopf algebroids have equivalent categories of comodules. We apply this to formal group laws, where we get considerable generalizations of the Miller-Ravenel and Hovey-Sadofsky change of rings theorems in algebraic topology.

Flattening and Subanalytic Sets in Rigid Analytic Geometry

Let K be an algebraically closed field endowed with a complete non-archimedean norm with valuation ring R. Let f: Y → X be a map of K-affinoid varieties. In this paper we study the analytic structure of the image f(Y) ⊂ X; such an image is a typical example of a subanalytic set. We show that the subanalytic sets are precisely the D-semianalytic sets, where D is the truncated division function first introduced by Denef and van den Dries. This result is most conveniently stated as a Quantifier Elimination result for the valuation ring R in an analytic expansion of the language of valued rings. To prove this we establish a Flattening Theorem for affinoid varieties in the style of Hironaka, which allows a reduction to the study of subanalytic sets arising from flat maps, that is, we show that a map of affinoid varieties can be rendered flat by using only finitely many local blowing ups. The case of a flat map is then dealt with by a small extension of a result of Raynaud and Gruson showing that the image of a flat map of affinoid varieties is open in the Grothendieck topology. Using Embedded Resolution of Singularities, we derive in the zero characteristic case, a Uniformization Theorem for subanalytic sets: a subanalytic set can be rendered semianalytic using only finitely many local blowing ups with smooth centres. As a corollary we obtain the fact that any subanalytic set in the plane R2 is semianalytic. 2000 Mathematical Subject Classification: 32P05, 32B20, 13C11, 12J25, 03C10.

Directed Graphs, Magic Squares, and Grothendieck Topologies

Directed Graphs, Magic Squares, and Grothendieck Topologies

Cup products in sheaf cohomology, pure injectives, and a substitute for projective resolutions

Two new definitions are given for cup products in the cohomology of sheaves on an arbitrary Grothendieck topology. These products are shown to have the usual properties. The existence of pure injectives is shown under very general conditions and it is shown how to do many constructions involving projective resolutions under more general conditions when there may not be enough projectives.

Coarse Geometry via Grothendieck Topologies

AbstractWe show that the coarse cohomology, defined by J. ROE for complete metric spaces, coincides with the sheaf cohomology of the constant sheaf IR for a certain Grothendieck topology. Therefore many properties of coarse cohomology are easy consequences of general principles.

The Primitive Topology of a Scheme

We define a Grothendieck topology on the category of schemes whose associated sheaf theory coincides in many cases with that of the Zariski topology. We also give some indications of possible advantages this new topology has over the Zariski topology.

Identification of certain structures as split opfibrations over Δop

The purpose of this paper is to show how certain interesting structures, models of coherent first-order theories on the one hand and Grothendieck topologies on the other, may be viewed as split opfibrations. In the course of doing so, two general classes of split opfibrations will be defined of which models and topologies comprise two special examples.

On $p$-Adic Zeta-Functions Associated to the Positive Topology of Algebraic Number Fields

In [S] we introduced a new Grothendieck topology (called positive topology) over the rings of integers of algebraic number fields. If K is a finite extension of Q then Spec(0^) furnished with the positive topology shows a behavior very similar to the etale site over a complete curve over a finite field.

Presheaves over lattices: application to twisted limits of finite groups.

The notion of “active sum of groups” appears as a special case of the notion of twisted (direct) limit of groups over partially ordered sets and lattices. This provides us with a sheaf-like theory; in fact up to introducing coverings in a lattice we obtain an extension of the theory of (pre-)sheaves over Grothendieck topologies. It is possible to introduce the point-lattice (or point-topology) and describe the transfer of structure properties from the base lattice to the point-lattice. Starting from general presheaf-action (of groups) one may then introduce a global twisted limit. Using particular functors it is then possible to define “structure sheaves” in our setting. Inspired by the theory of modules over rings, we then consider the special case of G-sets. This clears the way for application to the Burnside ring of finite groups.