Related papers: Finitary Spacetime Sheaves
A ringed finite space is a ringed space whose underlying topological space is finite. The category of ringed finite spaces contains, fully faithfully, the category of finite topological spaces and the category of affine schemes. Any ringed…
Some examples from the mathematics of shape are presented that question some of the almost hidden assumptions behind results on limiting behaviour of finitary approximations to space-time. These are presented so as to focus attention on the…
A ringed finite space is a ringed space whose underlying topological space is finite. The category of ringed finite spaces contains, fully faithfully, the category of finite topological spaces and the category of affine schemes. Any ringed…
An algebraic scheme is suggested in which discretized spacetime turns out to be a quantum observable. As an example, a toy model producing spacetimes of four points with different topologies is presented. The possibility of incorporating…
The schematic finite spaces are those finite ringed spaces where a theory of quasi-coherent modules can be developed with minimal natural conditions. We give various characterizations of these spaces and their natural morphisms. We show…
We study the homotopy theory of locally ordered spaces, that is manifolds with boundary whose charts are partially ordered in a compatible way. Their category is not particularly well-behaved with respect to colimits. However, this category…
The machinery is suggested to describe the varying spacetime topology on the level of its substitutes by finite topological spaces.
A ringed finite space is a ringed space whose underlying topological space is finite. The category of ringed finite spaces contains, fully faithfully, the category of finite topological spaces and the category of affine schemes. Any ringed…
We propose a mathematical model of quantum spacetime as an infinite-dimensional manifold locally homeomorphic to an appropriate Schwartz space. This extends and unifies both the standard function space construction of quantum mechanics and…
Finite metric spaces arise in many different contexts. Enormous bodies of data, scientific, commercial and others can often be viewed as large metric spaces. It turns out that the metric of graphs reveals a lot of interesting information.…
We survey indications from different branches of Physics that the fine scale structure of spacetime is not adequately described by a manifold. Based on the hints we accumulate, we propose a new structure, which we call a quantum topos. In…
A theorem about local in time existence of spacelike foliations with prescribed mean curvature in cosmological spacetimes will be proved. The time function of the foliation is geometrically defined and fixes the diffeomorphism invariance…
Free actions of finite groups on spheres give rise to topological spherical space forms. The existence and classification problems for space forms have a long history in the geometry and topology of manifolds. In this article, we present a…
Spacetime is modelled by binary relations - by the classes of the automorphisms $\GL(\C^2)$ of a complex 2-dimensional vector space with respect to the definite unitary subgroup $\U(2)$. In extension of Feynman propagators for particle…
What is the shape of space in a spacetime? One way of addressing this issue is to consider edgeless spacelike submanifolds of the spacetime. An alternative is to foliate the spacetime by timelike curves and consider the quotient obtained by…
There is an interplay between models, specified by variables and equations, and their connections to one another. This dichotomy should be reflected in the abstract as well. Without referring to the models directly -- only that a model…
We characterize the smallest finite spaces with the same homotopy groups of the spheres. Similarly, we describe the minimal finite models of any finite graph. We also develop new combinatorial techniques based on finite spaces to study…
A spacetime outlook on Computational Fluid Dynamics is advocated: models in fluid mechanics often have the spacetime correlation property, which should be inherited and preserved in the corresponding numerical algorithms. Starting from the…
The problem of classifying boundary points of space-time, for example singularities, regular points and points at infinity, is an unexpectedly subtle one. Due to the fact that whether or not two boundary points are identified or even…
The research effort reported in this paper is directed, in a broad sense, towards understanding the small-scale structure of spacetime. The fundamental question that guides our discussion is ``what is the physical content of spacetime…