Related papers: Topos Quantum Theory Reduced by Context-Selection …
We study sheaves in the context of a duality theory for lattice structure endowed with extra operations, and in the context of forcing in a topos. Using Sheaf duality theory of Comer for cylindric algebras, we give a representation theorem…
Given a general finite group $G$, we consider several categories built on it, their Grothendieck topologies and resulting sheaf categories. For a certain class of transporter categories and their quotients, equipped with atomic topology, we…
Contextuality is a key feature of quantum mechanics that provides an important non-classical resource for quantum information and computation. Abramsky and Brandenburger used sheaf theory to give a general treatment of contextuality in…
We firstly introduce some key concepts in category theory, such as quotient category, completion of limits, $\mathrm{Mor}$ category, and so on; then give the concept of topology algebras and sheaves, and discuss how to restore the structue…
We develop a resource theory of contextuality within the framework of symmetric monoidal categories, extending recent simplicial approaches to quantum contextuality. Building on the theory of simplicial distributions, which integrates…
We study various characterizations of higher sites over a given $\infty$-category $\mathcal{C}$ which are conceptually in line with their classical ordinary categorical counterparts, and extract some new results about $\infty$-topos theory…
We develop a general theory of extensions of flat functors along geometric morphisms of toposes, and apply it to the study of the class of theories whose classifying topos is equivalent to a presheaf topos. As a result, we obtain a…
For any site of definition $\mathcal C$ of a Grothendieck topos $\mathcal E$, we define a notion of a $\mathcal C$-ary Lawvere theory $\tau: \mathscr C \to \mathscr T$ whose category of models is a stack over $\mathcal E$. Our definitions…
Algorithmicists are well-aware that fast dynamic programming algorithms are very often the correct choice when computing on compositional (or even recursive) graphs. Here we initiate the study of how to generalize this folklore intuition to…
Kochen-Specker theorem rules out the non-contextual assignment of values to physical magnitudes. Here we enrich the usual orthomodular structure of quantum mechanical propositions with modal operators. This enlargement allows to refer…
We construct a topological space to study contextuality in quantum mechanics. The resulting space is a classifying space in the sense of algebraic topology. Cohomological invariants of our space correspond to physical quantities relevant to…
What is the role of topos theory in the topos models for quantum theory as used by Isham, Butterfield, Doring, Heunen, Landsman, Spitters and others? In other words, what is the interplay between physical motivation for the models and the…
In this paper, we introduce a new definition of sheaves on semicartesian quantales, providing first examples and categorical properties. We note that our sheaves are similar to the standard definition of a sheaf on a locale, however, we…
Topos theory occupies a singular place in contemporary mathematics: born from Grothendieck's algebraic geometry, it has emerged as a unifying language for geometry, topology, algebra, and logic. This book offers a progressive introduction…
All current approaches to quantum gravity employ essentially standard quantum theory including, in particular, continuum quantities such as the real or complex numbers. However, I wish to argue that this may be fundamentally wrong in so far…
There is a well-established homotopy theory of simplicial objects in a Grothendieck topos, and folklore says that the weak equivalences are axiomatisable in the geometric fragment of $L_{\omega_1, \omega}$. We show that it is in fact a…
The Godement cosimplicial resolution is available for a wide range of categories of sheaves. In this paper we investigate under which conditions of the Grothendieck site and the category of coefficients it can be used to obtain fibrant…
In this paper we study a model of intuitionistic higher-order logic which we call \emph{the Muchnik topos}. The Muchnik topos may be defined briefly as the category of sheaves of sets over the topological space consisting of the Turing…
This paper is the second in a series whose goal is to develop a fundamentally new way of constructing theories of physics. The motivation comes from a desire to address certain deep issues that arise when contemplating quantum theories of…
Topos theory, a branch of category theory, has been proposed as mathematical basis for the formulation of physical theories. In this article, we give a brief introduction to this approach, emphasising the logical aspects. Each topos serves…