Related papers: Topos Quantum Theory Reduced by Context-Selection …
We develop a "Soergel theory" for Bruhat-constructible perverse sheaves on the flag variety $G/B$ of a complex reductive group $G$, with coefficients in an arbitrary field $\Bbbk$. Namely, we describe the endomorphisms of the projective…
We introduce and compare two approaches to equivariant homotopy theory in a topological or ordinary Quillen model category. For the topological model category of spaces, we generalize Piacenza's result that the categories of topological…
The ALEA Coq library formalizes measure theory based on a variant of the Giry monad on the category of sets. This enables the interpretation of a probabilistic programming language with primitives for sampling from discrete distributions.…
We introduce the forcing model of IZFA (Intuitionistic Zermelo-Fraenkel set theory with Atoms) for every Grothendieck topology and prove that the topos of sheaves on every site is equivalent to the category of 'sets in this forcing model'.
This paper deals with topos-theoretic truth-value valuations of quantum propositions. Concretely, a mathematical framework of a specific type of modal approach is extended to the topos theory, and further, structures of the obtained…
In type theory, an oracle may be specified abstractly by a predicate whose domain is the type of queries asked of the oracle, and whose proofs are the oracle answers. Such a specification induces an oracle modality that captures a…
The topos approach to the formulation of physical theories includes a new form of quantum logic. We present this topos quantum logic, including some new results, and compare it to standard quantum logic, all with an eye to conceptual…
It is proved that for any Grothendieck site $X$, there exists a coreflection (called $\mathbf{cosheafification}$) from the category of precosheaves on $X$ with values in a category $\mathbf{K}$, to the full subcategory of cosheaves,…
We investigate Grothendieck topologies (in the sense of sheaf theory) on a poset $\P$ that are generated by some subset of $\P$. We show that such Grothendieck topologies exhaust all possibilities if and only if $\P$ is Artinian. If $\P$ is…
The article is devoted to a comparison of the \v{C}ech cohomology with the coefficients in a presheaf of Abelian groups and the topos cohomology of the sheaf generated by this presheaf for a poset with the Aleksandrov topology. The article…
In this paper, we investigate a sheaf-theoretic interpretation of stratification learning from geometric and topological perspectives. Our main result is the construction of stratification learning algorithms framed in terms of a sheaf on a…
Decision making often exhibits context dependence that challenges classical probability theory. While quantum cognition has successfully modeled such phenomena, it remains unclear whether quantum probability is merely a convenient…
If a Quillen model category can be specified using a certain logical syntax (intuitively, ``is algebraic/combinatorial enough''), so that it can be defined in any category of sheaves, then the satisfaction of Quillen's axioms over any site…
Butz and Moerdijk famously showed that every (Grothendieck) topos with enough points is equivalent to the category of sheaves on some topological groupoid. We give an alternative, more algebraic construction in the special case of a topos…
The first goal of this paper is to provide an abstract framework in which to formulate and study local duality in various algebraic and topological contexts. For any stable $\infty$-category $\mathcal{C}$ together with a collection of…
This book provides an inviting tour through sheaf theory, from the perspective of applied category theory and pitched at a less specialized audience than is typical with introductions to sheaves. The book makes it as easy as possible for…
It is shown that the principle of locality and noncommutative geometry can be connnected by a sheaf theoretical method. In this framework quantum spaces are introduced and examples in mathematical physics are given. With the language of…
Techniques from higher categories and higher-dimensional rewriting are becoming increasingly important for understanding the finer, computational properties of higher algebraic theories that arise, among other fields, in quantum…
Moerdijk's site description for equivariant sheaf toposes on open topological groupoids is used to give a proof for the (known, but apparently unpublished) proposition that if H is a strictly full subgroupoid of an open topological groupoid…
We propose a semantic foundation for logics for reasoning in settings that possess a distinction between equality of variables, a coarser equivalence of variables, and a notion of conditional independence between variables. We show that…