Related papers: Syntactic presentations for glued toposes and for …
We develop a general theory for class-sized symmetric systems as a natural extension of symmetric systems with respect to class forcing. In particular, adapting the usual notions of pretameness and tameness for class forcing, we present…
We introduce stratified toposes, which are toposes that are stratified by a suitable hierarchy of universes. The term `stratified topos' recalls the notion of stratified pseudotopos of Moerdijk and Palmgren (2002). However, the details of…
The notion of an existentially closed model is generalised to a property of geometric morphisms between toposes. We show that important properties of existentially closed models extend to existentially closed geometric morphisms, such as…
We propose the design of novel categorical generative AI architectures (GAIAs) using topos theory, a type of category that is ``set-like": a topos has all (co)limits, is Cartesian closed, and has a subobject classifier. Previous theoretical…
We introduce the theory of generalised ultracategories, these are relational extensions to ultracategories as defined by Lurie. An essential example of generalised ultracategories are topological spaces, and these play a fundamental role in…
We introduce judgemental theories and their calculi as a general framework to present and study deductive systems. As an exemplification of their expressivity, we approach dependent type theory and natural deduction as special kinds of…
We introduce the basic elements of the theory of parametrized $\infty$-categories and functors between them. These notions are defined as suitable fibrations of $\infty$-categories and functors between them. We give as many examples as we…
Complex systems of systems (SoS) are characterized by multiple interconnected subsystems. Typically, each subsystem is designed and analyzed using methodologies and formalisms that are specific to the particular subsystem model of…
It has been common wisdom among mathematicians that Extended Topological Field Theory in dimensions higher than two is naturally formulated in terms of n-categories with n> 1. Recently the physical meaning of these higher categorical…
The goal of this paper is to summarise the first steps in developing 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…
We consider complete lattices equipped with preorderings indexed by the ordinals less than a given (limit) ordinal subject to certain axioms. These structures, called stratified complete lattices, and weakly monotone functions over them,…
We give an explicit description of factorization algebras over the affine line, constructing them from the gluing data determined by its corresponding OPE algebra. We then generalize this construction to factorization monoids, obtaining a…
State of the art language models return a natural language text continuation from any piece of input text. This ability to generate coherent text extensions implies significant sophistication, including a knowledge of grammar and semantics.…
Toposes can be pictured as mathematical universes. Besides the standard topos, in which most of mathematics unfolds, there is a colorful host of alternate toposes in which mathematics plays out slightly differently. For instance, there are…
This expository note describes two convenient techniques in the context of homotopy type theory for proving and formalizing that a given map is an equivalence. The first technique decomposes the map as a series of basic equivalences, while…
A natural next step in the evolution of constraint-based grammar formalisms from rewriting formalisms is to abstract fully away from the details of the grammar mechanism---to express syntactic theories purely in terms of the properties of…
This is the author's PhD thesis. It is a contribution to categorical logic, in particular to the theory of realizability toposes. While the tools of categorical logic have proven very successful in analyzing and organizing proof theoretic…
We provide a new approach to categorical graph and hypergraph theory by using categorical syntax and semantics. For each monoid $M$ and action on a set $X$, there is an associated presheaf topos of $(X,M)$-graphs where each object can be…
We use Kan injectivity to axiomatise concepts in the 2-category of topoi. We showcase the expressivity of this language through many examples, and we establish some aspects of the formal theory of Kan extension in this 2-category (pointwise…
The first part of this dissertation defines "dependently typed algebraic theories", which are a strict subclass of the generalised algebraic theories (GATs) of Cartmell. We characterise dependently typed algebraic theories as finitary…