Related papers: Who needs category theory?
In topological quantum computing, information is encoded in "knotted" quantum states of topological phases of matter, thus being locked into topology to prevent decay. Topological precision has been confirmed in quantum Hall liquids by…
We discuss the motivation for pursuing research on the foundations of quantum theory.
This purpose of this book is twofold: to provide a general introduction to higher category theory (using the formalism of "quasicategories" or "weak Kan complexes"), and to apply this theory to the study of higher versions of Grothendieck…
Category theory provides an alternative to Hilbert's Formal Axiomatic method and goes beyond Mathematical Structuralism
The ongoing progress in quantum theory emphasizes the crucial role of the very basic principles of quantum theory. However, this is not properly followed in teaching quantum mechanics on the graduate and undergraduate levels of physics…
We propose a categorical foundation for the connection between pure and mixed states in quantum information and quantum computation. The foundation is based on distributive monoidal categories. First, we prove that the category of all…
Based on Gandy's principles for models of computation we give category-theoretic axioms describing locally deterministic updates to finite objects. Rather than fixing a particular category of states, we describe what properties such a…
Classical mathematics (involving such notions as infinitely small/large and continuity) is usually treated as fundamental while finite mathematics is treated as inferior which is used only in special applications. We first argue that the…
A certain amount of category theory is developed in an arbitrary finitely complete category with a factorization system on it, playing the role of the comprehensive factorization system on Cat. Those aspects related to the concepts of…
This paper presents a study of how the theory of categories leads to the creation of non classical logical systems. In particular, the case of the elementary topos of graphs, where there are three other truth values different from false and…
This paper is the fourth in a series whose goal is to develop a fundamentally new way of building theories of physics. The motivation comes from a desire to address certain deep issues that arise in the quantum theory of gravity. Our basic…
While its applications have made quantum theory arguably the most successful theory in physics, its interpretation continues to be the subject of lively debate within the community of physicists and philosophers concerned with conceptual…
The paper offers an argument against an intuitive reading of the Stone-von Neumann theorem as a categoricity result, thereby pointing out that, against what is usually taken to be the case, this theorem does not entail any model-theoretical…
The aim of these notes is to provide a succinct, accessible introduction to some of the basic ideas of category theory and categorical logic. The notes are based on a lecture course given at Oxford over the past few years. They contain…
"Ever since the advent of modern quantum mechanics in the late 1920's, the idea has been prevalent that the classical laws of probability cease, in some sense, to be valid in the new theory. [...] The primary object of this presentation is…
Category theory plays a special character in mathematics - it unifies distinct branches under the same formalism. Despite this integrative power in math, it also seems to provide the proper foundations to the experimental physicist. In this…
The growing complexity of modern practical problems puts high demands on the mathematical modelling. Given that various models can be used for modelling one physical phenomenon, the role of model comparison and model choice becomes…
In this article we present a pedagogical introduction of the main ideas and recent advances in the area of topological quantum computation. We give an overview of the concept of anyons and their exotic statistics, present various models…
This survey aims to provide a guide to the literature on topological 4-manifolds. Foundational theorems on 4-manifolds are stated, especially in the topological category. Precise references are given, with indications of the strategies…
We introduce a hierarchical classification of theories that describe systems with fundamentally limited information content. This property is introduced in an operational way and gives rise to the existence of mutually complementary…