Related papers: Axiomatic Method and Category Theory

Homotopy Type Theory is a new field of mathematics based on the surprising and elegant correspondence between Martin-Lofs constructive type theory and abstract homotopy theory. We have a powerful interplay between these disciplines - we can…

We survey several notions of Mackey functors and biset functors found in the literature and prove some old and new theorems comparing them. While little here will surprise the experts, we draw a conceptual and unified picture by making…

A noetherian form is an abstract self-dual framework suitable for establishing homomorphism theorems (such as the isomorphism theorems and homological diagram lemmas) for group-like structures. In this paper we identify and carry out an…

This paper presents a type theory in which it is possible to directly manipulate $n$-dimensional cubes (points, lines, squares, cubes, etc.) based on an interpretation of dependent type theory in a cubical set model. This enables new ways…

Axiomatic type theory is a dependent type theory without computation rules. The term equality judgements that usually characterise these rules are replaced by computation axioms, i.e., additional term judgements that are typed by identity…

The basic concepts of category theory are developed and examples of them are presented to illustrate them using measurement theory and probability theory tools. Motivated by Perrone's workarXiv:1912.10642 where notes on category theory are…

We give an introduction to the topics of our forthcoming work, in which we introduce and study new mathematical objects which we call "higher theories" of algebras, where inspiration for the term comes from William Lawvere's notion of…

We apply the algebraic theory of infinite classical lattices from Part I to write an axiomatic theory of measurements, based on Mackey's axioms for quantum mechanics. The axioms give a complete theory of measurements in the sense of Haag…

Many introductions to homotopy type theory and the univalence axiom gloss over the semantics of this new formal system in traditional set-based foundations. This expository article, written as lecture notes to accompany a 3-part mini course…

We introduce an abstract topos-theoretic framework for building Galois-type theories in a variety of different mathematical contexts; such theories are obtained from representations of certain atomic two-valued toposes as toposes of…

There are two ways to turn a categorical model for pure quantum theory into one for mixed quantum theory, both resulting in a category of completely positive maps. One has quantum systems as objects, whereas the other also allows classical…

In this paper we sketch how some fundamental concepts of modern topology (as well as logic and category theory) can be understood philosophically in the light of Hegel's Science Logic as well how modern topological concepts can provide…

In this paper, we analyze axiomatic issues of unconventional computations from a methodological and philosophical point of view. We explain how the new models of algorithms changed the algorithmic universe, making it open and allowing…

This book introduces a temporal type theory, the first of its kind as far as we know. It is based on a standard core, and as such it can be formalized in a proof assistant such as Coq or Lean by adding a number of axioms. Well-known…

Homotopy limits and colimits are homotopical replacements for the usual limits and colimits of category theory, which can be approached either using classical explicit constructions or the modern abstract machinery of derived functors. Our…

We survey Lawvere theories at the level of infinity categories, as an alternative framework for higher algebra (rather than infinity operads). From a pedagogical perspective, they make many key definitions and constructions less technical.…

This paper presents a novel connection between homotopical algebra and mathematical logic. It is shown that a form of intensional type theory is valid in any Quillen model category, generalizing the Hofmann-Streicher groupoid model 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…

In these lecture notes, an introduction to topological concepts and methods in studies of gauge field theories is presented. The three paradigms of topological objects, the Nielsen-Olesen vortex of the abelian Higgs model, the 't…

This article gives a conceptual introduction to the topos approach to the formulation of physical theories.