Related papers: Introducing categories to the practicing physicist
We discuss the role of propositions, truth, context and observers in scientific theories. We introduce the concept of generalized proposition and use it to define an algorithm for the classification of any scientific theory. The algorithm…
We revisit the standard axioms of domain theory with emphasis on their relation to the concept of partiality, explain how this idea arises naturally in probability theory and quantum mechanics, and then search for a mathematical setting…
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…
Categories of relations over a regular category form a family of models of quantum theory. Using regular logic, many properties of relations over sets lift to these models, including the correspondence between Frobenius structures and…
The development of mathematics has been characterized by the increasing interconnectivity of seemingly separate disciplines. Such interplay has been facilitated by a massive development in formalism; category theory has provided a common…
Harnessing the potential computational advantage of quantum computers for machine learning tasks relies on the uploading of classical data onto quantum computers through what are commonly referred to as quantum encodings. The choice of such…
According to the basic idea of category theory, any Einstein algebra, essentially an algebraic formulation of general relativity, can be considered from the point of view of any object of the category of smooth algebras; such an object is…
This monograph is a study of the category of polynomial endofunctors on the category of sets and its applications to modeling interaction protocols and dynamical systems. We assume basic categorical background and build the categorical…
Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism or other important equivalence in terms of relatively simple invariants. Where…
Modular categories are important algebraic structures in a variety of subjects in mathematics and physics. We provide an explicit, motivated and elementary definition of a modular category over a field of characteristic 0 as an equivalence…
Quantum theory provides an extremely accurate description of fundamental processes in physics. It thus seems likely that the theory is applicable beyond the, mostly microscopic, domain in which it has been tested experimentally. Here we…
There are many Lie groups used in physics, including the Lorentz group of special relativity, the spin groups (relativistic and non-relativistic) and the gauge groups of quantum electrodynamics and the weak and strong nuclear forces.…
Recall that the definition of the $K$-theory of an object C (e.g., a ring or a space) has the following pattern. One first associates to the object C a category A_C that has a suitable structure (exact, Waldhausen, symmetric monoidal, ...).…
We introduce the theory of enrichment over an internal monoidal category as a common generalization of both the standard theories of enriched and internal categories. The aim of the paper is to justify and contextualize the new notion by…
There is a long history of representing a quantum state using a quasi-probability distribution: a distribution allowing negative values. In this paper we extend such representations to deal with quantum channels. The result is a convex,…
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…
General Relativity describes gravity in geometrical terms. This suggests that quantizing such theory is the same as quantizing geometry. The subject can therefore be called quantum geometry and one may think that mathematicians are…
As part of a longitudinal study into identity development in upper-level physics students, we used a phenomenographic research method to examine students' perceptions of what it means to be a physicist. The results revealed four different…
Can we use mathematics, and in particular the abstract branch of category theory, to describe some basics of dance, and to highlight structural similarities between music and dance? We first summarize recent studies between mathematics and…
In physics, Feynman diagrams are used to reason about quantum processes. In the 1980s, it became clear that underlying these diagrams is a powerful analogy between quantum physics and topology: namely, a linear operator behaves very much…