Related papers: Introducing categories to the practicing physicist
Every smooth manifold contains particles which propagate. These form objects and morphisms of a category equipped with a functor to the category of Abelian groups, turning this into a 0+1 topological field theory. We investigate the…
We describe how graduate students categorize introductory mechanics problems based on the similarity of their solutions. Graduate students were asked at the end of a teaching assistant training class to categorize problems from their own…
It has long been known that every weak monoidal category A is equivalent via monoidal functors and monoidal natural transformations to a strict monoidal category st(A). We generalise the definition of weak monoidal category to give a…
We introduce some classes of genuine higher categories in homotopy type theory, defined as well-behaved subcategories of the category of types. We give several examples, and some techniques for showing other things are not examples. While…
Since categories are graphs with additional "structure", one should start from fuzzy graphs in order to define a theory of fuzzy categories. Thus is makes sense to introduce categories whose morphisms are associated with a plausibility…
Category theory unifies mathematical concepts, aiding comparisons across structures by incorporating objects and morphisms, which capture their interactions. It has influenced areas of computer science such as automata theory, functional…
The popular view according to which Category theory provides a support for Mathematical Structuralism is erroneous. Category-theoretic foundations of mathematics require a different philosophy of mathematics. While structural mathematics…
We present a simple categorical framework for the treatment of probabilistic theories, with the aim of reconciling the fields of Categorical Quantum Mechanics (CQM) and Operational Probabilistic Theories (OPTs). In recent years, both CQM…
Category theory has been successfully applied in various domains of science, shedding light on universal principles unifying diverse phenomena and thereby enabling knowledge transfer between them. Applications to machine learning have been…
A theoretical scheme, based on a probabilistic generalization of the Hamilton's principle, is elaborated to obtain an unified description of more general dynamical behaviors determined both from a lagrangian function and by mechanisms not…
There is debate as to whether quantum field theory is, at bottom, a quantum theory of fields or particles. One can take a field approach to the theory, using wave functionals over field configurations, or a particle approach, using wave…
Restriction categories were established to handle maps that are partially defined with respect to composition. Tensor topology realises that monoidal categories have an intrinsic notion of space, and deals with objects and maps that are…
This paper explores the interplay between category theory, topology, and the algebraic theory of finite groups. Our analysis unfolds in three stages. First, we establish the foundational universe of our objects: the complete and cocomplete…
Motivated by potential applications to theoretical computer science, in particular those areas where the Curry-Howard correspondence plays an important role, as well as by the ongoing search in pure mathematics for feasible approaches to…
In work of Fokkinga and Meertens a calculational approach to category theory is developed. The scheme has many merits, but sacrifices useful type information in the move to an equational style of reasoning. By contrast, traditional proofs…
The generality and pervasiness of category theory in modern mathematics makes it a frequent and useful target of formalization. It is however quite challenging to formalize, for a variety of reasons. Agda currently (i.e. in 2020) does not…
Quantum Theory is one of the pillars of modern science developed over the last hundred years. In this review paper we introduce, step by step, the quantum theory understood as a mathematical model describing quantum experiments. We start…
This work adapts the equivalent definitions of division algebras over a field into multiple types of division algebras in a monoidal category. Examples and consequences of these definitions are then established in various monoidal settings.
A general simplicity problem in category theory is proposed. A particular example, the simplest choice of generators of an algebra is specified and illustrated by an example.
Presenting systems of differential equations in the form of diagrams has become common in certain parts of physics, especially electromagnetism and computational physics. In this work, we aim to put such use of diagrams on a firm…