Related papers: From Finite Sets to Feynman Diagrams
We develop a unified categorical framework for gauging both continuous and finite symmetries in arbitrary spacetime dimensions. Our construction applies to geometric categories i.e. categories internal to stacks. This generalizes the…
We show that morphisms from n A_infinity-algebras to a single one are maps over an operad module with n+1 commuting actions of the operad A_infinity, whose algebras are conventional A_infinity-algebras. Similar statement holds for homotopy…
We present a unified categorical framework that connects the syntactic Henkin construction for the first-order Completeness Theorem with Lawvere's Fixed-Point Theorem. Concretely, we define two canonical functors from the category of…
Data integration and migration processes in polystores and multi-model database management systems highly benefit from data and schema transformations. Rigorous modeling of transformations is a complex problem. The data and schema…
We introduce a reducibility on classes of structures, essentially a uniform enumeration reducibility. This reducibility is inspired by the Friedman-Stanley paper on using Borel reductions to compare classes of countable structures. This…
We propose a generalization of quantization as a categorical way. For a fixed Poisson algebra quantization categories are defined as subcategories of R-module category with the structure of classical limits. We construct the generalized…
This paper proposes a formal cognitive framework for problem solving based on category theory. We introduce cognitive categories, which are categories with exactly one morphism between any two objects. Objects in these categories are…
The integration of knowledge extracted from different models described by domain experts or from models generated by machine learning algorithms is strongly conditioned by the lack of an appropriated framework to specify and integrate…
The recent trend in mathematics is towards a framework of abstract mathematical objects, rather than the more concrete approach of explicitly defining elements which objects were thought to consist of. A natural question to raise is whether…
We classify the propositional modal validities arising from the category of sets under its natural classes of morphisms. The resulting validities depend on the morphism class, the size of the world, and the permitted substitution instances.…
A concept of "evolving categories" is suggested to build a simple, scalable, mathematically consistent framework for representing in uniform way both data and algorithms. A state machine for executing algorithms becomes clear, rich and…
The category of I-spaces is the diagram category of spaces indexed by finite sets and injections. This is a symmetric monoidal category whose commutative monoids model all E-infinity spaces. Working in the category of I-spaces enables us to…
A finite-dimensional unital and associative algebra over $\mathbb{R}$, or what we shall call simply "an algebra" in this paper for short, generalities the construction by which we derive the complex numbers by "adjoining an element $i$" to…
We develop a novel formal theory of finite structures, based on a view of finite structures as a fundamental artifact of computing and programming, forming a common platform for computing both within particular finite structures, and in the…
In this article we discuss the operations of partitions (sequence of disjoint finite subsets) which are quotient, insertion, composition and Lie bracket. Moreover, we discuss applications of those operations for Feymman diagrams and…
The category of matchings between finite sets extends to the category of cobordisms of signed sets. A chain of cobordisms that starts and ends with unsigned sets A and B yields a matching from A to B. This is a convenient way to package the…
Condensed mathematics, developed by Clausen and Scholze over the last few years, proposes a generalization of topology with better categorical properties. It replaces the concept of a topological space by that of a condensed set, which can…
This paper introduces a category theory-based framework to redefine physical computing in light of advancements in quantum computing and non-standard computing systems. By integrating classical definitions within this broader perspective,…
In [KW14], the new concept of Feynman categories was introduced to simplify the discussion of operad--like objects. In this present paper, we demonstrate the usefulness of this approach, by introducing the concept of decorated Feynman…
We provide a foundation for working with homological and homotopical methods in categorical algebra. This involves two mutually complementary components, namely (a) the strategic selection of suitable axiomatic frameworks, some well known…