Related papers: Categorification of characteristic structures
We demonstrate that topological defects in a rational conformal field theory can be described by a classifying algebra for defects - a finite-dimensional semisimple unital commutative associative algebra whose irreducible representations…
Category Theory provides us with a clear notion of what is an internal structure. This will allow us to focus our attention on a certain type of relationship between context and structure.
Theory of representations of universal algebra is a natural development of the theory of universal algebra. Morphism of the representation is the map that conserve the structure of the representation. Exploring of morphisms of the…
The attempt is to give a formal concpet of system, and with this provide a definition of category, that will also satisfy the definition of a system. An axiomatic base is given, for constructing the group of integers. In the process, we…
It has become obvious in the recent development that the structural Ramsey property is a categorical property: it depends not only on the choice of objects, but also on the choice of morphisms involved. In this paper we explicitely put the…
We introduce a theory for encoding and manipulating algebraic data on categories via $\textit{concentration structures}$, which are equivalence relations on morphisms that satisfy certain axioms. For any category with a concentration…
Understanding how people represent categories is a core problem in cognitive science. Decades of research have yielded a variety of formal theories of categories, but validating them with naturalistic stimuli is difficult. The challenge is…
Category theory has foundational importance because it provides conceptual lenses to characterize what is important in mathematics. Originally the main lenses were universal mapping properties and natural transformations. In recent decades,…
We study and relate categories of modules, comodules and contramodules over a representation of a small category taking values in (co)algebras, in a manner similar to modules over a ringed space. As a result, we obtain a categorical…
We propose Universal Causality, an overarching framework based on category theory that defines the universal property that underlies causal inference independent of the underlying representational formalism used. More formally, universal…
The literature on concurrency theory offers a wealth of examples of characteristic-formula constructions for various behavioural relations over finite labelled transition systems and Kripke structures that are defined in terms of fixed…
This paper lays the groundwork for the theory of categorical diagonalization. Given a diagonalizable operator, tools in linear algebra (such as Lagrange interpolation) allow one to construct a collection of idempotents which project to each…
Standard methods of using categorical variables as predictors either endow them with an ordinal structure or assume they have no structure at all. However, categorical variables often possess structure that is more complicated than a linear…
Ornaments aim at taming the multiplication of special-purpose datatype in dependently-typed theory. In its original form, the definition of ornaments is tied to a particular universe of datatypes. Being a type theoretic object,…
Conditional independence has been widely used in AI, causal inference, machine learning, and statistics. We introduce categoroids, an algebraic structure for characterizing universal properties of conditional independence. Categoroids are…
We introduce a notion of complexity of diagrams (and in particular of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several…
We provide a construction for categorical representation learning and introduce the foundations of "$\textit{categorifier}$". The central theme in representation learning is the idea of $\textbf{everything to vector}$. Every object in a…
The study of representations is of fundamental importance to any form of communication, and our ability to exploit them effectively is paramount. This article presents a novel theory -- Representational Systems Theory -- that is designed to…
We study the categorical framework for the computation of persistent homology, without reliance on a particular computational algorithm. The computation of persistent homology is commonly summarized as a matrix theorem, which we call the…
In this note we provide a characterization, in terms of additional algebraic structure, of those intervals (certain cocategory objects) in a symmetric monoidal closed category E that are representable in the sense of inducing on E the…