Related papers: Formulating Categorical Concepts using Classes
Lexical Semantics is concerned with how words encode mental representations of the world, i.e., concepts . We call this type of concepts, classification concepts . In this paper, we focus on Visual Semantics , namely on how humans build…
The notion of class is ubiquitous in computer science and is central in many formalisms for the representation of structured knowledge used both in knowledge representation and in databases. In this paper we study the basic issues…
We study links between first-order formulas and arbitrary properties for families of theories, classes of structures and their isomorphism types. Possibilities for ranks and degrees for formulas and theories with respect to given properties…
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 study of complex systems through the lens of category theory consistently proves to be a powerful approach. We propose that cognition deserves the same category-theoretic treatment. We show that by considering a highly-compact cognitive…
The central focus is on clarifying the distinction between sets and proper classes. To this end we identify several categories of concepts (surveyable, definite, indefinite), and we attribute the classical set theoretic paradoxes to a…
Process theories combine a graphical language for compositional reasoning with an underlying categorical semantics. They have been successfully applied to fields such as quantum computation, natural language processing, linear dynamical…
Categorification is the process of finding category-theoretic analogs of set-theoretic concepts by replacing sets with categories, functions with functors, and equations between functions by natural isomorphisms between functors, which in…
We construct an iterative method for factorising small strict n-categories into a unique (up to isomorphism) collection of small 1- categories. Following this we develop the theory to include a large class of $\infty$-categories. We use…
In designing an intelligent system that must be able to explain its reasoning to a human user, or to provide generalizations that the human user finds reasonable, it may be useful to take into consideration psychological data on what types…
The purpose of this paper is to show that the dual notions of elements & distinctions are the basic analytical concepts needed to unpack and analyze morphisms, duality, and universal constructions in the Sets, the category of sets and…
Category theory can be used to state formulas in First-Order Logic without using set membership. Several notable results in logic such as proof of the continuum hypothesis can be elegantly rewritten in category theory. We propose in this…
The growing complexity of modern practical problems puts high demands on the mathematical modelling. Given that various models can be used for modelling one physical phenomenon, the role of model comparison and model choice becomes…
Class algebra provides a natural framework for sharing of ISA hierarchies between users that may be unaware of each other's definitions. This permits data from relational databases, object-oriented databases, and tagged XML documents to be…
Questions of set-theoretic size play an essential role in category theory, especially the distinction between sets and proper classes (or small sets and large sets). There are many different ways to formalize this, and which choice is made…
The unprecedented pace of machine learning research has lead to incredible advances, but also poses hard challenges. At present, the field lacks strong theoretical underpinnings, and many important achievements stem from ad hoc design…
Hierarchies allow feature sharing between objects at multiple levels of representation, can code exponential variability in a very compact way and enable fast inference. This makes them potentially suitable for learning and recognizing a…
We describe a number of geometric contexts where categorification appears naturally: coherent sheaves, constructible sheaves and sheaves of modules over quantizations. In each case, we discuss how "index formulas" allow us to easily perform…
We define a monoidal semantics for algebraic theories. The basis for the definition is provided by the analysis of the structural rules in the term calculus of algebraic languages. Models are described both explicitly, in a form that…
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…