Related papers: Introduction to Categories and Categorical Logic
This survey article is intended as an introduction to the recent categorical classification theorems of the three authors, restricting to the special case of the category of modules for a finite group.
In this paper we provide a unifying description of different types of semantics of modal logic found in the literature via the framework of topological categories. In the style of categorical logic, we establish an exact correspondence…
Category theory provides a means through which many far-ranging fields of mathematics can be related by their similar structure. In a paper by Robinson [2], this interconnectivity afforded by categorical perspectives allowed for the…
We propose a categorial grammar based on classical multiplicative linear logic. This can be seen as an extension of abstract categorial grammars (ACG) and is at least as expressive. However, constituents of {\it linear logic grammars (LLG)}…
The notion of a simplicial set originated in algebraic topology, and has also been utilized extensively in category theory, but until relatively recently was not used outside of those fields. However, with the increasing prominence of…
Modern categorical logic as well as the Kripke and topological models of intuitionistic logic suggest that the interpretation of ordinary "propositional" logic should in general be the logic of subsets of a given universe set. Partitions on…
For any length category, we establish a set of rules (necessary and sufficient) that ensure a partial order on the isomorphism classes of simple objects such that the category is equivalent to the category of finite dimensional…
Category theory gives a mathematical characterization of naturality but not of canonicity. The purpose of this paper is to develop the logical theory of canonical maps based on the broader demonstration that the dual notions of elements &…
Although the notion of a concept as a collection of objects sharing certain properties, and the notion of a conceptual hierarchy are fundamental to both Formal Concept Analysis and Description Logics, the ways concepts are described and…
The basic notion of how topoi can be utilized in physics is presented here. Topos and category theory serve as valuable tools which extend our ordinary set-theoretical conceptions, can further the study of quantum logic and give rise to new…
The goal of this note is to spell out the (apparently well-known and intuitively clear) notion of abelian category over an algebraic stack. In the future we will discuss the (much less evident) notion, when instead of an abelian category…
Categorical data are common in educational and social science research; however, methods for its analysis are generally not covered in introductory statistics courses. This chapter overviews fundamental concepts and methods in categorical…
We firstly introduce some key concepts in category theory, such as quotient category, completion of limits, $\mathrm{Mor}$ category, and so on; then give the concept of topology algebras and sheaves, and discuss how to restore the structue…
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…
We introduce the relational ontology log, or relational olog, a knowledge representation system based on the category of sets and relations. It is inspired by Spivak and Kent's olog, a recent categorical framework for knowledge…
We will investigate proof-theoretic and linguistic aspects of first-order linear logic. We will show that adding partial order constraints in such a way that each sequent defines a unique linear order on the antecedent formulas of a sequent…
These are general notes on tensor calculus which can be used as a reference for an introductory course on tensor algebra and calculus. A basic knowledge of calculus and linear algebra with some commonly used mathematical terminology is…
This paper aims to provide a careful and self-contained introduction to the theory of topological degree in Euclidean spaces. It is intended for people mostly interested in analysis and, in general, a heavy background in algebraic or…
A type theory is presented that combines (intuitionistic) linear types with type dependency, thus properly generalising both intuitionistic dependent type theory and full linear logic. A syntax and complete categorical semantics are…
A certain amount of category theory is developed in an arbitrary finitely complete category with a factorization system on it, playing the role of the comprehensive factorization system on Cat. Those aspects related to the concepts of…