Related papers: Categories as mathematical models
In this paper we develope a categorical theory of relations and use this formulation to define the notion of quantization for relations. Categories of relations are defined in the context of symmetric monoidal categories. They are shown to…
This monograph is a study of the category of polynomial endofunctors on the category of sets and its applications to modeling interaction protocols and dynamical systems. We assume basic categorical background and build the categorical…
We extend the framework of combinatorial model categories, so that the category of small presheaves over large indexing categories and ind-categories would be embraced by the new machinery called class-combinatorial model categories. The…
This short introductory category theory textbook is for readers with relatively little mathematical background (e.g. the first half of an undergraduate mathematics degree). At its heart is the concept of a universal property, important…
We introduce the notion of a definable category--a category equivalent to a full subcategory of a locally finitely presentable category that is closed under products, directed colimits and pure subobjects. Definable subcategories are…
For a collection of subcategories satisfying a fixed set of conditions, for example thick subcategories of a triangulated category, we define a topological space called classifying space of subcategories. We show that this space classifies…
We define a new model structure on the category of small categories, which is intimately related to the notion of coverings and fundamental groups of small categories. Fibrant objects in the model structure coincide with groupoids, and the…
Higher category theory is an exceedingly active area of research, whose rapid growth has been driven by its penetration into a diverse range of scientific fields. Its influence extends through key mathematical disciplines, notably homotopy…
A taxonomy is a standardized framework to classify and organize items into categories. Hierarchical taxonomies are ubiquitous, ranging from the classification of organisms to the file system on a computer. Characterizing the typical…
Categories such as animal or furniture are acquired at an early age and play an important role in processing, organizing, and communicating world knowledge. Categories exist across cultures: they allow to efficiently represent the…
Formal concept analysis has grown from a new branch of the mathematical field of lattice theory to a widely recognized tool in Computer Science and elsewhere. In order to fully benefit from this theory, we believe that it can be enriched…
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…
Stock and flow diagrams are already an important tool in epidemiology, but category theory lets us go further and treat these diagrams as mathematical entities in their own right. In this chapter we use communicable disease models created…
Categories, n-categories, double categories, and multicategories (among others) all have similar definitions as collections of cells with composition operations. We give an explicit description of the information required to define any…
We present a soundness theorem for a dependent type theory with context constants with respect to an indexed category of (finite, abstract) simplical complexes. The point of interest for computer science is that this category can be seen to…
The concept of n-categories and related subject is considered. An n-category is described as an n-graph with a composition. A new definition of operad is presented. Some illustrative examples are given.
We review several known categorification procedures, and introduce a functorial categorification of group extensions with applications to non-abelian group cohomology. Categorification of acyclic models and of topological spaces are briefly…
We examine various categorical structures that can and cannot be constructed. We show that total computable functions can be mimicked by constructible functors. More generally, whatever can be done by a Turing machine can be constructed by…
Generalising slightly the notions of a strict computability model and of a simulation between them, which were elaborated by Longley and Normann, we define canonical computability models over categories and appropriate Set-valued functors…
An n-category is some sort of algebraic structure consisting of objects, morphisms between objects, 2-morphisms between morphisms, and so on up to n-morphisms, together with various ways of composing them. We survey various concepts of…