Related papers: Basic Category Theory
In this note we characterize, within the framework of the theory of finite set, those categories of graphs that are {\em algebraic universal} in the sense that every concrete category embeds in them. The proof of the characterization is…
Category theory provides a collective description of many arrangements in mathematics, such as topological spaces, Banach spaces and game theory. Within this collective description, the perspective from any individual member of the…
This paper introduces the construction of a weakly globular double category of fractions for a category and studies its universal properties. It shows that this double category is locally small and considers a couple of concrete examples.
The basic concepts in category theory are representables, adjoints, limits, and monads. In this talk, we define the notion of a Kan extension and show that this notion encompasses these concepts.
Category theory is famous for its innovative way of thinking of concepts by their descriptions, in particular by establishing universal properties. Concepts that can be characterized in a universal way receive a certain quality seal, which…
Additive categories play a fundamental role in mathematics and related disciplines. Given an additive category equipped with a biadditive functor, one can construct its category of extensions, which encodes important structural information.…
This paper develops a categorical framework to clarify the relationship between the completeness and compactness theorems in classical first-order logic. Rather than claiming that different model constructions yield naturally isomorphic…
We present the first definition of strictly associative and unital $\infty$-category. Our proposal takes the form of a type theory whose terms describe the operations of such structures, and whose definitional equality relation enforces…
We present an intrinsic and concrete development of the subdivision of small categories, give some simple examples and derive its fundamental properties. As an application, we deduce an alternative way to compare the homotopy categories of…
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…
Quantum categories were introduced in [4] as generalizations of both bi(co)algebroids and small categories. We clarify details of that work. In particular, we show explicitly how the monadic definition of a quantum category unpacks to a set…
This is an introductory textbook to univalent mathematics and homotopy type theory, a mathematical foundation that takes advantage of the structural nature of mathematical definitions and constructions. It is common in mathematical practice…
This article serves as a preliminary introduction to the design of a new, open-source applied and computational category theory framework, named Categorica, built on top of the Wolfram Language. Categorica allows one to configure and…
Homological algebra is often understood as the translator between the world of topology and algebra. However, this branch of mathematics is worth studying by itself, given that it provides fascinating perspectives about other disciplines,…
The extension of ordinary category theory to $\infty$-categories at the start of the 21st century was a spectacular achievement pioneered by Joyal and Lurie with contributions from many others. Unfortunately, the technical arguments…
We begin with a context more general than set theory. The basic ingredients are essentially the object and functor primitives of category theory, and the logic is weak, requiring neither the Law of Excluded Middle nor quantification. Inside…
A 2-categorical generalisation of elementary topos is provided and some of the properties of the yoneda structure it generates are explored. Examples relevant to the globular approach to higher category theory are discussed. This paper also…
We introduce a categorical language in which it is possible to talk about DNA sequencing, alignment methods, CRISPR, homologous recombination, haplotypes, and genetic linkage. This language takes the form of a class of limit-sketches whose…
We introduce the category of finite strings and study its basic properties. The category is closely related to the augmented simplex category, and it models categories of linear representations. Each lattice of non-crossing partitions…
By means of several examples, we motivate that universal properties are the simplest way to solve a given mathematical problem, explaining in this way why they appear everywhere in mathematics. In particular, we present the co-universal…