Related papers: Diagrammatic category theory
We establish and advocate for a novel branch of category theory, centered around strong dinatural transformations (herein known as "paranatural transformations"). Paranatural transformations generalize natural transformations to…
In previous work, we defined the intersection graph of a chord diagram associated with a string link (as in the theory of finite type invariants). In this paper, we look at the case when this graph is a tree, and we show that in many cases…
This book is an invitation to discover advanced topics in category theory through concrete, real-world examples. It aims to give a tour: a gentle, quick introduction to guide later exploration. The tour takes place over seven sketches, each…
Invertibility is an important concept in category theory. In higher category theory, it becomes less obvious what the correct notion of invertibility is, as extra coherence conditions can become necessary for invertible structures to have…
There are many books designed to introduce category theory to either a mathematical audience or a computer science audience. In this book, our audience is the broader scientific community. We attempt to show that category theory can be…
We present our position on the elusive quest for a general-purpose framework for specifying and studying deep learning architectures. Our opinion is that the key attempts made so far lack a coherent bridge between specifying constraints…
We extend the basic concepts of Street's formal theory of monads from the setting of 2-categories to that of double categories. In particular, we introduce the double category Mnd(C) of monads in a double category C and define what it means…
As of today there exist consistent, gauge-invariant string field theories describing all string theories: bosonic open and closed strings, open superstrings, heterotic strings and type II strings. The construction of these theories require…
We work with a generalization of knot theory, in which one diagram is reachable from another via a finite sequence of moves if a fixed condition, regarding the existence of certain morphisms in an associated category, is satisfied for every…
This article tackles categorical coherence within a two-dimensional generalization of Lawvere's functorial semantics. 2-theories, a syntactical way of describing categories with structure, are presented. From the perspective here afforded,…
In the late 90s M. V. Nori constructed a category of motives in charakteristic 0. Using a directed graph with a representation into noetherian R-Modules, he defined a universal R-linear abelian category, called diagram category.The…
Category theory is a branch of mathematics that provides a formal framework for understanding the relationship between mathematical structures. To this end, a category not only incorporates the data of the desired objects, but also…
Diagrammatically speaking, grammatical calculi such as pregroups provide wires between words in order to elucidate their interactions, and this enables one to verify grammatical correctness of phrases and sentences. In this paper we also…
Two dimensional gauge theories with charged matter fields are useful toy models for studying gauge theory dynamics, and in particular for studying the duality of large $N$ gauge theories to perturbative string theories. A useful starting…
Categories are coreflectively embedded in multicategories via the "discrete cocone" construction, the right adjoint being given by the monoid construction. Furthermore, the adjunction lifts to the "cartesian level": preadditive categories…
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…
The contributions of this paper are twofold. Within the framework of Grothendieck's fibrational category theory, we present a web of fundamental 2-adjunctions surrounding the formation of the category of all small diagrams in a given…
String diagrams are a powerful and intuitive graphical syntax for terms of symmetric monoidal categories (SMCs). They find many applications in computer science and are becoming increasingly relevant in other fields such as physics and…
We introduce an abstract concept of quantum field theory on categories fibered in groupoids over the category of spacetimes. This provides us with a general and flexible framework to study quantum field theories defined on spacetimes with…
Tree-width is an invaluable tool for computational problems on graphs. But often one would like to compute on other kinds of objects (e.g. decorated graphs or even algebraic structures) where there is no known tree-width analogue. Here we…