Cartesian double theories: A double-categorical framework for categorical doctrines
M Lambert, E Patterson - Advances in Mathematics, 2024 - Elsevier
The categorified theories known as “doctrines” specify a category equipped with extra
structure, analogous to how ordinary theories specify a set with extra structure. We introduce …
structure, analogous to how ordinary theories specify a set with extra structure. We introduce …
Constructing symmetric monoidal bicategories functorially
LW Hansen, M Shulman - arXiv preprint arXiv:1910.09240, 2019 - arxiv.org
We present a method of constructing monoidal, braided monoidal, and symmetric monoidal
bicategories from corresponding types of monoidal double categories that satisfy a lifting …
bicategories from corresponding types of monoidal double categories that satisfy a lifting …
Products in double categories, revisited
E Patterson - arXiv preprint arXiv:2401.08990, 2024 - arxiv.org
Products in double categories, as found in cartesian double categories, are an elegant
concept with numerous applications, yet also have a few puzzling aspects. In this paper, we …
concept with numerous applications, yet also have a few puzzling aspects. In this paper, we …
Structured and decorated cospans from the viewpoint of double category theory
E Patterson - arXiv preprint arXiv:2304.00447, 2023 - arxiv.org
Structured and decorated cospans are broadly applicable frameworks for building
bicategories or double categories of open systems. We streamline and generalize these …
bicategories or double categories of open systems. We streamline and generalize these …
Double categories of relations
M Lambert - arXiv preprint arXiv:2107.07621, 2021 - arxiv.org
A double category of relations is essentially a cartesian equipment with strong, discrete and
functorial tabulators and for which certain local products satisfy a Frobenius Law. A double …
functorial tabulators and for which certain local products satisfy a Frobenius Law. A double …
Formal category theory in augmented virtual double categories
SR Koudenburg - arXiv preprint arXiv:2205.04890, 2022 - arxiv.org
Abridged abstract: In this article we develop formal category theory within augmented virtual
double categories. Notably we formalise the notions of Kan extension and Yoneda …
double categories. Notably we formalise the notions of Kan extension and Yoneda …
Cartesian Gray-Monoidal Double Categories
E Morehouse - arXiv preprint arXiv:2302.07810, 2023 - arxiv.org
In this paper we present cartesian structure for symmetric Gray-monoidal double categories.
To do this we first introduce locally cubical Gray categories, which are three-dimensional …
To do this we first introduce locally cubical Gray categories, which are three-dimensional …
Transposing cartesian and other structure in double categories
E Patterson - arXiv preprint arXiv:2404.08835, 2024 - arxiv.org
The cartesian structure possessed by morphisms like relations, spans, and profunctors is
elegantly expressed by universal properties in double categories. Though cartesian double …
elegantly expressed by universal properties in double categories. Though cartesian double …
Representing Knowledge and Querying Data using Double-Functorial Semantics
M Lambert, E Patterson - arXiv preprint arXiv:2403.19884, 2024 - arxiv.org
Category theory offers a mathematical foundation for knowledge representation and
database systems. Popular existing approaches model a database instance as a functor into …
database systems. Popular existing approaches model a database instance as a functor into …
Double categories of relations relative to factorisation systems
K Hoshino, H Nasu - arXiv preprint arXiv:2310.19428, 2023 - arxiv.org
We relativise double categories of relations to stable orthogonal factorisation systems.
Furthermore, we present the characterisation of the relative double categories of relations in …
Furthermore, we present the characterisation of the relative double categories of relations in …