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A new approach to the construction of general persistent polyhierarchical classifications is proposed. It is based on implicit description of category polyhierarchy by a generating polyhierarchy of classification criteria. Similarly to…
This document is centered around a main idea: simplicial categories, by which we mean simplicial objects in the category of categories, can be treated as a two-fold categorical structure and their double category theory is homotopically…
Cartesian difference categories are a recent generalisation of Cartesian differential categories which introduce a notion of "infinitesimal" arrows satisfying an analogue of the Kock-Lawvere axiom, with the axioms of a Cartesian…
We have generalised the notion of categorical theory in model theory to the context of coherent theories. We prove a duality result between the full sub-2-category of pretopoi which are categorical, and the 2-category of profinite monoids.…
Written to be contributed as the "mathematical modeling" chapter of a book, edited by Elaine Landry, to be titled "Categories for the Working Philosopher". In this chapter, category theory is presented as a mathematical modeling framework…
Cartesian differential categories come equipped with a differential combinator that formalizes the derivative from multi-variable differential calculus, and also provide the categorical semantics of the differential $\lambda$-calculus. An…
Recently, there has been growing interest in bicategorical models of programming languages, which are "proof-relevant" in the sense that they keep distinct account of execution traces leading to the same observable outcomes, while assigning…
Many structures of interest in two-dimensional category theory have aspects that are inherently strict. This strictness is not a limitation, but rather plays a fundamental role in the theory of such structures. For instance, a monoidal…
It is well-known that pseudo functors from bicategories of spans are equivalent to Beck-Chevalley bifibrations, and therefore capture the relationships underlying the adjunctions suitable as semantics for existential quantification. This…
As an example of the categorical apparatus of pseudo algebras over 2-theories, we show that pseudo algebras over the 2-theory of categories can be viewed as pseudo double categories with folding or as appropriate 2-functors into…
In this work, we present a brief but insightful overview of the gauge theories, which are defined on $ n $-dimensional lattices by using finite gauge groups, in order to show how they can be interpreted as a Hamiltonian system with…
A general procedure is presented which associates to a finite crossed module a premodular category, generalizing the representation categories of a finite group and of its double, and the extent to which the resulting category fails to be…
Whereas formal category theory is classically considered within a $2$-category, in this paper a double-dimensional approach is taken. More precisely we develop such theory within the setting of augmented virtual double categories, a notion…
We define a monoidal semantics for algebraic theories. The basis for the definition is provided by the analysis of the structural rules in the term calculus of algebraic languages. Models are described both explicitly, in a form that…
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…
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…
Higher-order quantum theory is an extension of quantum theory where one introduces transformations whose input and output are transformations, thus generalizing the notion of channels and quantum operations. The generalization then goes…
Matrix theory, foundational in diverse fields such as mathematics, physics, and computational sciences, typically categorizes matrices based strictly on their invertibility-determined by a sharply defined singular or nonsingular…
fc-multicategories are a very general kind of two-dimensional structure, encompassing bicategories, monoidal categories, double categories and ordinary multicategories. We define them and explain how they provide a natural setting for two…
A differential category is an additive symmetric monoidal category, that is, a symmetric monoidal category enriched over commutative monoids, with an algebra modality, axiomatizing smooth functions, and a deriving transformation on this…