Related papers: Symmetry in the cubical Joyal model structure

We prove that each of the model structures for ($n$-trivial, saturated) comical sets on the category of marked cubical sets having only faces and degeneracies (without connections) is Quillen equivalent to the corresponding model structure…

We construct a model structure on the category of cubical sets with connections whose cofibrations are the monomorphisms and whose fibrant objects are defined by the right lifting property with respect to inner open boxes, the cubical…

We give a new construction of the Joyal model structure on the category of simplicial sets, and we provide a simple characterization of the fibrations in it. We characterize the inner anodyne maps in terms of categorical equivalences and…

We show that the category of simplicial sets is a co-reflective subcategory of the category of cubical sets with connections, with the inclusion given by a version of the straightening functor. We show that using the co-reflector, one can…

This paper is part of a series of three articles with the objective of investigating a stratified version of the homotopy hypothesis in terms of semi-model structures that interact well with classical examples of stratified spaces, such as…

It is shown that the Joyal quasi-category model structure for simplicial sets extends to a model structure on simplicial presheaves, for which the weak equivalences are local (or stalkwise) Joyal equivalences.

The category of Cartesian cubical sets is introduced and endowed with a Quillen model structure using ideas coming from recent constructions of cubical systems of univalent type theory.

We introduce a new cubical model for homotopy types. More precisely, we'll define a category Qs with the following features: Qs is a PROP containing the classical box category as a subcategory, the category Qs-Set of presheaves of sets on…

We propose a new model for the theory of $(\infty,n)$-categories (including the case $n=\infty$) in the category of marked cubical sets with connections, similar in flavor to complicial sets of Verity. The model structure characterizing our…

This paper continues the research of the author on the homology of cubical and semi-cubical sets with coefficients in systems of objects. The main result is the theorem that the homology of cubical sets with coefficients in contravariant…

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 categorical structures analogous to classical, locally…

It is common to encounter symmetric monoidal categories $\mathcal{C}$ for which every object is equipped with an algebraic structure, in a way that is compatible with the monoidal product and unit in $\mathcal{C}$. We define this formally…

A symmetric monoidal category naturally arises as the mathematical structure that organizes physical systems, processes, and composition thereof, both sequentially and in parallel. This structure admits a purely graphical calculus. This…

It is proved that the category of simplicial complete bornological spaces over $\mathbb R$ carries a combinatorial monoidal model structure satisfying the monoid axiom. For any commutative monoid in this category the category of modules is…

In these notes we describe models of globular weak $(\infty,m)$-categories ($m\in\mathbb{N}$) in the Grothendieck style, i.e for each $m\in\mathbb{N}$ we define a globular coherator $\Theta^{\infty}_{\mathbb{M}^m}$ whose set-models are…

In categorical realizability, it is common to construct categories of assemblies and categories of modest sets from applicative structures. These categories have structures corresponding to the structures of applicative structures. In the…

This article gives a solid theoretical grounding to the observation that cubical structures arise naturally when working with parametricity. We claim that cubical models are cofreely parametric. We use categories, lex categories or clans as…

We construct a compact closed category out of any symmetric monoidal category by freely adding adjoints to its objects. The morphisms of the completion are defined as string diagrams annotated by objects and morphisms from the original…

We show that the category of algebraically cofibrant objects in a combinatorial and simplicial model category A has a model structure that is left-induced from that on A. In particular it follows that any presentable model category is…

We prove that the marked triangulation functor from the category of marked cubical sets equipped with a model structure for ($n$-trivial, saturated) comical sets to the category of marked simplicial set equipped with a model structure for…