Related papers: Simplicial sets inside cubical sets
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 a 2005 paper, Casacuberta, Scevenels and Smith construct a homotopy idempotent functor $E$ on the category of simplicial sets with the property that whether it can be expressed as localization with respect to a map $f$ is independent of…
This is an expository introduction to simplicial sets and simplicial homotopy theory with particular focus on relating the combinatorial aspects of the theory to their geometric/topological origins. It is intended to be accessible to…
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…
Both simplicial sets and simplicial spaces are used pervasively in homotopy theory as presentations of spaces, where in both cases we extract the "underlying space" by taking geometric realization. We have a good handle on the category of…
Given a small simplicial category $\C$ whose underlying ordinary category is equipped with a Grothendieck topology $\tau$, we construct a model structure on the category of simplicially enriched presheaves on $\C$ where the weak…
We begin by recalling the essentially global character of universes in various models of homotopy type theory, which prevents a straightforward axiomatization of their properties using the internal language of the presheaf toposes from…
We produce a highly structured way of associating a simplicial category to a model category which improves on work of Dwyer and Kan and answers a question of Hovey. We show that model categories satisfying a certain axiom are Quillen…
In this article, we construct a cofibrantly generated model structure on the category of spaces stratified over a fixed poset, and show that it is Quillen-equivalent to a category of diagrams of simplicial sets. Then, considering all those…
Semi-simplicial and semi-cubical sets are commonly defined as presheaves over respectively, the semi-simplex or semi-cube category. Homotopy Type Theory then popularized an alternative definition, where the set of n-simplices or n-cubes are…
The mathematical representation of uncertainty has led to a proliferation of preference structures, such as interval-valued fuzzy sets, intuitionistic fuzzy sets, and various granular models. While these extensions are often studied…
Cubical type theory is an extension of Martin-L\"of type theory recently proposed by Cohen, Coquand, M\"ortberg and the author which allows for direct manipulation of $n$-dimensional cubes and where Voevodsky's Univalence Axiom is provable.…
An expansion set is a set $\mathcal{B}$ such that each $b \in \mathcal{B}$ is equipped with a set of expansions $\mathcal{E}(b)$. The theory of expansion sets offers a systematic approach to the construction of classifying spaces for…
Homotopy type theory is a modern foundation for mathematics that introduces the univalence axiom and is particularly suitable for the study of homotopical mathematics and its formalization via proof assistants. In order to better comprehend…
We develop a denotational semantics for general reference types in an impredicative version of guarded homotopy type theory, an adaptation of synthetic guarded domain theory to Voevodsky's univalent foundations. We observe for the first…
Compact sets in constructive mathematics capture our intuition of what computable subsets of the plane (or any other complete metric space) ought to be. A good representation of compact sets provides an efficient means of creating and…
Much of the homotopical and homological structure of the categories of chain complexes and topological spaces can be deduced from the existence and properties of the 'simple' functors Tot : {double chain complexes} -> {chain complexes} and…
Simplicial sets generalise many categories of graphs. In this paper, we give a complete characterisation of the Lawvere-Tierney topologies on (semi-)simplicial sets, on bicolored graphs, and on fuzzy sets. We apply our results to establish…
In this note we interpret Voevodsky's Univalence Axiom in the language of (abstract) model categories. We then show that any posetal locally Cartesian closed model category $Qt$ in which the mapping $Hom^{(w)}(Z\times B,C):Qt\longrightarrow…
We prove that every categorical model of dependent type theory with dependent sums and products, intensional identity types and univalent universes presents via its $\infty$-localisation an elementary $\infty$-topos, that is, a finitely…