Related papers: Sets in homotopy type theory

Using dependent type theory to formalise the syntax of dependent type theory is a very active topic of study and goes under the name of "type theory eating itself" or "type theory in type theory." Most approaches are at least loosely based…

A quasi-schemoid is a small category with a particular partition of the set of morphisms. We define a homotopy relation on the category of quasi-schemoids and study its fundamental properties. As a homotopy invariant, the homotopy set of…

In this paper, we define indexed type theories which are related to indexed ($\infty$-)categories in the same way as (homotopy) type theories are related to ($\infty$-)categories. We define several standard constructions for such theories…

We present the first definition of strictly associative and unital $\infty$-category. Our proposal takes the form of a type theory whose terms describe the operations of such structures, and whose definitional equality relation enforces…

We study the connection between the Goodwillie tower of the identity and the lower central series of the loop group on connected spaces. We define the simplicial theory of homotopy n-nilpotent groups. This notion interpolates between…

Let $M$ be a smooth manifold. We use Chern-Weil theory to study the characteristic classes of principal $G$-bundles built from continuous families of $\pi_{1}(M)$-representations, where $G$ is a compact Lie group. We then relate these…

The lambda calculus is a universal programming language. It can represent the computable functions, and such offers a formal counterpart to the point of view of functions as rules. Terms represent functions and this allows for the…

We define the notion of {\em classifying space} of a topological stack and show that every topological stack \X has a classifying space X which is a topological space well-defined up to weak homotopy equivalence. Under a certain…

As Goresky and MacPherson intersection homology is not the homology of a space, there is no preferred candidate for intersection homotopy groups. Here, they are defined as the homotopy groups of a simplicial set which P. Gajer associates to…

It is known that the existence of localization with respect to an arbitrary (possibly proper) class of maps in the category of simplicial sets is implied by a large-cardinal axiom called Vopenka's principle.In this article we extend the…

Univalence was first defined in the setting of homotopy type theory by Voevodsky, who also (along with Kapulkin and Lumsdaine) adapted it to a model categorical setting, which was subsequently generalized to locally Cartesian closed…

Let V be a cofibrantly generated closed symmetric monoidal model category and M a model V-category. We say that a weighted colimit W*D of a diagram D weighted by W is a homotopy weighted colimit if the diagram D is pointwise cofibrant and…

In this paper we consider simplicial families, that is, simplicial objects indexed by a simplicial set. We develop a method to construct family hypercover refinements of a cover family based on the notion of \emph{n-spans} that we introduce…

An appropriate framework is put forward for the construction of $\lambda$-models with $\infty$-groupoid structure, which we call \textit{homotopic $\lambda$-models}, through the use of an $\infty$-category with cartesian closure and enough…

Nearness theory comes into play in homotopy theory because the notion of closeness between points is essential in determining whether two spaces are homotopy equivalent. While nearness theory and homotopy theory have different focuses and…

Let $A$ be a commutative noetherian ring, containing a field $k$, with $1/2\in k$, $\dim A=d$, and let $P$ be a projective $A$-module or $rank(P)=n$. In continuation of \cite{MM}, we study Homotopy obstructions for $P$ to split off a free…

Higher-dimensional rewriting systems are tools to analyse the structure of formally reducing terms to normal forms, as well as comparing the different reduction paths that lead to those normal forms. This higher structure can be captured by…

In this paper we lay the foundations of an $\infty$-categorical theory of Stokes data.

Covering spaces are a fundamental tool in algebraic topology because of the close relationship they bear with the fundamental groups of spaces. Indeed, they are in correspondence with the subgroups of the fundamental group: this is known as…

We introduce a notion of a filtered model structure and use this notion to produce various model structures on pro-categories. This framework generalizes several known examples. We give several examples, including a homotopy theory for…