相关论文: Topological and simplicial models of identity type…
Most categorical models for dependent types have traditionally been heavily set based: contexts form a category, and for each we have a set of types in said context -- and for each type a set of terms of said type. This is the case for…
The study of equality types is central to homotopy type theory. Characterizing these types is often tricky, and various strategies, such as the encode-decode method, have been developed. We prove a theorem about equality types of…
Diagrammatic sets admit a notion of internal equivalence in the sense of coinductive weak invertibility, with similar properties to its analogue in strict $\omega$-categories. We construct a model structure whose fibrant objects are…
A monoidal model category is a model category with a compatible closed monoidal structure. Such things abound in nature; simplicial sets and chain complexes of abelian groups are examples. Given a monoidal model category, one can consider…
We introduce a topology on the space of all isomorphism types represented in a given class of countable models, and use this topology as an aid in classifying the isomorphism types. This mixes ideas from effective descriptive set theory and…
Gambino and Garner proved that the syntactic category of a dependent type theory with identity types can be endowed with a weak factorization system structure, called identity type weak factorization system. In this paper we consider an…
In this paper we establish a natural definition of Lusternik-Schnirelmann category for simplicial complexes via the well known notion of contiguity. This category has the property of being homotopy invariant under strong equivalences, and…
Diffeological spaces are generalizations of smooth manifolds. In this paper, we study the homotopy theory of diffeological spaces. We begin by proving basic properties of the smooth homotopy groups that we will need later. Then we introduce…
In previous work, we showed that there are appropriate model category structures on the category of simplicial categories and on the category of Segal precategories, and that they are Quillen equivalent to one another and to Rezk's complete…
We give a self-contained introduction to accessible categories and how they shed light on both model- and set-theoretic questions. We survey for example recent developments on the study of presentability ranks, a notion of cardinality…
We prove the existence of a model structure on the category of stratified simplicial sets whose fibrant objects are precisely $n$-complicial sets, which are a proposed model for $(\infty,n)$-categories, based on previous work of Verity and…
Riehl and Shulman introduced simplicial type theory (STT), a variant of homotopy type theory which aimed to study not just homotopy theory, but its fusion with category theory: $(\infty,1)$-category theory. While notoriously technical,…
Many formal languages of contemporary mathematical music theory -- particularly those employing category theory -- are powerful but cumbersome: ideas that are conceptually simple frequently require expression through elaborate categorical…
In this article the author endows the functor category [B(Z2),Gpd] with the structure of a type-theoretic fibration category with a univalent universe using the so-called injective model structure. It gives us a new model of Martin-L\"of…
Univalent homotopy type theory (HoTT) may be seen as a language for the category of $\infty$-groupoids. It is being developed as a new foundation for mathematics and as an internal language for (elementary) higher toposes. We develop the…
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…
The aim of this paper is to show that the most elementary homotopy theory of $\mathbf{G}$-spaces is equivalent to a homotopy theory of simplicial sets over $\mathbf{BG}$, where $\mathbf{G}$ is a fixed group. Both homotopy theories are…
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…
Quillen's notion of small object and the Gabriel-Ulmer notion of finitely presentable or generated object are fundamental in homotopy theory and categorical algebra. Do these notions always lead to rather uninteresting classes of objects in…
In this paper we propose an approach to homotopical algebra where the basic ingredient is a category with two classes of distinguished morphisms: strong and weak equivalences. These data determine the cofibrant objects by an extension…