相关论文: The univalence axiom in posetal model categories

We construct a univalent universe in the sense of Voevodsky in some suitable model categories for homotopy types (obtained from Grothendieck's theory of test categories). In practice, this means for instance that, appart from the homotopy…

We develop category theory within Univalent Foundations, which is a foundational system for mathematics based on a homotopical interpretation of dependent type theory. In this system, we propose a definition of "category" for which equality…

We use a category-theoretic formulation of Aczel's Fullness Axiom from Constructive Set Theory to derive the local cartesian closure of an exact completion. As an application, we prove that such a formulation is valid in the homotopy…

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…

In this short note we give a glimpse of homotopy type theory, a new field of mathematics at the intersection of algebraic topology and mathematical logic, and we explain Vladimir Voevodsky's univalent interpretation of it. This…

After developing the basic theory of locally cartesian localizations of presentable locally cartesian closed infinity-categories, we establish the representability of equivalences and show that univalent families, in the sense of Voevodsky,…

We give a model of set theory based on multisets in homotopy type theory. The equality of the model is the identity type. The underlying type of iterative sets can be formulated in Martin-L\"of type theory, without Higher Inductive Types…

We present Voevodsky's construction of a model of univalent type theory in the category of simplicial sets. To this end, we first give a general technique for constructing categorical models of dependent type theory, using universes to…

It is a well-known theorem of homotopy type theory, originally due to Voevodsky, that function extensionality holds inside any univalent universe. We consider a weaker variant of the univalence axiom, asserting that the wild category formed…

When working in Homotopy Type Theory and Univalent Foundations, the traditional role of the category of sets, Set, is replaced by the category hSet of homotopy sets (h-sets); types with h-propositional identity types. Many of the properties…

Recent discoveries have been made connecting abstract homotopy theory and the field of type theory from logic and theoretical computer science. This has given rise to a new field, which has been christened "homotopy type theory". In this…

We construct a new model category presenting the homotopy theory of presheaves on "inverse EI $(\infty,1)$-categories", which contains universe objects that satisfy Voevodsky's univalence axiom. In addition to diagrams on ordinary inverse…

Let $G$ be a discrete group. We prove that the category of $G$-posets admits a model structure that is Quillen equivalent to the standard model structure on $G$-spaces. As is already true nonequivariantly, the three classes of maps defining…

In this note we show that Voevodsky's univalence axiom holds in the model of type theory based on symmetric cubical sets. We will also discuss Swan's construction of the identity type in this variation of cubical sets. This proves that we…

This paper gives a uniform-theoretic refinement of classical homotopy theory. Both cubical sets (with connections) and uniform spaces admit classes of weak equivalences, special cases of classical weak equivalences, appropriate for the…

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…

Building on work of Marta Bunge in the one-categorical case, we characterize when a given model category is Quillen equivalent to a presheaf category with the projective model structure. This involves introducing a notion of homotopy atoms,…

We will give a detailed account of why the simplicial sets model of the univalence axiom due to Voevodsky also models W-types. In addition, we will discuss W-types in categories of simplicial presheaves and an application to models of set…

We describe a homotopical version of the relational and gluing models of type theory, and generalize it to inverse diagrams and oplax limits. Our method uses the Reedy homotopy theory on inverse diagrams, and relies on the fact that Reedy…

We show that Voevodsky's univalence axiom for intensional type theory is valid in categories of simplicial presheaves on elegant Reedy categories. In addition to diagrams on inverse categories, as considered in previous work of the author,…