Related papers: Categorical structures for type theory in univalen…

We give a collection of results regarding path types, identity types and univalent universes in certain models of type theory based on presheaves. The main result is that path types cannot be used directly as identity types in any…

We give a model of dependent type theory with one univalent universe and propositional truncation interpreting a type as a stack, generalising the groupoid model of type theory. As an application, we show that countable choice cannot be…

Inspired by recent work on the categorical semantics of dependent type theories, we investigate the following question: When is logical structure (crucially, dependent-product and subobject-classifier structure) induced from a category to…

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…

Staton has shown that there is an equivalence between the category of presheaves on (the opposite of) finite sets and partial bijections and the category of nominal restriction sets: see [2, Exercise 9.7]. The aim here is to see that this…

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…

A hierarchy of type universes is a rudimentary ingredient in the type theories of many proof assistants to prevent the logical inconsistency resulting from combining dependent functions and the type-in-type rule. In this work, we argue that…

Our aim is to give a fairly complete account on the construction of compatible model structures on exact categories and symmetric monoidal exact categories, in some cases generalizing previously known results. We describe the close…

This is the third installment in a series of papers on algebraic set theory. In it, we develop a uniform approach to sheaf models of constructive set theories based on ideas from categorical logic. The key notion is that of a "predicative…

The intended model of the homotopy type theories used in Univalent Foundations is the infinity-category of homotopy types, also known as infinity-groupoids. The problem of higher structures is that of constructing the homotopy types needed…

If F is a type-definable family of commensurable subsets, subgroups or sub-vector spaces in a metric structure, then there is an invariant subset, subgroup or sub-vector space commensurable with F. This in particular applies to…

We introduce a reducibility on classes of structures, essentially a uniform enumeration reducibility. This reducibility is inspired by the Friedman-Stanley paper on using Borel reductions to compare classes of countable structures. This…

We continue the investigation of tabular algebras with trace (a certain class of associative ${\Bbb Z}[v, v^{-1}]$-algebras equipped with distinguished bases) by determining the extent to which the tabular structure may be recovered from a…

In [1] we introduced the concept of structured space, which is a topological space that locally resembles some algebraic structures. In [2] we proceeded the study of these spaces, developing two cohomology theories. The aim of this paper is…

In this note, we review a construction of category with families (CwF) in a presheaf category. When the base category of a presheaf category is a CwF, we internalize this CwF structure in the CwF of the presheaf category. This note assumes…

Standard methods of using categorical variables as predictors either endow them with an ordinal structure or assume they have no structure at all. However, categorical variables often possess structure that is more complicated than a linear…

We investigate combinations of structures by families of structures relative to families of unary predicates and equivalence relations. Conditions preserving $\omega$-categoricity and Ehrenfeuchtness under these combinations are…

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…

On the transversals of a subgroup of a group, using the binary operation of the group, structural mappings are defined. Based on these mappings, the notion of the hypergroup over the group is introduced, which generalizes the notion of the…

The main result of this paper may be stated as a construction of "almost representations" for the canonical presheaves of object extensions of length n on the C-systems defined by locally cartesian closed universe categories with binary…