Michael A. Warren

Homotopy theoretic models of identity types

This paper presents a novel connection between homotopical algebra and mathematical logic. It is shown that a form of intensional type theory is valid in any Quillen model category, generalizing the Hofmann-Streicher groupoid model of Martin-Loef type theory.

The Local Universes Model: An Overlooked Coherence Construction for Dependent Type Theories

We present a new coherence theorem for comprehension categories, providing strict models of dependent type theory with all standard constructors, including dependent products, dependent sums, identity types, and other inductive types. Precisely, we take as input a “weak model”: a comprehension category, equipped with structure corresponding to the desired logical constructions. We assume throughout that the base category is close to locally Cartesian closed: specifically, that products and certain exponentials exist. Beyond this, we require only that the logical structure should be weakly stable—a pure existence statement, not involving any specific choice of structure, weaker than standard categorical Beck--Chevalley conditions, and holding in the now standard homotopy-theoretic models of type theory. Given such a comprehension category, we construct an equivalent split one whose logical structure is strictly stable under reindexing. This yields an interpretation of type theory with the chosen constructors. The model is adapted from Voevodskys use of universes for coherence, and at the level of fibrations is a classical construction of Giraud. It may be viewed in terms of local universes or delayed substitutions.

Lawvere-Tierney sheaves in Algebraic Set Theory

We present a solution to the problem of defining a counter- part in Algebraic Set Theory of the construction of internal sheaves in Topos Theory. Our approach is general in that we consider sheaves as determined by Lawvere-Tierney coverages, rather than by Grothendieck coverages, and assume only a weakening of the axioms for small maps originally introduced by Joyal and Moerdijk, thus subsuming the exist- ing topos-theoretic results.

Coalgebras in a category of classes

Abstract In this paper the familiar construction of the category of coalgebras for a cartesian comonad is extended to the setting of “algebraic set theory”. In particular, it is shown that, under suitable assumptions, several kinds of categories of classes are stable under the formation of coalgebras for a cartesian comonad, internal presheaves and comma categories.

A univalent formalization of the p -adic numbers

The goal of this paper is to report on a formalization of the p -adic numbers in the setting of the second authors univalent foundations program. This formalization, which has been verified in the Coq proof assistant, provides an approach to the p -adic numbers in constructive algebra and analysis.