The Frobenius Condition, Right Properness, and Uniform Fibrations

TL;DR

This paper advances the theory of weak factorization systems by providing a new construction method for algebraic weak factorization systems with fibrations satisfying the Frobenius condition, with applications to model structures and type theory.

Contribution

It introduces a novel method for constructing algebraic weak factorization systems with fibrations satisfying the Frobenius condition, and applies it to prove right properness and properties in type theory models.

Findings

Proves the Quillen model structure for Kan complexes is right proper without topological realization.

Constructively shows preservation of Kan fibrations under pushforward along Kan fibrations.

Extends and subsumes previous work on cubical sets and type-theoretic models.

Abstract

We develop further the theory of weak factorization systems and algebraic weak factorization systems. In particular, we give a method for constructing (algebraic) weak factorization systems whose right maps can be thought of as (uniform) fibrations and that satisfy the (functorial) Frobenius condition. As applications, we obtain a new proof that the Quillen model structure for Kan complexes is right proper, avoiding entirely the use of topological realization and minimal fibrations, and we solve an open problem in the study of Voevodsky's simplicial model of type theory, proving a constructive version of the preservation of Kan fibrations by pushforward along Kan fibrations. Our results also subsume and extend work by Coquand and others on cubical sets.