The Frobenius Condition, Right Properness, and Uniform Fibrations

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.

v5: reverted definition of uniform fibration to that in v3 to mirror Coquand et al., parts of development restructured accordingly; added references to some technical statements in Section 4; slightly simplified Def. 6.1; added comparison to work of Bourke/Garner (Rmk. 8.5); improved exposition and fixed various typos; 40 pages. Accepted for publication in Journal of Pure and Applied Algebra

References in corpus (2)

• The identity type weak factorisation system
• Minimal fibrations of dendroidal sets

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