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. Huber, André Joyal, Andrew Pitts, Emily Riehl, and Andrew Swan for helpful discussions and comments on earlier versions of the paper, to Marco Larrea Schiavon for pointing out an erroneous statement of Proposition 4.2 in an earlier version of this paper, and to Fosco Loregian for pointing us to a useful reference. ⋔ . Of course, a wfs could be free on many different classes of maps. By Quillen's small object argument [34], every set I determines a wfs (L, R) free on I. These are what we call cofibrantly generated wfs's. The next, very simple, proposition provides a convenient way of checking that a cofibrantly generated wfs satisfies the Frobenius condition and will be used in the proof of Theorem 3.8.Proposition 1.4. Let (L, R) be a wfs that is free on a class of maps I. Then the following are equivalent:(i) (L, R) has the Frobenius property. (ii) The pushforward along an R-map preserves R-maps.

A homotopy-theoretic model of function extensionality in the effective topos

“…Also, if we consider cubical sets as in the work by Coquand and others (Cohen et al 2018), we recover their notion of a Kan cubical set and obtain a model structure on these Kan cubical sets. This result is inspired by earlier works of Orton and Pitts (2016) and Gambino and Sattler (2017), which in turn is based on the work of Cohen et al (2018) and earlier work by Cisinski (2002); indeed, with a few exceptions most steps in our proof of the model structure can be found in these earlier sources. The main innovation is that we do not assume cocompleteness of the underlying topos, so that our result can be applied to realizability toposes as well.…”

“…This theorem subsumes the classical model structure on simplicial sets in which the fibrant objects are the Kan complexes and it also includes the work on cubical sets by Coquand and others [9]. This result is inspired by earlier work by Orton and Pitts [23] and Gambino and Sattler [12], which in turn is based on the work of Cohen, Coquand, Huber, and Mörtberg [9] and earlier work by Cisinski [8]; indeed, with a few exceptions most steps in our proof of the model structure can be found in these earlier sources. The main innovation is that we do not assume cocompleteness of the underlying topos, so that our result can be applied to realizability toposes as well.…”

“…(This is Lemma 3.5. (ii) in [12].) If u is a strong homotopy equivalence, then u is a retract of u⊗∂ 0 by Proposition 3.11.…”

“…This means that, unlike Cisinski, Gambino and Sattler, we do not rely on the small object argument to build our factorisations. (Other differences are that we work in a non-algebraic setting, as in part 1, but not part 2 of Gambino and Sattler (2017), and we work mostly externally with toposes, not inside a type theory as in Orton and Pitts (2016); related to this is that for us being a fibration, for instance, is a property of morphism, not additional structure. )…”

Equipping weak equivalences with algebraic structure

“…In 2-category theory, such coherence conditions enable us to capture lax structures such as Grothendieck fibrations and lax morphisms in two dimensional universal algebra [6,7] that lie far beyond the expressive power of wfs. They also naturally arise in homotopy type theory as a means to construct good categorical models -see, for instance, [36,13,4].…”