The Constructive Kan–Quillen Model Structure: Two New Proofs

Abstract

We present two new proofs of Simon Henry’s result that the category of simplicial sets admits a constructive counterpart of the classical Kan–Quillen model structure. Our proofs are entirely self-contained and avoid complex combinatorial arguments on anodyne extensions. We also give new constructive proofs of the left and right properness of the model structure.

The effective model structure and -groupoid objects

“…Novel aspects. This paper differs significantly from our work in [22,26,47] in both scope and technical aspects. Regarding scope, apart from generalising the existence of the model structure from the case E = Set to that of a general countably lextensive category E, here we discuss a number of topics that are not even mentioned for the case E = Set in our earlier work, such as the structure and characterisation of the ∞-category associated to the effective model structure, the discussion of descent and the connections with Elmendorf's theorem.…”

“…The initial motivation for this work was the desire to establish whether our earlier work on the constructive Kan-Quillen model structure [21,22,26,47] could be developed further so as to obtain a new model structure on categories of simplicial sheaves. Indeed, in [22,26], we worked with simplicial sets without using the law of excluded middle and the axiom of choice, thus opening the possibility of replacing them with simplicial objects in a Grothendieck topos. As we explored this idea, we realised that the resulting argument admitted not only a clean presentation in terms of enriched weak factorisation systems [44,Chapter 13] but also a vast generalisation.…”

“…Regarding the technical aspects, even if the general strategy for proving the existence of the effective model structure is inspired by the case E = Set in [22], several new ideas are necessary to implement it to the general case, as we explain below. This strategy involves three steps.…”

Towards a constructive simplicial model of Univalent Foundations

“…Another proof of Theorem 4.1 is given in [19, Section 4] modifying appropriately the arguments in [18]. Remark We define explicitly how the $$\Pi$$‐types of the comprehension category of cofibrant simplicial sets are defined.…”

“…However, this model structure coincides with the standard one as soon as the law of excluded middle is assumed. Two other proofs of the existence of this constructive model structure are obtained in [19].…”