Model structures on simplicial presheaves of piecewise-linear manifolds

Question

Let $\mathbf{PL}$ denote the category of piecewise-linear manifolds. The goal is to embed $\mathbf{PL}$ into a category of simplicial presheaves, endow it with a model structure, and then localize it with respect to a Grothendieck topology and the interval.

Consider the category of simplicial presheaves $\mathbf{sPre}(\mathbf{PL})$, which consists of functors $F: \mathbf{PL}^{op} \to \mathbf{sSet}$, where $\mathbf{sSet}$ is the category of simplicial sets. The Yoneda embedding $y: \mathbf{PL} \to \mathbf{sPre}(\mathbf{PL})$ sends each piecewise-linear manifold $M$ to the representable presheaf $y(M) = \mathrm{Hom}_{\mathbf{PL}}(-, M)$.

Endow $\mathbf{sPre}(\mathbf{PL})$ with the projective model structure, where weak equivalences and fibrations are defined objectwise, and cofibrations are determined by the left lifting property.

Choose a Grothendieck topology $\tau$ on $\mathbf{PL}$, for example, the topology generated by open covers. Let $\mathcal{H}_{\tau}$ be the class of $\tau$-hypercovers in $\mathbf{PL}$. Define the $\tau$-local equivalences in $\mathbf{sPre}(\mathbf{PL})$ as follows:

$$W_{\tau} = \left\{ \eta: F \to G \ | \ \eta^*: \pi_0(\mathbf{sPre}(\mathbf{PL})(y(-), G)) \to \pi_0(\mathbf{sPre}(\mathbf{PL})(y(-), F)) \text{ is an isomorphism of } \tau\text{-sheaves} \right\}$$

Perform a left Bousfield localization of the projective model structure on $\mathbf{sPre}(\mathbf{PL})$ with respect to the class $W_{\tau}$. Denote the resulting model category by $\mathbf{sPre}(\mathbf{PL})_{\tau}$.

Let $I$ be the interval object in $\mathbf{PL}$ (e.g., $I = [0, 1]$). Define the class of $I$-homotopy equivalences as:

$$W_I = \left\{ \eta: F \to G \ | \ \eta \times \mathrm{id}_I: F \times I \to G \times I \text{ is a weak equivalence in } \mathbf{sPre}(\mathbf{PL})_{\tau} \right\}$$

Perform another left Bousfield localization of $\mathbf{sPre}(\mathbf{PL})_{\tau}$ with respect to $W_I$ to obtain the model category $\mathbf{sPre}(\mathbf{PL})_{\tau, I}$.

Question: Is the model category $\mathbf{sPre}(\mathbf{PL})_{\tau, I}$ Quillen equivalent to the Quillen model structure on simplicial sets?

To answer this question, one would need to construct a Quillen adjunction between the two model categories and prove that it is a Quillen equivalence.

Any insights, partial results, or related work would be greatly appreciated. Thank you!

Answer 1

The original reference for such results is Proposition 3.3.3 on page 120 in

• Fabien Morel, Vladimir Voevodsky, A^1-homotopy theory of schemes, Publications mathématiques de l’I.H.É.S., tome 90 (1999), p. 45-143.

In fact, they prove a more general $G$-equivariant version. Another early reference is Proposition 8.3 in Dugger's Universal homotopy theories.

These proofs rely on the existence of partitions of unity, which holds for all three classical categories: smooth, piecewise-linear, and topological manifolds. Nothing specific to any category other than partitions of unity is used in the proof.

Modern expositions of the Morel–Voevodsky theorem appear in many papers, e.g., arXiv:1310.7930 (Section 4.4), arXiv:1912.10544 (Section 2), arXiv:2007.06039, and the book arXiv:2109.12250 (Sections 4, 5).