# Model structures on simplicial presheaves of piecewise-linear manifolds

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!

• I literally just posted a paper on arxiv that does this kind of thing. Might be of interest to you: arxiv.org/abs/2405.15511 Commented Jun 2 at 19:53
• @DavidWhite-gonefromMO: Thank you Dr. White.
– user528837
Commented Jun 2 at 20:05

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