What is the connection between Lurie's definition of shape and Čech homotopy?

Question

It seems there are many subtly different notions of the shape of a topological space (and, more generally, toposes).

For instance, Lurie [Higher topos theory] defines this one:

Definition 1. The shape of a topos $\mathcal{E}$ is the pro-object in $\mathcal{S}$ representing the endofunctor $p_* p^* : \mathcal{S} \to \mathcal{S}$, where $p$ is the unique geometric morphism from the $\infty$-sheaf topos $\mathcal{E}$ to the $(\infty, 1)$-category $\mathcal{S}$ of $\infty$-groupoids.

Here is another, perhaps closer to what is studied in classical shape theory:

Definition 2. The shape of a topological space $X$ is the pro-object $\varprojlim_{\mathcal{U} : \textrm{Cov} (X)} \mathrm{B} \mathcal{U}$ where $\mathcal{U}$ ranges over the poset $\textrm{Cov} (X)$ of covering sieves $\mathcal{U}$ of non-empty open subspaces of $X$ and $\mathrm{B} \mathcal{U}$ is the geometric realisation of the nerve of $\mathcal{U}$.

There is a canonical comparison from definition 1 to definition 2. Roughly speaking, it comes from the Grothendieck plus construction for sheafification. The sheaf $p^* K$ is the sheafification of the constant presheaf with value $K$, so there is a comparison morphism $\varprojlim_{U : \mathcal{U}} K \to p_* p^* K$ for each covering sieve $\mathcal{U}$. We have $\varprojlim_{U : \mathcal{U}} K \cong \mathcal{S} (\mathrm{B} \mathcal{U}, K)$, and taking the colimit over all covering sieves $\mathcal{U}$ yields a morphism $${\textstyle \varinjlim}_{\mathcal{U} : \textrm{Cov} (X)^\textrm{op}} \mathcal{S} (\mathrm{B} \mathcal{U}, K) \to {\textstyle \varinjlim}_{\mathcal{U} : \textrm{Cov} (X)^\textrm{op}} p_* p^* K \cong p_* p^* K$$ which yields a morphism from the pro-object representing $p_* p^*$ to $\varprojlim_{\mathcal{U} : \textrm{Cov} (X)^\textrm{op}} \mathrm{B} \mathcal{U}$.

Question. Is this an equivalence?

When $K$ is discrete, the constant presheaf with value $K$ is almost separated – there is a problem over $\varnothing$, but it can be ignored – so $\varinjlim_{\mathcal{U} : \textrm{Cov} (X)^\textrm{op}} \mathcal{S} (\mathrm{B} \mathcal{U}, K) \to p_* p^* K$ is in fact a bijection in this case. I am less clear about the situation for non-discrete $K$ – do we need to iterate the plus construction even for constant presheaves? Can we fix the problem by replacing $\textrm{Cov} (X)$ with some suitable category of hypercovers?

Answer 1

The plus construction has to be iterated, yes. The topological space from this answer provides a simple counterexample. Let $X=\{a,b,c,d\}$ with opens $\{a\},\{b\},\{a,b\},\{a,b,c\},\{a,b,d\}$. Then the plus construction does not change the global sections of the constant presheaf on $X$ with fiber $K$. However, the global sections of the associated sheaf is the free loop space $\mathcal LK=K\times_{K\times K}K$. So the shape of $X$ is a circle, and in this case Definition 2 "does not work".

The problem can more or less be fixed by using hypercovers, because the analogue of the plus construction with hypercovers always produces the hypersheafification (this is essentially Verdier's hypercovering theorem). So the analogue of Definition 2 gives the shape of the $\infty$-topos of hypersheaves on $X$.