We all know that the homotopy fiber of a continuous function $f: A \to B$ has a simple expression in terms of points and paths connecting them, that is $$ \textrm{hofib}_y(f) = \{ (x,\gamma) \in A \times B^{I}: \gamma(0) = f(x), \gamma(1) = y\}$$ I think of this process of relaxation as the dual to the "cylinder relaxation" that one does in homotopy pushouts. This makes sense since path objects are dual to cylinder objects when coming to cofibration-fibration factorizations, and these are used to resolve the diagram.
I would like to give an explicit resolution when we take the homotopy limit over a cover. Specifically, let $M$ be a manifold (that is, nice space) and $\mathcal{U}$ a cover of $M$ which is closed under finite intersections. Consider a presheaf of spaces $F$ on $M$. We are interested in computing the "homotopy global sections" coming from $\mathcal{U}$: $$ F^h(M) := \textrm{holim}_{U \in \mathcal{U} } F(U) $$ I have the impression this can be resolved in the following way. Let $N(\mathcal{U})$ be the nerve of the cover, meaning $$N_n(\mathcal{U}) = \{(U_0, \ldots, U_n) \in \mathcal{U}^{n+1} : U_0 \subset \ldots \subset U_n \} $$ Extend the presheaf $F$ in the following way: let $F(U_0, \ldots, U_n)$ be $$ F(U_0, \ldots, U_n) := \{ (a_0, \ldots, a_n) \in F(U_0)^{\Delta^n} \times F(U_1)^{\Delta^{n-1}} \times \ldots \times F(U_n)^{\Delta^0} : d_0a_i = a_{i+1}| U_{i+1} \} $$ I find the example $n=2$ to describe well the picture behind. An element there is a triangle $012$ in the smallest subset, then a segment in the middle one lifting the segment $12$, then a point in the biggest one lifting $2$.
In this way, $F$ becomes a functor from the category of elements of the simplicial set $N(\mathcal{U})$ to Top. In other words, if I take a simplex $(U_0, \ldots, U_n)$ and exclude a subset from the chain, I have a map
$$ F(U_0, \ldots, U_n) \to F(U_0, \ldots, \hat{U}_i, \ldots, U_n )$$ given by forgetting the $i$-th vertex of the simplex, and lifting at the $i$-th stage not to $U_i$ but to $U_{i+1}$ directly.
We are ready for the question.
Can I compute the homotopy limit in the following way? $$ \textrm{holim}_{U \in \mathcal{U} } F(U) \simeq \textrm{lim}_{ U_{\bullet} \in N(\mathcal{U} ) } F( U_{\bullet} ) $$ If yes, is there a reference?
I suspect it does come from some more general result on simplicial resolution, but I prefer to stick to my case not to make improper generalizations.
