Given a category $\mathcal{C}$ with a notion of covering $\{ U\_{i} \rightarrow X \}$ for an object $X$ (say $\mathcal{C}$ is a Grothendieck site), we can form the Cech nerve


$$ \cdots \coprod\_{i}{U\_{ijk}} \substack{\rightarrow \\ \rightarrow \\ \rightarrow}\coprod\_{i}{U\_{ij}} \rightrightarrows \coprod\_{i} U\_{i}$$


(In the notation, I've suppressed degeneracy maps going from right to left.) 


This can be viewed in two ways. 1. As a simplicial object in simplicial presheaves, by considering each $U\_{i}$ as a simplicial presheaf constant in the simplicial direction. I'll denote that by $U\_{\bullet}$. 2. As a simplicial object in presheaves and hence a simplicial presheaf. I'll denote that by $\check{U}\_{\bullet}$.


The latter $\check{U}\_{\bullet}$ can be shown to be level-wise weakly equivalent to $\operatorname{colim}(\coprod\_{i}{U\_{ij}} \rightrightarrows \coprod\_{i} U\_{i})$, where we consider this as a simplicial presheaf constant in the simplicial direction. 


On the other hand, we could compute $\operatorname{hocolim}(U\_{\bullet})$, and from things I read, this is supposed to be identified with/weakly equivalent to $\check{U}\_{\bullet}$. One reason I am having difficulty seeing this is that I don't really understand $\operatorname{hocolim}(U\_{\bullet})$. Since each object in $U\_{\bullet}$ is cofibrant, I would guess I could just take the usual colimit, but this seems to produce something constant in the simplicial direction, which is clearly wrong.


So the question is:


How to see if there is a weak equivalence $\operatorname{hocolim}(U\_{\bullet}) \rightarrow \check{U}\_{\bullet}$?


Probably if I read through the many pages of material suggested in the comments to [this question](https://mathoverflow.net/questions/87658/computing-homotopy-colimits-in-a-nice-simplicial-model-category), I'd be able to figure this out. But a more direct answer would make that reading more fruitful for me, I think. At least, pointing out what things I need to know to figure this out would help.



Cech nerve as homotopy colimit?
More generally, let $X\_\cdot$ be a simplicial presheaf. As such, we can consider it as a simplicial object in presheaves, which in particular may be thought of as a simplicial object in simplicial presheaves $X'\_\cdot.$ So we have:


$$X\_\cdot:\Delta^{op} \to Set^{C^{op}}$$ and $$X'\_\cdot =\left( \mspace{3mu} \cdot \mspace{3mu}\right)^{(id)} \circ X\_\cdot:\Delta^{op} \to Set\_{\Delta}^{C^{op}}$$ where $$\left( \mspace{3mu} \cdot \mspace{3mu}\right)^{(id)}:Set^{C^{op}} \to Set\_{\Delta}^{C^{op}}$$ is the evident inclusion of presheaves into simplicial presheaves.


The homotopy colimit of $X'\_\cdot$ in simplicial presheaves is computed "object-wise". Hence, for all $c \in C$ we have $$hocolim\left(X'\right)(C)=hocolim \left( X'\left(C\right)\right).$$ The right-hand side is the homotopy colimit of a simplicial object in simplicial sets, and can be computed by taking the diagonal of the corresponding bisimplicial set. But the diagonal of $X'\left(C\right)\_\cdot$ is simply $X\left(C\right)\_\cdot$ since $X'\_\cdot$ ` in constant in one simplicial direction. Hence $$hocolim\left(X'\right)=X.$$


