Let $\pi \_1:SS\to Grpd$ denote the fundamental groupoid functor, from simplicial sets to groupoids, and let $N:Grpd\to SS$ denote the nerve functor. Then $\pi \_1$ is left adjoint to $N.$


On simplicial sets we consider the usual model category and on groupoids the model category structure inherited by the one given by Thomason. It is known that $\pi \_1$ preserves weak equivalences, cofibrations, fibrations and colimits, and that $N$ preserves weak equivalences, fibrations and limits. From these facts one can easily deduce that $\pi \_1$ preserves homotopy colimits and $N$ preserves homotopy limits. I wonder whether the following questions have positive answers:


1. Does $N$ preserve homotopy colimits?
2. Does $\pi \_1$ preserve homotopy limits?



Does the nerve functor (resp. fundamental groupoid functor) preserve homotopy colimits (resp. homotopy limits)?
Just for fun, here's a purely abstract way of seeing neither of these can happen:


The functors you wrote down both preserve weak equivalences, so are morphisms of relative categories, and induce adjunctions of the associated $\left(\infty,1\right)$-categories, which are $\infty$-groupoids and $1$-groupoids (which is actually a $(2,1)$-category) respectively. The adjunction is precisely the one exhibiting $1$-groupoids as a reflective subcategory of $\infty$-groupoids. The full and faithful inclusion is modeled by $N$. The $\left(\infty,1\right)$-category of $\infty$-groupoids is freely generated by the (any) contractible $\infty$-groupoid (i.e. the one point space), (i.e. it's the free colimit cocompletion of the terminal category). Since the $\left(2,1\right)$-category of groupoids is cocomplete, and contains the terminal groupoid, if the inclusion preserved colimits, it would have to be essentially surjective:


If $X$ is any $\infty$-groupoid, you can write $X$ as the colimit of the terminal functor $X \to \infty\mbox{-}\mathbf{Gpd}$ (the constant functor with value the point) (this is basically just the Yoneda Lemma for $\infty$-categories), and this functor (obviously) factors through the inclusion $\mathbf{Gpd} \hookrightarrow \infty\mbox{-}\mathbf{Gpd}$.


If the left adjoint $\tau\_1$ of this inclusion (which is modeled by $\Pi\_1$) preserved even **finite** limits, then this would exhibit the $(2,1)$-category of groupoids as a left-exact localization of $\infty\mbox{-}\mathbf{Gpd}=\mathbf{Psh}\_\infty\left(\*\right),$ i.e. it would exhibit $\mathbf{Gpd}$ as an $\infty$-topos. This means there would exist a unique Grothendieck topology $J$ on the terminal category such that $\mathbf{Gpd}$ sat somewhere between $\mathbf{Sh}\_\infty\left(\*,J\right)$ and its hypercompletion $\mathbf{HSh}\_\infty\left(\*,J\right).$ But, there is a unique Grothendieck topology on the terminal category (the trivial one!), so we'd have $$\mathbf{HSh}\_\infty\left(\*,J\right)=\mathbf{Sh}\_\infty\left(\*,J\right)=\mathbf{Psh}\_\infty\left(\*\right).$$ So we'd have to have $\mathbf{Gpd}\simeq \mathbf{Psh}\_\infty\left(\*\right)=\infty\mbox{-}\mathbf{Gpd}$ which is clearly nonsense. 


