I have a theoretical question about comparing two objects that I have recently come across.


For concreteness, let us work over the category $C$ of schemes over $k$. Let $G$ be an algebraic group over $k$. One can construct the stack $BG$ as a fibered category in groupoids and as a simplicial scheme $(BG)\_n=G^n$ (with certain face and degeneracy maps which I don't specify). What is the precise relation between these two versions of $BG$? Can I obtain one from the other?


I think I might have heard that "the two constructions are equivalent because the simplicial $BG$ has no higher homotopy groups". Does this make sense? How does this implication work?


Any answer that could help me better understand the relation between these two $BG$s is very welcome.



$BG$ the stack, $BG$ the simplicial presheaf
The two constructions are not quite equivalent. Let me write $\mathbf BG$ for the stack and $B\_\bullet G$ for the simplicial scheme to better distinguish between them. There is a third relevant player, $BG$, which is the presheaf of ∞-groupoids on $C$ presented by $B\_\bullet G$.


The precise relation between these three objects is the following:


1. $\mathbf BG$ is the fppf sheafification of $BG$
2. $BG$ is the colimit of the simplicial object $B\_\bullet G$
3. $B\_\bullet G$ is the Čech nerve (= $0$-coskeleton) of either $\mathrm{Spec}(k) \to \mathbf BG$ or $\mathrm{Spec}(k) \to BG$


In more detail, algebraic stacks over $k$ (in the sense of Artin, say) form a full subcategory of the $2$-category of fppf sheaves of groupoids (classically, "stacks in groupoids") on $C$, which is itself a full subcategory of the ∞-category of presheaves of ∞-groupoids on $C$. Thus, both $\mathbf BG$ and $BG$ live in this ∞-category (in fact they both belong to the subcategory of presheaves of groupoids), and one is the fppf sheafification of the other. The étale sheafification suffices if $G$ is smooth.


The reason $\mathbf BG$ and $BG$ are not the same is that there is a unique homotopy class of map $X\to BG$ from any scheme $X$, but homotopy classes of maps $X\to \mathbf BG$ are in bijection with isomorphism classes of $G$-torsors on $X$. In fact $BG$ is the full subpresheaf of $\mathbf BG$ spanned by the trivial $G$-torsors.


(Added details about 2.) 
$BG$ being the colimit of $B\_\bullet G$ is the manner in which simplicial sets give rise to ∞-groupoids. Of course, one must define ∞-groupoids at some point and one way to do that is to start with simplicial sets and invert weak equivalences, so that we have a localization functor {simplicial sets} → {∞-groupoids}. Once higher category theory is set up however, it turns out that this functor is the composition of the inclusion of simplicial sets into simplicial ∞-groupoids and of the colimit over $\Delta^{op}$ functor (this is the standard fact that a simplicial set is canonically the homotopy colimit of itself). In practice this is often a more useful way to think about it.


