Let $\mathcal{M}$ be an algebraic stack, and let $X$ be its coarse moduli space (assume it exists as a scheme).


We know that $h\_X(Spec(k))=\mathcal{M}(Spec(k))$ if $k$ is algebraically closed. Is there anything intelligent we can say about $h\_X(U)$ for a general scheme $U$?


For example, can you come up with an algorithm for knowing what $h\_X(U)$ is that would be considerably easier than constructing $X$?



If X is the coarse moduli space of the algebraic stack M, is there a nice description of Hom(_,X)?
This is more an answer to the comment of *unknowngoogle*.  

• The object *U* ↦ π0(ℳ(*U*)) that you described is the initial **presheaf** to which the stack ℳ maps.  

• One can also consider the sheafification of *U* ↦ π0(ℳ(*U*)),
which is the initial **sheaf** to which ℳ maps.  

• Finally, there is the (possibly non-existent) initial **representable sheaf** (=scheme) to which ℳ maps.


In general, all those are different.  

They are already different in the case of the affine line modded out by the involution *x* ↦ -*x*.


