Definition(s) of Stack

Question

I've started to learn about stacks, and a question arose in my attempts of looking at the very definition of a stack by several points of view. First, I recall some background and fix the notation (which is a mix of Fantechi's Stacks for everybody, Edidin's Notes on the construction of the moduli space space of curves, and my personal notation: I'm sorry about that).

Let $\mathfrak S:=\underline S:=Sch/S$ be the "base" category of schemes over a fixed scheme $S$, and let $Gpd$ be the category of groupoids. We say that a groupoid fibration $\pi:\mathfrak X\to \mathfrak S$ is a stack if two things happen: $(i)$ every descent datum is effective, and $(ii)$ isomorphisms are a sheaf for $\mathfrak X$ (with respect to the étale topology).

(It may help rephrase $(ii)$ as follows: for every $S$-scheme $B$ and for every two objects $X,Y$ in the fiber $\mathfrak X_B$, the presheaf $\mathcal I_B^{X,Y}:\underline B\to \textrm{Sets}$ is a sheaf on the (big) étale site associated to $B$. Here $\mathcal I_B^{X,Y}$ takes a $B$-scheme $f:B'\to B$ to the set of isomorphisms $f^\ast X\cong f^\ast Y$ in $\mathfrak X_{B'}$.)

My question: can we say that to give a stack is the same as to give a sheaf of groupoids $\mathcal F:\mathfrak S\to Gpd$ on the étale site associated to $S$?

My attempts. I'll sketch how I began to prove that the answer is yes, and convince myself that the answer is no.

First, given $\mathcal F$, we construct a groupoid fibration $\pi:\mathfrak X\to\mathfrak S$ fiberwise, by attaching $\mathfrak X_B:=\mathcal F(B)$ over $B\in\mathfrak S$. We then need to check $(i)$ and $(ii)$. Ok, let us stop here for the moment.

Conversely, if we have $\pi$, we may define $\mathcal F$ by $\mathcal F(B):=\mathfrak X_B$ on objects and by $\mathcal F(f:B'\to B)=(f^\ast:\mathfrak X_B\to\mathfrak X_{B'})$ on arrows. This seems to be natural. Let us stop here.

In both directions, there is a problem: I have to use (or prove) exactness of the sequence $$ \mathcal F(B)\to\prod_i\mathcal F(B_i)\rightrightarrows \prod_{i,j}\mathcal F(B_i\times_BB_j), $$ for $\{B_i\to B\}$ a covering of $B$. But does exactness make sense in $Gpd$? For instance, is $Gpd$ abelian?

Also, it seems to me that $(i)$ has nothing to do with the sheaf condition: I feel like $(i)$ doesn't help me to prove anything when it is assumed, and cannot be proven when starting with $\mathcal F$.

Any correction/insight is welcome. Thank you!

Answer 1

If we have a sheaf of groupoids, then in particular the presheaf of objects must be a sheaf as well. Thus one way of showing that a stack is not just a sheaf of groupoids is to show that the presheaf of objects it gives is not a sheaf.

Rather than working with complicated sites like the big étale site, let me construct an example for the standard site for a single topological space $X$. Consider the stack $\textbf{Pic}$ of line bundles over $X$: as a fibred category, its objects are pairs $(U, L)$ where $U \subseteq X$ is open and $L$ is a (real) line bundle over $U$, and its morphisms are fibrewise linear isomorphisms. (Warning: The fibre $\textbf{Pic}(U)$ is a groupoid, but it is not the Picard group of $U$ in general!) It is a standard exercise to check that $\textbf{Pic}$ is a stack: this amounts to showing that line bundles can be glued together.

For convenience, we assume $\textbf{Pic}$ is skeletal, so that all isomorphisms in $\textbf{Pic}$ are automorphisms. Let $\textrm{Pic}$ be the presheaf that assigns to each open $U \subseteq X$ the set of isomorphism classes of line bundles over $U$. Obviously, $\textrm{Pic}$ is the presheaf of objects of $\textbf{Pic}$. Now, $\textrm{Pic}$ is not a sheaf in general: by definition, if $L$ is a line bundle over $X$ and $\mathfrak{U}$ is a sufficiently fine open cover of $X$, then $L$ pulls back along $\mathfrak{U}$ to the trivial line bundle; but the Möbius strip is a non-trivial line bundle over $X = S^1$, so in this case we see that $\textrm{Pic}$ fails to even be a separated presheaf.

Now, consider the sheaf $\mathscr{O}_X^\times$ of continuous non-vanishing (real-valued) functions on $X$. This is a sheaf of groups and gives rise to a category $\mathbf{G}$ fibred in groupoids over the standard site of $X$ via the Grothendieck construction. Then $\textbf{G}$ is not a stack in general: again, for $X = S^1 \subseteq \mathbb{C}$, consider the open cover $\mathfrak{U} = \{ X \setminus \{ +1 \}, X \setminus \{ -1 \} \}$ and the descent data induced by the evident trivialisation of the Möbius strip over $\mathfrak{U}$. In this case, the failure of $\textbf{G}$ to be a stack is directly connected to the non-triviality of $\check{H}^1 (X, \mathscr{O}^\times_X)$!