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Suppose that $C$ is a Grothendieck site, and $\mathscr{X}$ is a stack over $C$ (which is NOT equivalent to a sheaf). Let $$\pi_{\mathscr{X}}:\int_{C} \mathscr{X}\to C$$ denote the associated fibered category. The underlying category $\int_{C} \mathscr{X}$ carries an induced Grothendieck topology such that $$St\left(\int_{C} \mathscr{X}\right) \simeq St\left(C\right)/\mathscr{X}$$ where the latter is the slice 2-category over $\mathscr{X}$.

Now the functor $\pi_{\mathscr{X}}$ induces the functor $$\left(\pi_{\mathscr{X}}\right)_{!} :Sh\left(\int_{C} \mathscr{X}\right) \to Sh(C).$$ It also induces a $2$-functor $$\left(\pi_{\mathscr{X}}\right)_{!} :St\left(\int_{C} \mathscr{X}\right) \to St(C)$$ by taking the weak left Kan extension of $y \circ \pi_{\mathscr{X}}$ along Yoneda, where $y$ here denotes the Yoneda embedding $C \to St(C)$ of $C$. Under the equivalence $$St\left(\int_{C} \mathscr{X}\right) \simeq St\left(C\right)/\mathscr{X},$$ $\left(\pi_{\mathscr{X}}\right)_{!}$ corresponds to the projection $$St(C)/\mathscr{X} \to St(C),$$ since this is weak colimit preserving and agrees with $\left(\pi_{\mathscr{X}}\right)_{!}$ on representables.

This implies that $$\left(\pi_{\mathscr{X}}\right)_{!}\left(1\right)\simeq \mathscr{X}.$$ But the terminal object $1$ is a sheaf, so we should have $\left(\pi_{\mathscr{X}}\right)_{!}\left(1\right) \in Sh(C)$, but we should also have it equivalent to $\mathscr{X}$ which is NOT equivalent to a sheaf. What am I missing?

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Silly off-topic question: why the integral? :S – Mattia Talpo Jan 9 '12 at 14:04
@Mattia: This is the Grothendieck construction; it expresses a pseudo-functor $C^{op} \to \mathrm{Cat}$ as a fibered category over $C$. You learn more about it in Angelo Vistoli's notes on descent theory. – Martin Brandenburg Jan 9 '12 at 15:07
@Martin: ok thanks, I know about the construction you're mentioning, I was just wondering about the symbol. Is there some reason why it should be analogue to an integral? – Mattia Talpo Jan 9 '12 at 15:14
@Mattia: One may think of it as adding all of the images of the pseudo-functor together to get the source category of the fibration. I don't know of a more precise analogy. – S. Carnahan Jan 10 '12 at 6:51
up vote 1 down vote accepted

Here's what's wrong:

The inclusion $j:Sh(C) \to St(C)$ does not preserve colimits. Notice that $j$ has a right-adjoint given by $\pi_0,$ at least making sheaves reflective. To see that $j$ does not preserve colimits, take for instance the colimit of $\pi_\mathscr{X}$, first by composing with the Yoneda embedding into stacks, and second by composing it with the Yoneda embedding into sheaves. The former is $j$ composed with the latter. If $y$ denotes the Yoneda embedding into stacks, the former is the colimit of $y \circ \pi_{\mathscr{X}}$ which is canonically equivalent to $\mathscr{X},$ whereas the latter is the colimit of $\pi_0 \circ y \circ \pi_\mathscr{X}$ which is equivalent to $\pi_0(\mathscr{X})$.

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Note that that inclusion doesn't preserve colimits even when $C$ is the trivial site (when it is $Set \to Gpd$). – Mike Shulman Jan 13 '12 at 17:15

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