The Stacks project

Lemma 8.12.12. Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a functor satisfying the assumptions of Sites, Lemma 7.21.8. Let $f : \mathcal{D} \to \mathcal{C}$ be the corresponding morphism of sites. Then

  1. for every stack $p : \mathcal{S} \to \mathcal{C}$ the canonical functor $\mathcal{S} \to f_*f^{-1}\mathcal{S}$ is an equivalence of stacks,

  2. given stacks $\mathcal{S}, \mathcal{S}'$ over $\mathcal{C}$ the construction $f^{-1}$ induces an equivalence

    \[ \mathop{Mor}\nolimits _{\textit{Stacks}/\mathcal{C}}(\mathcal{S}, \mathcal{S}') \longrightarrow \mathop{Mor}\nolimits _{\textit{Stacks}/\mathcal{D}}(f^{-1}\mathcal{S}, f^{-1}\mathcal{S}') \]

    of morphism categories.

Proof. Note that by Lemma 8.12.10 we have an equivalence of categories

\[ \mathop{Mor}\nolimits _{\textit{Stacks}/\mathcal{D}}(f^{-1}\mathcal{S}, f^{-1}\mathcal{S}') = \mathop{Mor}\nolimits _{\textit{Stacks}/\mathcal{C}}(\mathcal{S}, f_*f^{-1}\mathcal{S}') \]

Hence (2) follows from (1).

To prove (1) we are going to use Lemma 8.4.8. This lemma tells us that we have to show that $can : \mathcal{S} \to f_*f^{-1}\mathcal{S}$ is fully faithful and that all objects of $f_*f^{-1}\mathcal{S}$ are locally in the essential image.

We quickly describe the functor $can$, see proof of Lemma 8.12.8. To do this we introduce the functor $c'' : \mathcal{S} \to u_{pp}\mathcal{S}$ defined by $c''(x/U) = (U, \text{id} : u(U) \to u(U), x)$, and $c''(\alpha /a) = (a, u(a), \alpha )$. We set $c' : \mathcal{S} \to u_ p\mathcal{S}$ equal to the composition of $c''$ and the canonical functor $u_{pp}\mathcal{S} \to u_ p\mathcal{S}$. We set $c : \mathcal{S} \to f^{-1}\mathcal{S}$ equal to the composition of $c'$ and the canonical functor $u_ p\mathcal{S} \to f^{-1}\mathcal{S}$. Then $can : \mathcal{S} \to f_*f^{-1}\mathcal{S}$ is the functor which to $x/U$ associates the pair $(U, c(x))$ and to $\alpha /a$ the morphism $(a, c(\alpha ))$.

Fully faithfulness. To prove this we are going to use Lemma 8.4.7. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Let $x, y \in \mathcal{S}_ U$. First off, as $u$ is fully faithful, we have

\[ \mathop{Mor}\nolimits _{(f_*f^{-1}\mathcal{S})_ U}(can(x), can(y)) = \mathop{Mor}\nolimits _{(f^{-1}\mathcal{S})_{u(U)}}(c(x), c(y)) \]

directly from the definition of $f_*$. Similar holds after pulling back to any $U'/U$. Because $f^{-1}\mathcal{S}$ is the stackification of $u_ p\mathcal{S}$, and since $u$ is continuous and cocontinuous the presheaf

\[ U'/U \longmapsto \mathop{Mor}\nolimits _{(f^{-1}\mathcal{S})_{u(U')}}(c(x|_{U'}), c(y|_{U'})) \]

is the sheafification of the presheaf

\[ U'/U \longmapsto \mathop{Mor}\nolimits _{(u_ p\mathcal{S})_{u(U')}}(c'(x|_{U'}), c'(y|_{U'})) \]

Hence to finish the proof of fully faithfulness it suffices to show that for any $U$ and $x, y$ the map

\[ \mathop{Mor}\nolimits _{\mathcal{S}_ U}(x, y) \longrightarrow \mathop{Mor}\nolimits _{(u_ p\mathcal{S})_ U}(c'(x), c'(y)) \]

is bijective. A morphism $f : x \to y$ in $u_ p\mathcal{S}$ over $u(U)$ is given by an equivalence class of diagrams

\[ \xymatrix{ (U', \phi : u(U) \to u(U'), x') \ar[d]_{(c, \text{id}_{u(U)}, \gamma )} \ar[r]_{(a, b, \alpha )} & (U, \text{id} : u(U) \to u(U), y) \\ (U, \text{id} : u(U) \to u(U), x) } \]

with $\gamma $ strongly cartesian and $b = \text{id}_{u(U)}$. But since $u$ is fully faithful we can write $\phi = u(c')$ for some morphism $c' : U \to U'$ and then we see that $a \circ c' = \text{id}_ U$ and $c \circ c' = \text{id}_{U'}$. Because $\gamma $ is strongly cartesian we can find a morphism $\gamma ' : x \to x'$ lifting $c'$ such that $\gamma \circ \gamma ' = \text{id}_ x$. By definition of the equivalence classes defining morphisms in $u_ p\mathcal{S}$ it follows that the morphism

\[ \xymatrix{ (U, \text{id} : u(U) \to u(U), x) \ar[rr]_{(\text{id}, \text{id}, \alpha \circ \gamma ')} & & (U, \text{id} : u(U) \to u(U), y) } \]

of $u_{pp}\mathcal{S}$ induces the morphism $f$ in $u_ p\mathcal{S}$. This proves that the map is surjective. We omit the proof that it is injective.

Finally, we have to show that any object of $f_*f^{-1}\mathcal{S}$ locally comes from an object of $\mathcal{S}$. This is clear from the constructions (details omitted). $\square$


Comments (2)

Comment #1215 by JuanPablo on

I had problem with the last omitted proof: "an object of comes locally from an object of ".

Say take an object in . Then I need to find a cover in , and elements in with in .

For this I think I need the existence of a cover such that comes from . (This is condition (4) in Lemma 7.28.1, tag 03A0).

( I have not seen necessity, but it is sufficient and if the morphism is in the image of a morphism the it is also necessary).

Comment #1217 by JuanPablo on

Ah, is assumed fully faithful (which is stronger than the property I stated before). Please ignore the comment above.

There are also:

  • 2 comment(s) on Section 8.12: Functoriality for stacks

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 04WS. Beware of the difference between the letter 'O' and the digit '0'.