The Stacks project

Lemma 7.28.7. Assume given sites $\mathcal{C}', \mathcal{C}, \mathcal{D}', \mathcal{D}$ and functors

\[ \xymatrix{ \mathcal{C}' \ar[r]_{v'} & \mathcal{C} \\ \mathcal{D}' \ar[r]^ v \ar[u]^{u'} & \mathcal{D} \ar[u]_ u } \]

With notation as in Sections 7.14 and 7.21 assume

  1. $u$ and $u'$ are continuous giving rise to morphisms of sites $f$ and $f'$,

  2. $v$ and $v'$ are cocontinuous giving rise to morphisms of topoi $g$ and $g'$,

  3. $u \circ v = v' \circ u'$, and

  4. $v$ and $v'$ are continuous as well as cocontinuous.

Then1 $f'_* \circ (g')^{-1} = g^{-1} \circ f_*$ and $g'_! \circ (f')^{-1} = f^{-1} \circ g_!$.

Proof. Namely, we have

\[ f'_*(g')^{-1}\mathcal{F} = (u')^ p((v')^ p\mathcal{F})^\# = (u')^ p(v')^ p\mathcal{F} \]

The first equality by definition and the second by Lemma 7.21.5. We have

\[ g^{-1}f_*\mathcal{F} = (v^ pu^ p\mathcal{F})^\# = ((u')^ p(v')^ p\mathcal{F})^\# = (u')^ p(v')^ p\mathcal{F} \]

The first equality by definition, the second because $u \circ v = v' \circ u'$, the third because we already saw that $(u')^ p(v')^ p\mathcal{F}$ is a sheaf. This proves $f'_* \circ (g')^{-1} = g^{-1} \circ f_*$ and the equality $g'_! \circ (f')^{-1} = f^{-1} \circ g_!$ follows by uniqueness of left adjoints. $\square$

[1] In this generality we don't know $f \circ g'$ is equal to $g \circ f'$ as morphisms of topoi (there is a canonical $2$-arrow from the first to the second which may not be an isomorphism).

Comments (0)

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Your email address will not be published.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 0FN2. Beware of the difference between the letter 'O' and the digit '0'.