The Stacks project

Lemma 100.3.6. Let $p : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. The following are equivalent

  1. the diagonal $\Delta : \mathcal{X} \to \mathcal{X} \times _\mathcal {Y} \mathcal{X}$ is limit preserving, and

  2. for every directed limit $U = \mathop{\mathrm{lim}}\nolimits U_ i$ of affine schemes over $S$ the functor

    \[ \mathop{\mathrm{colim}}\nolimits \mathcal{X}_{U_ i} \longrightarrow \mathcal{X}_ U \times _{\mathcal{Y}_ U} \mathop{\mathrm{colim}}\nolimits \mathcal{Y}_{U_ i} \]

    is fully faithful.

In particular, if $p$ is limit preserving, then $\Delta $ is too.

Proof. Let $U = \mathop{\mathrm{lim}}\nolimits U_ i$ be a directed limit of affine schemes over $S$. We claim that the functor

\[ \mathop{\mathrm{colim}}\nolimits \mathcal{X}_{U_ i} \longrightarrow \mathcal{X}_ U \times _{\mathcal{Y}_ U} \mathop{\mathrm{colim}}\nolimits \mathcal{Y}_{U_ i} \]

is fully faithful if and only if the functor

\[ \mathop{\mathrm{colim}}\nolimits \mathcal{X}_{U_ i} \longrightarrow \mathcal{X}_ U \times _{(\mathcal{X} \times _\mathcal {Y} \mathcal{X})_ U} \mathop{\mathrm{colim}}\nolimits (\mathcal{X} \times _\mathcal {Y} \mathcal{X})_{U_ i} \]

is an equivalence. This will prove the lemma. Since $(\mathcal{X} \times _\mathcal {Y} \mathcal{X})_ U = \mathcal{X}_ U \times _{\mathcal{Y}_ U} \mathcal{X}_ U$ and $(\mathcal{X} \times _\mathcal {Y} \mathcal{X})_{U_ i} = \mathcal{X}_{U_ i} \times _{\mathcal{Y}_{U_ i}} \mathcal{X}_{U_ i}$ this is a purely category theoretic assertion which we discuss in the next paragraph.

Let $\mathcal{I}$ be a filtered index category. Let $(\mathcal{C}_ i)$ and $(\mathcal{D}_ i)$ be systems of groupoids over $\mathcal{I}$. Let $p : (\mathcal{C}_ i) \to (\mathcal{D}_ i)$ be a map of systems of groupoids over $\mathcal{I}$. Suppose we have a functor $p : \mathcal{C} \to \mathcal{D}$ of groupoids and functors $f : \mathop{\mathrm{colim}}\nolimits \mathcal{C}_ i \to \mathcal{C}$ and $g : \mathop{\mathrm{colim}}\nolimits \mathcal{D}_ i \to \mathcal{D}$ fitting into a commutative diagram

\[ \xymatrix{ \mathop{\mathrm{colim}}\nolimits \mathcal{C}_ i \ar[d]_ p \ar[r]_ f & \mathcal{C} \ar[d]^ p \\ \mathop{\mathrm{colim}}\nolimits \mathcal{D}_ i \ar[r]^ g & \mathcal{D} } \]

Then we claim that

\[ A : \mathop{\mathrm{colim}}\nolimits \mathcal{C}_ i \longrightarrow \mathcal{C} \times _\mathcal {D} \mathop{\mathrm{colim}}\nolimits \mathcal{D}_ i \]

is fully faithful if and only if the functor

\[ B : \mathop{\mathrm{colim}}\nolimits \mathcal{C}_ i \longrightarrow \mathcal{C} \times _{\Delta , \mathcal{C} \times _\mathcal {D} \mathcal{C}, f \times _ g f} \mathop{\mathrm{colim}}\nolimits (\mathcal{C}_ i \times _{\mathcal{D}_ i} \mathcal{C}_ i) \]

is an equivalence. Set $\mathcal{C}' = \mathop{\mathrm{colim}}\nolimits \mathcal{C}_ i$ and $\mathcal{D}' = \mathop{\mathrm{colim}}\nolimits \mathcal{D}_ i$. Since $2$-fibre products commute with filtered colimits we see that $A$ and $B$ become the functors

\[ A' : \mathcal{C}' \to \mathcal{C} \times _\mathcal {D} \mathcal{D}' \quad \text{and}\quad B' : \mathcal{C}' \longrightarrow \mathcal{C} \times _{\Delta , \mathcal{C} \times _\mathcal {D} \mathcal{C}, f \times _ g f} (\mathcal{C}' \times _{\mathcal{D}'} \mathcal{C}') \]

Thus it suffices to prove that if

\[ \xymatrix{ \mathcal{C}' \ar[d]_ p \ar[r]_ f & \mathcal{C} \ar[d]^ p \\ \mathcal{D}' \ar[r]^ g & \mathcal{D} } \]

is a commutative diagram of groupoids, then $A'$ is fully faithful if and only if $B'$ is an equivalence. This follows from Categories, Lemma 4.35.9 (with trivial, i.e., punctual, base category) because

\[ \mathcal{C} \times _{\Delta , \mathcal{C} \times _\mathcal {D} \mathcal{C}, f \times _ g f} (\mathcal{C}' \times _{\mathcal{D}'} \mathcal{C}') = \mathcal{C}' \times _{A', \mathcal{C} \times _\mathcal {D} \mathcal{D}', A'} \mathcal{C}' \]

This finishes the proof. $\square$

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 0CMW. Beware of the difference between the letter 'O' and the digit '0'.