The Stacks project

Lemma 7.47.11. Let $\mathcal{C}$ be a category. Let $\{ \mathcal{F}_ i \} _{i\in I}$ be a collection of presheaves of sets on $\mathcal{C}$. For each $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ denote $J(U)$ the set of sieves $S$ with the following property: For every morphism $V \to U$, the maps

\[ \mathop{Mor}\nolimits _{\textit{PSh}(\mathcal{C})}(h_ V, \mathcal{F}_ i) \longrightarrow \mathop{Mor}\nolimits _{\textit{PSh}(\mathcal{C})}(S \times _ U V, \mathcal{F}_ i) \]

are bijective for all $i \in I$. Then $J$ defines a topology on $\mathcal{C}$. This topology is the finest topology in which all of the $\mathcal{F}_ i$ are sheaves.

Proof. If we show that $J$ is a topology, then the last statement of the lemma immediately follows. The first and third axioms of a topology are immediately verified. Thus, assume that we have an object $U$, and sieves $S, S'$ of $U$ such that $S \in J(U)$, and for all $V \to U$ in $S(V)$ we have $S' \times _ U V \in J(V)$. We have to show that $S' \in J(U)$. In other words, we have to show that for any $f : W \to U$, the maps

\[ \mathcal{F}_ i(W) = \mathop{Mor}\nolimits _{\textit{PSh}(\mathcal{C})}(h_ W, \mathcal{F}_ i) \longrightarrow \mathop{Mor}\nolimits _{\textit{PSh}(\mathcal{C})}(S' \times _ U W, \mathcal{F}_ i) \]

are bijective for all $i \in I$. Pick an element $i \in I$ and pick an element $\varphi \in \mathop{Mor}\nolimits _{\textit{PSh}(\mathcal{C})}(S' \times _ U W, \mathcal{F}_ i)$. We will construct a section $s \in \mathcal{F}_ i(W)$ mapping to $\varphi $.

Suppose $\alpha : V \to W$ is an element of $S \times _ U W$. According to the definition of pullbacks we see that the composition $f \circ \alpha : V \to W \to U$ is in $S$. Hence $S' \times _ U V$ is in $J(W)$ by assumption on the pair of sieves $S, S'$. Now we have a commutative diagram of presheaves

\[ \xymatrix{ S' \times _ U V \ar[r] \ar[d] & h_ V \ar[d] \\ S' \times _ U W \ar[r] & h_ W } \]

The restriction of $\varphi $ to $S' \times _ U V$ corresponds to an element $s_{V, \alpha } \in \mathcal{F}_ i(V)$. This we see from the definition of $J$, and because $S' \times _ U V$ is in $J(W)$. We leave it to the reader to check that the rule $(V, \alpha ) \mapsto s_{V, \alpha }$ defines an element $\psi \in \mathop{Mor}\nolimits _{\textit{PSh}(\mathcal{C})}(S \times _ U W, \mathcal{F}_ i)$. Since $S \in J(U)$ we see immediately from the definition of $J$ that $\psi $ corresponds to an element $s$ of $\mathcal{F}_ i(W)$.

We leave it to the reader to verify that the construction $\varphi \mapsto s$ is inverse to the natural map displayed above. $\square$

Comments (2)

Comment #3436 by Remy on

"The first and second axioms of a topology are immediately verified." This should be "first and third".

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 00Z9. Beware of the difference between the letter 'O' and the digit '0'.