Lemma 7.47.11 (00Z9)—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

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

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

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

are bijective for all $i \in I$. Pick an element $i \in I$ and pick an element $\varphi \in \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(S' \times _ U W, \mathcal{F}_ i)$. 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 $\psi \in \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(S \times _ U W, \mathcal{F}_ i)$. 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 (7)

Comment #3436 by Remy on July 25, 2018 at 08:26

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

Comment #3493 by Johan on August 05, 2018 at 11:34

Thanks and fixed here.

Comment #4290 by Kazuki Masugi on July 04, 2019 at 11:16

I can't make sure that $\varphi \mapsto s$ is right inverse to the map in the statement. Would you please describe the precise way to verify?

Comment #4291 by Laurent Moret-Bailly on July 04, 2019 at 11:52

I think one can simplify the exposition by reducing (via 00Z7(1)) to the case of just one presheaf.

Comment #4455 by Johan on September 01, 2019 at 11:07

Unless somebody tells me that this is wrong I am going to leave it as is. This lemma and all of the material on sieves is never used later in the Stacks project. I would be happy to accept a careful write up of this proof if somebody has the time and interest in doing so. Of course the suggestion of Laurent is a good one: there is an immediate reduction to the case of a single presheaf which should simplify the proof (and the notation used in the proof).

Comment #10925 by Léo Navarro Chafloque on October 20, 2025 at 09:06

"Hence $S'\times_U V$ is in $J(W)$ ". This should be $J(V)$ . Same issue after "This we see from the definition of $J$ , and because $S'\times_U V$ is in $J(W)$ ". Should also be $J(V)$ . In the statement of this lemma, "(...) denote $J(U)$ the set if sieves $S$ on $U$ with the following (...)" for clarity.

Comment #10928 by Léo Navarro Chafloque on October 20, 2025 at 11:23

Here is an argument showing that the construction in the proof is a right inverse to the natural map. Sorry for occasionally dropping the index $_i$ on the sheaf; sometimes I was not able to make it display correctly in the preview.

Take any $h\colon Z\to W$ in the sieve $S'\times_U W$ . The goal is to show that $\varphi_{Z,h}=\mathcal{F}_i(h)(s)\in \mathcal{F}_i(Z).$ To check an equality of sections in $\mathcal{F}_i(Z)$ , we might as well check it locally on the sieve $S\times_U Z$ since this sieve belongs to $J(Z)$ . By the last sentence we actually mean that we want to invoke the bijection

$\mathcal{F}_i(Z)\to \text{Mor}(S\times_U Z,\mathcal{F}_i).$

More precisely, using the above bijection, it suffices to show that for any $h'\colon V\to Z$ in the sieve $S\times_U Z$ , we have $\varphi_{V,hh'}=\mathcal{F}_i (hh')(s).$ But now, this holds by the very construction of $s$ . Indeed, $s$ was defined above as the unique section corresponding to $\psi\colon S\times_U W\to \mathcal{F}_i$ . But $\psi$ was defined by the property that for any element of $S\times_U W$ , for instance $hh'\colon V\to W$ , we have (first equality is the definition of $s$ and the second by that of $\psi$ ) $\mathcal{F}(hh')(s)=\psi_{V,hh'}=s_{V,hh'}\in \mathcal{F}_i (V).$

Finally $s_{V, hh'}$ was defined as the unique section in $\mathcal{F}_i(V)$ such that for any $\sigma \colon C\to V$ in the sieve $S'\times_U V$ we have $\varphi_{C,hh'\sigma}={\mathcal{F}}(\sigma)(s_{V,hh'}).$

In order to conclude, note that $\text{id}_{V}$ is in the sieve $S'\times_U V$ which is therefore the maximal sieve $h_{V}$ on $V$ . Indeed, a morphism $C\to V$ is in the sieve $S'\times_U V$ if and only if it is in $S'$ when composed with $fhh'$ . But $fh\colon Z\to U$ is in $S'$ by initial assumption. So any precomposition of $fhh'$ is in $S'$ by functoriality of sieves.

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4 comment(s) on Section 7.47: Topologies

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