The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

7.48 The topology defined by a site

Suppose that $\mathcal{C}$ is a category, and suppose that $\text{Cov}_1(\mathcal{C})$ and $\text{Cov}_2(\mathcal{C})$ are sets of coverings that define the structure of a site on $\mathcal{C}$. In this situation it can happen that the categories of sheaves (of sets) for $\text{Cov}_1(\mathcal{C})$ and $\text{Cov}_2(\mathcal{C})$ are the same, see for example Lemma 7.8.5.

It turns out that the category of sheaves on $\mathcal{C}$ with respect to some topology $J$ determines and is determined by the topology $J$. This is a nontrivial statement which we will address later, see Theorem 7.50.2.

Accepting this for the moment it makes sense to study the topology determined by a site.

Lemma 7.48.1. Let $\mathcal{C}$ be a site with coverings $\text{Cov}(\mathcal{C})$. For every object $U$ of $\mathcal{C}$, let $J(U)$ denote the set of sieves $S$ on $U$ with the following property: there exists a covering $\{ f_ i : U_ i \to U\} _{i\in I} \in \text{Cov}(\mathcal{C})$ so that the sieve $S'$ generated by the $f_ i$ (see Definition 7.47.3) is contained in $S$.

  1. This $J$ is a topology on $\mathcal{C}$.

  2. A presheaf $\mathcal{F}$ is a sheaf for this topology (see Definition 7.47.10) if and only if it is a sheaf on the site (see Definition 7.7.1).

Proof. To prove the first assertion we just note that axioms (1), (2) and (3) of the definition of a site (Definition 7.6.2) directly imply the axioms (3), (2) and (1) of the definition of a topology (Definition 7.47.6). As an example we prove $J$ has property (2). Namely, let $U$ be an object of $\mathcal{C}$, let $S, S'$ be sieves on $U$ such that $S \in J(U)$, and such that for every $V \to U$ in $S(V)$ we have $S' \times _ U V \in J(V)$. By definition of $J(U)$ we can find a covering $\{ f_ i : U_ i \to U\} $ of the site such that $S$ the image of $h_{U_ i} \to h_ U$ is contained in $S$. Since each $S'\times _ U U_ i$ is in $J(U_ i)$ we see that there are coverings $\{ U_{ij} \to U_ i\} $ of the site such that $h_{U_{ij}} \to h_{U_ i}$ is contained in $S' \times _ U U_ i$. By definition of the base change this means that $h_{U_{ij}} \to h_ U$ is contained in the subpresheaf $S' \subset h_ U$. By axiom (2) for sites we see that $\{ U_{ij} \to U\} $ is a covering of $U$ and we conclude that $S' \in J(U)$ by definition of $J$.

Let $\mathcal{F}$ be a presheaf. Suppose that $\mathcal{F}$ is a sheaf in the topology $J$. We will show that $\mathcal{F}$ is a sheaf on the site as well. Let $\{ f_ i : U_ i \to U\} _{i\in I}$ be a covering of the site. Let $s_ i \in \mathcal{F}(U_ i)$ be a family of sections such that $s_ i|_{U_ i \times _ U U_ j} = s_ j|_{U_ i \times _ U U_ j}$ for all $i, j$. We have to show that there exists a unique section $s \in \mathcal{F}(U)$ restricting back to the $s_ i$ on the $U_ i$. Let $S \subset h_ U$ be the sieve generated by the $f_ i$. Note that $S \in J(U)$ by definition. In stead of constructing $s$, by the sheaf condition in the topology, it suffices to construct an element

\[ \varphi \in \mathop{Mor}\nolimits _{\textit{PSh}(\mathcal{C})}(S, \mathcal{F}). \]

Take $\alpha \in S(T)$ for some object $T \in \mathcal{U}$. This means exactly that $\alpha : T \to U$ is a morphism which factors through $f_ i$ for some $i\in I$ (and maybe more than $1$). Pick such an index $i$ and a factorization $\alpha = f_ i \circ \alpha _ i$. Define $\varphi (\alpha ) = \alpha _ i^* s_ i$. If $i'$, $\alpha = f_ i \circ \alpha _{i'}'$ is a second choice, then $\alpha _ i^* s_ i = (\alpha _{i'}')^* s_{i'}$ exactly because of our condition $s_ i|_{U_ i \times _ U U_ j} = s_ j|_{U_ i \times _ U U_ j}$ for all $i, j$. Thus $\varphi (\alpha )$ is well defined. We leave it to the reader to verify that $\varphi $, which in turn determines $s$ is correct in the sense that $s$ restricts back to $s_ i$.

Let $\mathcal{F}$ be a presheaf. Suppose that $\mathcal{F}$ is a sheaf on the site $(\mathcal{C}, \text{Cov}(\mathcal{C}))$. We will show that $\mathcal{F}$ is a sheaf for the topology $J$ as well. Let $U$ be an object of $\mathcal{C}$. Let $S$ be a covering sieve on $U$ with respect to the topology $J$. Let

\[ \varphi \in \mathop{Mor}\nolimits _{\textit{PSh}(\mathcal{C})}(S, \mathcal{F}). \]

We have to show there is a unique element in $\mathcal{F}(U) = \mathop{Mor}\nolimits _{\textit{PSh}(\mathcal{C})}(h_ U, \mathcal{F})$ which restricts back to $\varphi $. By definition there exists a covering $\{ f_ i : U_ i \to U\} _{i\in I} \in \text{Cov}(\mathcal{C})$ such that $f_ i : U_ i \in U$ belongs to $S(U_ i)$. Hence we can set $s_ i = \varphi (f_ i) \in \mathcal{F}(U_ i)$. Then it is a pleasant exercise to see that $s_ i|_{U_ i \times _ U U_ j} = s_ j|_{U_ i \times _ U U_ j}$ for all $i, j$. Thus we obtain the desired section $s$ by the sheaf condition for $\mathcal{F}$ on the site $(\mathcal{C}, \text{Cov}(\mathcal{C}))$. Details left to the reader. $\square$

Definition 7.48.2. Let $\mathcal{C}$ be a site with coverings $\text{Cov}(\mathcal{C})$. The topology associated to $\mathcal{C}$ is the topology $J$ constructed in Lemma 7.48.1 above.

Let $\mathcal{C}$ be a category. Let $\text{Cov}_1(\mathcal{C})$ and $\text{Cov}_2(\mathcal{C})$ be two coverings defining the structure of a site on $\mathcal{C}$. It may very well happen that the topologies defined by these are the same. If this happens then we say $\text{Cov}_1(\mathcal{C})$ and $\text{Cov}_2(\mathcal{C})$ define the same topology on $\mathcal{C}$. And if this happens then the categories of sheaves are the same, by Lemma 7.48.1.

It is usually the case that we only care about the topology defined by a collection of coverings, and we view the possibility of choosing different sets of coverings as a tool to study the topology.

Remark 7.48.3. Enlarging the class of coverings. Clearly, if $\text{Cov}(\mathcal{C})$ defines the structure of a site on $\mathcal{C}$ then we may add to $\mathcal{C}$ any set of families of morphisms with fixed target tautologically equivalent (see Definition 7.8.2) to elements of $\text{Cov}(\mathcal{C})$ without changing the topology.

Remark 7.48.4. Shrinking the class of coverings. Let $\mathcal{C}$ be a site. Consider the set

\[ \mathcal{S} = P(\text{Arrows}(\mathcal{C})) \times \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}) \]

where $P(\text{Arrows}(\mathcal{C}))$ is the power set of the set of morphisms, i.e., the set of all sets of morphisms. Let $\mathcal{S}_\tau \subset \mathcal{S}$ be the subset consisting of those $(T, U) \in \mathcal{S}$ such that (a) all $\varphi \in T$ have target $U$, (b) the collection $\{ \varphi \} _{\varphi \in T}$ is tautologically equivalent (see Definition 7.8.2) to some covering in $\text{Cov}(\mathcal{C})$. Clearly, considering the elements of $\mathcal{S}_\tau $ as the coverings, we do not get exactly the notion of a site as defined in Definition 7.6.2. The structure $(\mathcal{C}, \mathcal{S}_\tau )$ we get satisfies slightly modified conditions. The modified conditions are:

  1. $\text{Cov}(\mathcal{C}) \subset P(\text{Arrows}(\mathcal{C})) \times \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$,

  2. If $V \to U$ is an isomorphism then $(\{ V \to U\} , U) \in \text{Cov}(\mathcal{C})$.

  3. If $(T, U) \in \text{Cov}(\mathcal{C})$ and for $f : U' \to U$ in $T$ we are given $(T_ f, U') \in \text{Cov}(\mathcal{C})$, then setting $T' = \{ f \circ f' \mid f \in T,\ f' \in T_ f\} $, we get $(T', U) \in \text{Cov}(\mathcal{C})$.

  4. If $(T, U) \in \text{Cov}(\mathcal{C})$ and $g : V \to U$ is a morphism of $\mathcal{C}$ then

    1. $U' \times _{f, U, g} V$ exists for $f : U' \to U$ in $T$, and

    2. setting $T' = \{ \text{pr}_2 : U' \times _{f, U, g} V \to V \mid f : U' \to U \in T\} $ for some choice of fibre products we get $(T', V) \in \text{Cov}(\mathcal{C})$.

And it is easy to verify that, given a structure satisfying (0') – (3') above, then after suitably enlarging $\text{Cov}(\mathcal{C})$ (compare Sets, Section 3.11) we get a site. Obviously there is little difference between this notion and the actual notion of a site, at least from the point of view of the topology. There are two benefits: because of condition (0') above the coverings automatically form a set, and because of (0') the totality of all structures of this type forms a set as well. The price you pay for this is that you have to keep writing “tautologically equivalent” everywhere.


Comments (2)

Comment #370 by Baptiste Calmès on

Shouldn't the list of conditions be numbered (0'),...,(3') instead of (1),...,(4)? (see paragraph after the list, in which, I guess, the idea is that the numbering should match the numbering of the definition of a covering.)

Comment #371 by on

This is one of the bugs on the website: the code to parse LaTeX to HTML is not aware of all the possible ways to renumber lists. As you can see in the source code (https://github.com/stacks/stacks-project/blob/master/sites.tex#L9148-9192, although the line numbers are of course susceptible to change) the PDF output should (and does) have the correct numbering.

I'll fix this particular instance later today or next week, including some of the other possible ways of renumbering lists. But it's impossible to get all the possibilities because that would require writing a real TeX parser.


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