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*}

7.8 Families of morphisms with fixed target

This section is meant to introduce some notions regarding families of morphisms with the same target.

Definition 7.8.1. Let $\mathcal{C}$ be a category. Let $\mathcal{U} = \{ U_ i \to U\} _{i\in I}$ be a family of morphisms of $\mathcal{C}$ with fixed target. Let $\mathcal{V} = \{ V_ j \to V\} _{j\in J}$ be another.

  1. A morphism of families of maps with fixed target of $\mathcal{C}$ from $\mathcal{U}$ to $\mathcal{V}$, or simply a morphism from $\mathcal{U}$ to $\mathcal{V}$ is given by a morphism $U \to V$, a map of sets $\alpha : I \to J$ and for each $i\in I$ a morphism $U_ i \to V_{\alpha (i)}$ such that the diagram

    \[ \xymatrix{ U_ i \ar[r] \ar[d] & V_{\alpha (i)} \ar[d] \\ U \ar[r] & V } \]

    is commutative.

  2. In the special case that $U = V$ and $U \to V$ is the identity we call $\mathcal{U}$ a refinement of the family $\mathcal{V}$.

A trivial but important remark is that if $\mathcal{V} = \{ V_ j \to V\} _{j \in J}$ is the empty family of maps, i.e., if $J = \emptyset $, then no family $\mathcal{U} = \{ U_ i \to V\} _{i \in I}$ with $I \not= \emptyset $ can refine $\mathcal{V}$!

Definition 7.8.2. Let $\mathcal{C}$ be a category. Let $\mathcal{U} = \{ \varphi _ i : U_ i \to U\} _{i\in I}$, and $\mathcal{V} = \{ \psi _ j : V_ j \to U\} _{j\in J}$ be two families of morphisms with fixed target.

  1. We say $\mathcal{U}$ and $\mathcal{V}$ are combinatorially equivalent if there exist maps $\alpha : I \to J$ and $\beta : J\to I$ such that $\varphi _ i = \psi _{\alpha (i)}$ and $\psi _ j = \varphi _{\beta (j)}$.

  2. We say $\mathcal{U}$ and $\mathcal{V}$ are tautologically equivalent if there exist maps $\alpha : I \to J$ and $\beta : J\to I$ and for all $i\in I$ and $j \in J$ commutative diagrams

    \[ \xymatrix{ U_ i \ar[rd] \ar[rr] & & V_{\alpha (i)} \ar[ld] & & V_ j \ar[rd] \ar[rr] & & U_{\beta (j)} \ar[ld] \\ & U & & & & U & } \]

    with isomorphisms as horizontal arrows.

Lemma 7.8.3. Let $\mathcal{C}$ be a category. Let $\mathcal{U} = \{ \varphi _ i : U_ i \to U\} _{i\in I}$, and $\mathcal{V} = \{ \psi _ j : V_ j \to U\} _{j\in J}$ be two families of morphisms with the same fixed target.

  1. If $\mathcal{U}$ and $\mathcal{V}$ are combinatorially equivalent then they are tautologically equivalent.

  2. If $\mathcal{U}$ and $\mathcal{V}$ are tautologically equivalent then $\mathcal{U}$ is a refinement of $\mathcal{V}$ and $\mathcal{V}$ is a refinement of $\mathcal{U}$.

  3. The relation “being combinatorially equivalent” is an equivalence relation on all families of morphisms with fixed target.

  4. The relation “being tautologically equivalent” is an equivalence relation on all families of morphisms with fixed target.

  5. The relation “$\mathcal{U}$ refines $\mathcal{V}$ and $\mathcal{V}$ refines $\mathcal{U}$” is an equivalence relation on all families of morphisms with fixed target.

Proof. Omitted. $\square$

In the following lemma, given a category $\mathcal{C}$, a presheaf $\mathcal{F}$ on $\mathcal{C}$, a family $\mathcal{U} = \{ U_ i \to U\} _{i\in I}$ such that all fibre products $U_ i \times _ U U_{i'}$ exist, we say that the sheaf condition for $\mathcal{F}$ with respect to $\mathcal{U}$ holds if the diagram (7.7.1.1) is an equalizer diagram.

Lemma 7.8.4. Let $\mathcal{C}$ be a category. Let $\mathcal{U} = \{ \varphi _ i : U_ i \to U\} _{i\in I}$, and $\mathcal{V} = \{ \psi _ j : V_ j \to U\} _{j\in J}$ be two families of morphisms with the same fixed target. Assume that the fibre products $U_ i \times _ U U_{i'}$ and $V_ j \times _ U V_{j'}$ exist. If $\mathcal{U}$ and $\mathcal{V}$ are tautologically equivalent, then for any presheaf $\mathcal{F}$ on $\mathcal{C}$ the sheaf condition for $\mathcal{F}$ with respect to $\mathcal{U}$ is equivalent to the sheaf condition for $\mathcal{F}$ with respect to $\mathcal{V}$.

Proof. First, note that if $\varphi : A \to B$ is an isomorphism in the category $\mathcal{C}$, then $\varphi ^* : \mathcal{F}(B) \to \mathcal{F}(A)$ is an isomorphism. Let $\beta : J \to I$ be a map and let $\psi _ j : V_ j \to U_{\beta (j)}$ be isomorphisms over $U$ which are assumed to exist by hypothesis. Let us show that the sheaf condition for $\mathcal{V}$ implies the sheaf condition for $\mathcal{U}$. Suppose given sections $s_ i \in \mathcal{F}(U_ i)$ such that

\[ s_ i|_{U_ i \times _ U U_{i'}} = s_{i'}|_{U_ i \times _ U U_{i'}} \]

in $\mathcal{F}(U_ i \times _ U U_{i'})$ for all pairs $(i, i') \in I \times I$. Then we can define $s_ j = \psi _ j^*s_{\beta (j)}$. For any pair $(j, j') \in J \times J$ the morphism $\psi _ j \times _{\text{id}_ U} \psi _{j'} : V_ j \times _ U V_{j'} \to U_{\beta (j)} \times _ U U_{\beta (j')}$ is an isomorphism as well. Hence by transport of structure we see that

\[ s_ j|_{V_ j \times _ U V_{j'}} = s_{j'}|_{V_ j \times _ U V_{j'}} \]

as well. The sheaf condition w.r.t. $\mathcal{V}$ implies there exists a unique $s$ such that $s|_{V_ j} = s_ j$ for all $j \in J$. By the first remark of the proof this implies that $s|_{U_ i} = s_ i$ for all $i \in \mathop{\mathrm{Im}}(\beta )$ as well. Suppose that $i \in I$, $i \not\in \mathop{\mathrm{Im}}(\beta )$. For such an $i$ we have isomorphisms $U_ i \to V_{\alpha (i)} \to U_{\beta (\alpha (i))}$ over $U$. This gives a morphism $U_ i \to U_ i \times _ U U_{\beta (\alpha (i))}$ which is a section of the projection. Because $s_ i$ and $s_{\beta (\alpha (i))}$ restrict to the same element on the fibre product we conclude that $s_{\beta (\alpha (i))}$ pulls back to $s_ i$ via $U_ i \to U_{\beta (\alpha (i))}$. Thus we see that also $s_ i = s|_{U_ i}$ as desired. $\square$

Lemma 7.8.5. Let $\mathcal{C}$ be a category. Let $\text{Cov}_ i$, $i = 1, 2$ be two sets of families of morphisms with fixed target which each define the structure of a site on $\mathcal{C}$.

  1. If every $\mathcal{U} \in \text{Cov}_1$ is tautologically equivalent to some $\mathcal{V} \in \text{Cov}_2$, then $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}, \text{Cov}_2) \subset \mathop{\mathit{Sh}}\nolimits (\mathcal{C}, \text{Cov}_1)$. If also, every $\mathcal{U} \in \text{Cov}_2$ is tautologically equivalent to some $\mathcal{V} \in \text{Cov}_1$ then the category of sheaves are equal.

  2. Suppose that for each $\mathcal{U} \in \text{Cov}_1$ there exists a $\mathcal{V} \in \text{Cov}_2$ such that $\mathcal{V}$ refines $\mathcal{U}$. In this case $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}, \text{Cov}_2) \subset \mathop{\mathit{Sh}}\nolimits (\mathcal{C}, \text{Cov}_1)$. If also for every $\mathcal{U} \in \text{Cov}_2$ there exists a $\mathcal{V} \in \text{Cov}_1$ such that $\mathcal{V}$ refines $\mathcal{U}$, then the categories of sheaves are equal.

Proof. Part (1) follows directly from Lemma 7.8.4 and the definitions.

We advise the reader to skip the proof of (2) on a first reading. Let $\mathcal{F}$ be a sheaf of sets for the site $(\mathcal{C}, \text{Cov}_2)$. Let $\mathcal{U} \in \text{Cov}_1$, say $\mathcal{U} = \{ U_ i \to U\} _{i\in I}$. Choose a refinement $\mathcal{V} \in \text{Cov}_2$ of $\mathcal{U}$, say $\mathcal{V} = \{ V_ j \to U\} _{j\in J}$ and refinement given by $\alpha : J \to I$ and $f_ j : V_ j \to U_{\alpha (j)}$.

First let $s, s' \in \mathcal{F}(U)$. If for all $i \in I$ we have $s|_{U_ i} = s'|_{U_ i}$, then we also have $s|_{V_ j} = s'|_{V_ j}$ for all $j \in J$. This implies that $s = s'$ by the sheaf condition for $\mathcal{F}$ with respect to $\text{Cov}_2$. Hence we see that the unicity in the sheaf condition for $\mathcal{F}$ and the site $(\mathcal{C}, \text{Cov}_1)$ holds.

Next, suppose given $s_ i \in \mathcal{F}(U_ i)$ such that $s_ i|_{U_ i \times _ U U_{i'}} = s_{i'}|_{U_ i \times _ U U_{i'}}$ for all $i, i' \in I$. Set $s_ j = f_ j^*(s_{\alpha (j)}) \in \mathcal{F}(V_ j)$. Since the morphisms $f_ j$ are morphisms over $U$ we obtain induced morphisms $f_{jj'} : V_ j \times _ U V_{j'} \to U_{\alpha (i)} \times _ U U_{\alpha (i')}$ compatible with the $f_ j, f_{j'}$ via the projection maps. It follows that

\[ s_ j|_{V_ j \times _ U V_{j'}} = f_{jj'}^*(s_{\alpha (j)}|_{U_{\alpha (j)} \times _ U U_{\alpha (j')}}) = f_{jj'}^*(s_{\alpha (j')}|_{U_{\alpha (j)} \times _ U U_{\alpha (j')}}) = s_{j'}|_{V_ j \times _ U V_{j'}} \]

for all $j, j' \in J$. Hence, by the sheaf condition for $\mathcal{F}$ with respect to $\text{Cov}_2$, we get a section $s \in \mathcal{F}(U)$ which restricts to $s_ j$ on each $V_ j$. We are done if we show $s$ restricts to $s_{i_0}$ on $U_{i_0}$ for any $i_0 \in I$. For each $i_0 \in I$ the family $\mathcal{U}' = \{ U_ i \times _ U U_{i_0} \to U_{i_0}\} _{i \in I}$ is an element of $\text{Cov}_1$ by the axioms of a site. Also, the family $\mathcal{V}' = \{ V_ j \times _ U U_{i_0} \to U_{i_0}\} _{j \in J}$ is an element of $\text{Cov}_2$. Then $\mathcal{V}'$ refines $\mathcal{U}'$ via $\alpha : J \to I$ and the maps $f'_ j = f_ j \times \text{id}_{U_{i_0}}$. The element $s_{i_0}$ restricts to $s_ i|_{U_ i \times _ U U_{i_0}}$ on the members of the covering $\mathcal{U}'$ and hence via $(f_ j')^*$ to the elements $s_ j|_{V_ j \times _ U U_{i_0}}$ on the members of the covering $\mathcal{V}'$. By construction of $s$ this is the same as the family of restrictions of $s|_{U_{i_0}}$ to the members of the covering $\mathcal{V}'$. Hence by the sheaf condition for $\mathcal{F}$ with respect to $\text{Cov}_2$ we see that $s|_{U_{i_0}} = s_{i_0}$ as desired. $\square$

Lemma 7.8.6. Let $\mathcal{C}$ be a category. Let $\text{Cov}(\mathcal{C})$ be a proper class of coverings satisfying conditions (1), (2) and (3) of Definition 7.6.2. Let $\text{Cov}_1, \text{Cov}_2 \subset \text{Cov}(\mathcal{C})$ be two subsets of $\text{Cov}(\mathcal{C})$ which endow $\mathcal{C}$ with the structure of a site. If every covering $\mathcal{U} \in \text{Cov}(\mathcal{C})$ is combinatorially equivalent to a covering in $\text{Cov}_1$ and combinatorially equivalent to a covering in $\text{Cov}_2$, then $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}, \text{Cov}_1) = \mathop{\mathit{Sh}}\nolimits (\mathcal{C}, \text{Cov}_2)$.

Proof. This is clear from Lemmas 7.8.5 and 7.8.3 above as the hypothesis implies that every covering $\mathcal{U} \in \text{Cov}_1 \subset \text{Cov}(\mathcal{C})$ is combinatorially equivalent to an element of $\text{Cov}_2$, and similarly with the roles of $\text{Cov}_1$ and $\text{Cov}_2$ reversed. $\square$


Comments (4)

Comment #862 by Amit Hogadi on

minor typo: In 7.8.5 line 4, 'quivalent' should be 'equivalent'

Comment #2308 by Dominic Wynter on

Minor typo: in the comment after the first definition, the definition of should read , and not .


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