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.
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.
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.
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 $\psi _ j = \varphi _{\beta (j)}$.
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.
If $\mathcal{U}$ and $\mathcal{V}$ are combinatorially equivalent then they are tautologically equivalent.
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 $\mathcal{U}$.
The relation “being combinatorially equivalent” is an equivalence relation on all families of morphisms with fixed target.
The relation “being tautologically equivalent” is an equivalence relation on all families of morphisms with fixed target.
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 $\chi _ j : V_ j \to U_{\beta (j)}$ 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 $s_ j = \chi _ j^*s_{\beta (j)}$. For any pair $(j, j') \in J \times J$ the morphism $\chi _ j \times _{\text{id}_ U} \chi _{j'} : V_ j \times _ U V_{j'} \to U_{\beta (j)} \times _ U U_{\beta (j')}$ 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 $\mathcal{V} = \{ V_ j \to U\} _{j \in J} \to \mathcal{U} = \{ U_ i \to U\} _{i \in I}$ be a morphism of families of maps with fixed target of $\mathcal{C}$ given by $\text{id} : U \to U$, $\alpha : J \to I$ and $f_ j : V_ j \to U_{\alpha (j)}$. Let $\mathcal{F}$ be a presheaf on $\mathcal{C}$. If $\mathcal{F}(U) \to \prod _{j \in J} \mathcal{F}(V_ j)$ is injective then $\mathcal{F}(U) \to \prod _{i \in I} \mathcal{F}(U_ i)$ is injective.
Proof.
Omitted.
$\square$
Lemma 7.8.6. Let $\mathcal{C}$ be a category. Let $\mathcal{V} = \{ V_ j \to U\} _{j \in J} \to \mathcal{U} = \{ U_ i \to U\} _{i \in I}$ be a morphism of families of maps with fixed target of $\mathcal{C}$ given by $\text{id} : U \to U$, $\alpha : J \to I$ and $f_ j : V_ j \to U_{\alpha (j)}$. Let $\mathcal{F}$ be a presheaf on $\mathcal{C}$. If
the fibre products $U_ i \times _ U U_{i'}$, $U_ i \times _ U V_ j$, $V_ j \times _ U V_{j'}$ exist,
$\mathcal{F}$ satisfies the sheaf condition with respect to $\mathcal{V}$, and
for every $i \in I$ the map $\mathcal{F}(U_ i) \to \prod _{j \in J} \mathcal{F}(V_ j \times _ U U_ i)$ is injective.
Then $\mathcal{F}$ satisfies the sheaf condition with respect to $\mathcal{U}$.
Proof.
By Lemma 7.8.5 the map $\mathcal{F}(U) \to \prod \mathcal{F}(U_ i)$ is injective. 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 $\mathcal{V}$, 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 $s_ i$ on $U_ i$ for any $i \in I$. Since $\mathcal{F}$ satisfies (3) it suffices to show that $s$ and $s_ i$ restrict to the same element over $U_ i \times _ U V_ j$ for all $j \in J$. To see this we use
\[ s|_{U_ i \times _ U V_ j} = s_ j|_{U_ i \times _ U V_ j} = (\text{id} \times f_ j)^*s_{\alpha (j)}|_{U_ i \times _ U U_{\alpha (j)}} = (\text{id} \times f_ j)^*s_ i|_{U_ i \times _ U U_{\alpha (j)}} = s_ i|_{U_ i \times _ U V_ j} \]
as desired.
$\square$
Lemma 7.8.7. 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 $\mathcal{C}$.
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.
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.
Proof of (2). Let $\mathcal{F}$ be a sheaf of sets for the site $(\mathcal{C}, \text{Cov}_2)$. Let $\mathcal{U} \in \text{Cov}_1$, say $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$. By assumption we may 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 $f_ j : V_ j \to U_{\alpha (j)}$. Observe that $\mathcal{F}$ satisfies the sheaf condition for $\mathcal{V}$ and for the coverings $\{ V_ j \times _ U U_ i \to U_ i\} _{j \in J}$ as these are in $\text{Cov}_2$. Hence $\mathcal{F}$ satisfies the sheaf condition for $\mathcal{U}$ by Lemma 7.8.6.
$\square$
Lemma 7.8.8. 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 7.8.7 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.
$\square$
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