## 3.11 Coverings of a site

Suppose that $\mathcal{C}$ is a category (as in Categories, Definition 4.2.1) and that $\text{Cov}(\mathcal{C})$ is a proper class of coverings satisfying properties (1), (2), and (3) of Sites, Definition 7.6.2. We list them here:

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

2. If $\{ U_ i \to U\} _{i\in I} \in \text{Cov}(\mathcal{C})$ and for each $i$ we have $\{ V_{ij} \to U_ i\} _{j\in J_ i} \in \text{Cov}(\mathcal{C})$, then $\{ V_{ij} \to U\} _{i \in I, j\in J_ i} \in \text{Cov}(\mathcal{C})$.

3. If $\{ U_ i \to U\} _{i\in I}\in \text{Cov}(\mathcal{C})$ and $V \to U$ is a morphism of $\mathcal{C}$, then $U_ i \times _ U V$ exists for all $i$ and $\{ U_ i \times _ U V \to V \} _{i\in I} \in \text{Cov}(\mathcal{C})$.

For an ordinal $\alpha$, we set $\text{Cov}(\mathcal{C})_\alpha = \text{Cov}(\mathcal{C}) \cap V_\alpha$. Given an ordinal $\alpha$ and a cardinal $\kappa$, we set $\text{Cov}(\mathcal{C})_{\kappa , \alpha }$ equal to the set of elements $\mathcal{U} = \{ \varphi _ i : U_ i \to U\} _{i\in I} \in \text{Cov}(\mathcal{C})_\alpha$ such that $|I| \leq \kappa$.

We recall the following notion, see Sites, Definition 7.8.2. Two families of morphisms, $\{ \varphi _ i : U_ i \to U\} _{i\in I}$ and $\{ \psi _ j : W_ j \to U\} _{j\in J}$, with the same target of $\mathcal{C}$ are called combinatorially equivalent if there exist maps $\alpha : I \to J$ and $\beta : J\to I$ such that $\varphi _ i = \psi _{\alpha (i)}$ and $\psi _ j = \varphi _{\beta (j)}$. This defines an equivalence relation on families of morphisms having a fixed target.

Lemma 3.11.1. With notations as above. Let $\text{Cov}_0 \subset \text{Cov}(\mathcal{C})$ be a set contained in $\text{Cov}(\mathcal{C})$. There exist a cardinal $\kappa$ and a limit ordinal $\alpha$ with the following properties:

1. We have $\text{Cov}_0 \subset \text{Cov}(\mathcal{C})_{\kappa , \alpha }$.

2. The set of coverings $\text{Cov}(\mathcal{C})_{\kappa , \alpha }$ satisfies (1), (2), and (3) of Sites, Definition 7.6.2 (see above). In other words $(\mathcal{C}, \text{Cov}(\mathcal{C})_{\kappa , \alpha })$ is a site.

3. Every covering in $\text{Cov}(\mathcal{C})$ is combinatorially equivalent to a covering in $\text{Cov}(\mathcal{C})_{\kappa , \alpha }$.

Proof. To prove this, we first consider the set $\mathcal{S}$ of all sets of morphisms of $\mathcal{C}$ with fixed target. In other words, an element of $\mathcal{S}$ is a subset $T$ of $\text{Arrows}(\mathcal{C})$ such that all elements of $T$ have the same target. Given a family $\mathcal{U} = \{ \varphi _ i : U_ i \to U\} _{i\in I}$ of morphisms with fixed target, we define $Supp(\mathcal{U}) = \{ \varphi \in \text{Arrows}(\mathcal{C}) \mid \exists i\in I, \varphi = \varphi _ i\}$. Note that two families $\mathcal{U} = \{ \varphi _ i : U_ i \to U\} _{i\in I}$ and $\mathcal{V} = \{ V_ j \to V\} _{j \in J}$ are combinatorially equivalent if and only if $Supp(\mathcal{U}) = Supp(\mathcal{V})$. Next, we define $\mathcal{S}_\tau \subset \mathcal{S}$ to be the subset $\mathcal{S}_\tau = \{ T \in \mathcal{S} \mid \exists \ \mathcal{U} \in \text{Cov}(\mathcal{C}) \ T = Supp(\mathcal{U})\}$. For every element $T \in \mathcal{S}_\tau$, set $\beta (T)$ to equal the least ordinal $\beta$ such that there exists a $\mathcal{U} \in \text{Cov}(\mathcal{C})_\beta$ such that $T = \text{Supp}(\mathcal{U})$. Finally, set $\beta _0 = \sup _{T \in S_\tau } \beta (T)$. At this point it follows that every $\mathcal{U} \in \text{Cov}(\mathcal{C})$ is combinatorially equivalent to some element of $\text{Cov}(\mathcal{C})_{\beta _0}$.

Let $\kappa$ be the maximum of $\aleph _0$, the cardinality $|\text{Arrows}(\mathcal{C})|$,

$\sup \nolimits _{\{ U_ i \to U\} _{i\in I} \in \text{Cov}(\mathcal{C})_{\beta _0}} |I|, \quad \text{and}\quad \sup \nolimits _{\{ U_ i \to U\} _{i\in I} \in \text{Cov}_0} |I|.$

Since $\kappa$ is an infinite cardinal, we have $\kappa \otimes \kappa = \kappa$. Note that obviously $\text{Cov}(\mathcal{C})_{\beta _0} = \text{Cov}(\mathcal{C})_{\kappa , \beta _0}$.

We define, by transfinite induction, a function $f$ which associates to every ordinal an ordinal as follows. Let $f(0) = 0$. Given $f(\alpha )$, we define $f(\alpha + 1)$ to be the least ordinal $\beta$ such that the following hold:

1. We have $\alpha + 1 \leq \beta$ and $f(\alpha ) \leq \beta$.

2. If $\{ U_ i \to U\} _{i\in I} \in \text{Cov}(\mathcal{C})_{\kappa , f(\alpha )}$ and for each $i$ we have $\{ W_{ij} \to U_ i\} _{j\in J_ i} \in \text{Cov}(\mathcal{C})_{\kappa , f(\alpha )}$, then $\{ W_{ij} \to U\} _{i \in I, j\in J_ i} \in \text{Cov}(\mathcal{C})_{\kappa , \beta }$.

3. If $\{ U_ i \to U\} _{i\in I} \in \text{Cov}(\mathcal{C})_{\kappa , \alpha }$ and $W \to U$ is a morphism of $\mathcal{C}$, then $\{ U_ i \times _ U W \to W \} _{i\in I} \in \text{Cov}(\mathcal{C})_{\kappa , \beta }$.

To see $\beta$ exists we note that clearly the collection of all coverings $\{ W_{ij} \to U\}$ and $\{ U_ i \times _ U W \to W \}$ that occur in (2) and (3) form a set. Hence there is some ordinal $\beta$ such that $V_\beta$ contains all of these coverings. Moreover, the index set of the covering $\{ W_{ij} \to U\}$ has cardinality $\sum _{i \in I} |J_ i| \leq \kappa \otimes \kappa = \kappa$, and hence these coverings are contained in $\text{Cov}(\mathcal{C})_{\kappa , \beta }$. Since every nonempty collection of ordinals has a least element we see that $f(\alpha + 1)$ is well defined. Finally, if $\alpha$ is a limit ordinal, then we set $f(\alpha ) = \sup _{\alpha ' < \alpha } f(\alpha ')$.

Pick an ordinal $\beta _1$ such that $\text{Arrows}(\mathcal{C}) \subset V_{\beta _1}$, $\text{Cov}_0 \subset V_{\beta _0}$, and $\beta _1 \geq \beta _0$. By construction $f(\beta _1) \geq \beta _1$ and we see that the same properties hold for $V_{f(\beta _1)}$. Moreover, as $f$ is nondecreasing this remains true for any $\beta \geq \beta _1$. Next, choose any ordinal $\beta _2 > \beta _1$ with cofinality $\text{cf}(\beta _2) > \kappa$. This is possible since the cofinality of ordinals gets arbitrarily large, see Proposition 3.7.2. We claim that the pair $\kappa$, $\alpha = f(\beta _2)$ is a solution to the problem posed in the lemma.

The first and third property of the lemma holds by our choices of $\kappa$, $\beta _2 > \beta _1 > \beta _0$ above.

Since $\beta _2$ is a limit ordinal (as its cofinality is infinite) we get $f(\beta _2) = \sup _{\beta < \beta _2} f(\beta )$. Hence $\{ f(\beta ) \mid \beta < \beta _2\} \subset f(\beta _2)$ is a cofinal subset. Hence we see that

$V_\alpha = V_{f(\beta _2)} = \bigcup \nolimits _{\beta < \beta _2} V_{f(\beta )}.$

Now, let $\mathcal{U} \in \text{Cov}_{\kappa , \alpha }$. We define $\beta (\mathcal{U})$ to be the least ordinal $\beta$ such that $\mathcal{U} \in \text{Cov}_{\kappa , f(\beta )}$. By the above we see that always $\beta (\mathcal{U}) < \beta _2$.

We have to show properties (1), (2), and (3) defining a site hold for the pair $(\mathcal{C}, \text{Cov}_{\kappa , \alpha })$. The first holds because by our choice of $\beta _2$ all arrows of $\mathcal{C}$ are contained in $V_{f(\beta _2)}$. For the third, we use that given a covering $\mathcal{U} = \{ U_ i \to U\} _{i \in I} \in \text{Cov}(\mathcal{C})_{\kappa , \alpha }$ we have $\beta (\mathcal{U}) < \beta _2$ and hence any base change of $\mathcal{U}$ is by construction of $f$ contained in $\text{Cov}(\mathcal{C})_{\kappa , f(\beta + 1)}$ and hence in $\text{Cov}(\mathcal{C})_{\kappa , \alpha }$.

Finally, for the second condition, suppose that $\{ U_ i \to U\} _{i\in I} \in \text{Cov}(\mathcal{C})_{\kappa , f(\alpha )}$ and for each $i$ we have $\mathcal{W}_ i = \{ W_{ij} \to U_ i\} _{j\in J_ i} \in \text{Cov}(\mathcal{C})_{\kappa , f(\alpha )}$. Consider the function $I \to \beta _2$, $i \mapsto \beta (\mathcal{W}_ i)$. Since the cofinality of $\beta _2$ is $> \kappa \geq |I|$ the image of this function cannot be a cofinal subset. Hence there exists a $\beta < \beta _1$ such that $\mathcal{W}_ i \in \text{Cov}_{\kappa , f(\beta )}$ for all $i \in I$. It follows that the covering $\{ W_{ij} \to U\} _{i\in I, j \in J_ i}$ is an element of $\text{Cov}(\mathcal{C})_{\kappa , f(\beta + 1)} \subset \text{Cov}(\mathcal{C})_{\kappa , \alpha }$ as desired. $\square$

Remark 3.11.2. It is likely the case that, for some limit ordinal $\alpha$, the set of coverings $\text{Cov}(\mathcal{C})_\alpha$ satisfies the conditions of the lemma. This is after all what an application of the reflection principle would appear to give (modulo caveats as described at the end of Section 3.8 and in Remark 3.9.3).

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