The Stacks project

25.11 Hypercoverings of spaces

The theory above is mildly interesting even in the case of topological spaces. In this case we can work out what a hypercovering is and see what the result actually says.

Let $X$ be a topological space. Consider the site $X_{Zar}$ of Sites, Example 7.6.4. Recall that an object of $X_{Zar}$ is simply an open of $X$ and that morphisms of $X_{Zar}$ correspond simply to inclusions. So what is a hypercovering of $X$ for the site $X_{Zar}$?

Let us first unwind Definition 25.2.1. An object of $\text{SR}(X_{Zar}, X)$ is simply given by a set $I$ and for each $i \in I$ an open $U_ i \subset X$. Let us denote this by $\{ U_ i\} _{i \in I}$ since there can be no confusion about the morphism $U_ i \to X$. A morphism $\{ U_ i\} _{i \in I} \to \{ V_ j\} _{j \in J}$ between two such objects is given by a map of sets $\alpha : I \to J$ such that $U_ i \subset V_{\alpha (i)}$ for all $i \in I$. When is such a morphism a covering? This is the case if and only if for every $j \in J$ we have $V_ j = \bigcup _{i\in I, \ \alpha (i) = j} U_ i$ (and is a covering in the site $X_{Zar}$).

Using the above we get the following description of a hypercovering in the site $X_{Zar}$. A hypercovering of $X$ in $X_{Zar}$ is given by the following data

  1. a simplicial set $I$ (see Simplicial, Section 14.11), and

  2. for each $n \geq 0$ and every $i \in I_ n$ an open set $U_ i \subset X$.

We will denote such a collection of data by the notation $(I, \{ U_ i\} )$. In order for this to be a hypercovering of $X$ we require the following properties

  • for $i \in I_ n$ and $0 \leq a \leq n$ we have $U_ i \subset U_{d^ n_ a(i)}$,

  • for $i \in I_ n$ and $0 \leq a \leq n$ we have $U_ i = U_{s^ n_ a(i)}$,

  • we have
    \begin{equation} \label{hypercovering-equation-covering-X} X = \bigcup \nolimits _{i \in I_0} U_ i, \end{equation}
  • for every $i_0, i_1 \in I_0$, we have
    \begin{equation} \label{hypercovering-equation-covering-two} U_{i_0} \cap U_{i_1} = \bigcup \nolimits _{i \in I_1, \ d^1_0(i) = i_0, \ d^1_1(i) = i_1} U_ i, \end{equation}
  • for every $n \geq 1$ and every $(i_0, \ldots , i_{n + 1}) \in (I_ n)^{n + 2}$ such that $d^ n_{b - 1}(i_ a) = d^ n_ a(i_ b)$ for all $0\leq a < b\leq n + 1$ we have
    \begin{equation} \label{hypercovering-equation-covering-general} U_{i_0} \cap \ldots \cap U_{i_{n + 1}} = \bigcup \nolimits _{i \in I_{n + 1}, \ d^{n + 1}_ a(i) = i_ a, \ a = 0, \ldots , n + 1} U_ i, \end{equation}
  • each of the open coverings (, (, and ( is an element of $\text{Cov}(X_{Zar})$ (this is a set theoretic condition, bounding the size of the index sets of the coverings).

Conditions ( and ( should be familiar from the chapter on sheaves on spaces for example, and condition ( is the natural generalization.

Remark 25.11.1. One feature of this description is that if one of the multiple intersections $U_{i_0} \cap \ldots \cap U_{i_{n + 1}}$ is empty then the covering on the right hand side may be the empty covering. Thus it is not automatically the case that the maps $I_{n + 1} \to (\text{cosk}_ n\text{sk}_ n I)_{n + 1}$ are surjective. This means that the geometric realization of $I$ may be an interesting (non-contractible) space.

In fact, let $I'_ n \subset I_ n$ be the subset consisting of those simplices $i \in I_ n$ such that $U_ i \not= \emptyset $. It is easy to see that $I' \subset I$ is a subsimplicial set, and that $(I', \{ U_ i\} )$ is a hypercovering. Hence we can always refine a hypercovering to a hypercovering where none of the opens $U_ i$ is empty.

Remark 25.11.2. Let us repackage this information in yet another way. Namely, suppose that $(I, \{ U_ i\} )$ is a hypercovering of the topological space $X$. Given this data we can construct a simplicial topological space $U_\bullet $ by setting

\[ U_ n = \coprod \nolimits _{i \in I_ n} U_ i, \]

and where for given $\varphi : [n] \to [m]$ we let morphisms $U(\varphi ) : U_ n \to U_ m$ be the morphism coming from the inclusions $U_ i \subset U_{\varphi (i)}$ for $i \in I_ n$. This simplicial topological space comes with an augmentation $\epsilon : U_\bullet \to X$. With this morphism the simplicial space $U_\bullet $ becomes a hypercovering of $X$ along which one has cohomological descent in the sense of [Exposé Vbis, SGA4]. In other words, $H^ n(U_\bullet , \epsilon ^*\mathcal{F}) = H^ n(X, \mathcal{F})$. (Insert future reference here to cohomology over simplicial spaces and cohomological descent formulated in those terms.) Suppose that $\mathcal{F}$ is an abelian sheaf on $X$. In this case the spectral sequence of Lemma 25.5.3 becomes the spectral sequence with $E_1$-term

\[ E_1^{p, q} = H^ q(U_ p, \epsilon _ q^*\mathcal{F}) \Rightarrow H^{p + q}(U_\bullet , \epsilon ^*\mathcal{F}) = H^{p + q}(X, \mathcal{F}) \]

comparing the total cohomology of $\epsilon ^*\mathcal{F}$ to the cohomology groups of $\mathcal{F}$ over the pieces of $U_\bullet $. (Insert future reference to this spectral sequence here.)

In topology we often want to find hypercoverings of $X$ which have the property that all the $U_ i$ come from a given basis for the topology of $X$ and that all the coverings ( and ( are from a given cofinal collection of coverings. Here are two example lemmas.

Lemma 25.11.3. Let $X$ be a topological space. Let $\mathcal{B}$ be a basis for the topology of $X$. There exists a hypercovering $(I, \{ U_ i\} )$ of $X$ such that each $U_ i$ is an element of $\mathcal{B}$.

Proof. Let $n \geq 0$. Let us say that an $n$-truncated hypercovering of $X$ is given by an $n$-truncated simplicial set $I$ and for each $i \in I_ a$, $0 \leq a \leq n$ an open $U_ i$ of $X$ such that the conditions defining a hypercovering hold whenever they make sense. In other words we require the inclusion relations and covering conditions only when all simplices that occur in them are $a$-simplices with $a \leq n$. The lemma follows if we can prove that given a $n$-truncated hypercovering $(I, \{ U_ i\} )$ with all $U_ i \in \mathcal{B}$ we can extend it to an $(n + 1)$-truncated hypercovering without adding any $a$-simplices for $a \leq n$. This we do as follows. First we consider the $(n + 1)$-truncated simplicial set $I'$ defined by $I' = \text{sk}_{n + 1}(\text{cosk}_ n I)$. Recall that

\[ I'_{n + 1} = \left\{ \begin{matrix} (i_0, \ldots , i_{n + 1}) \in (I_ n)^{n + 2} \text{ such that} \\ d^ n_{b - 1}(i_ a) = d^ n_ a(i_ b) \text{ for all }0\leq a < b\leq n + 1 \end{matrix} \right\} \]

If $i' \in I'_{n + 1}$ is degenerate, say $i' = s^ n_ a(i)$ then we set $U_{i'} = U_ i$ (this is forced on us anyway by the second condition). We also set $J_{i'} = \{ i'\} $ in this case. If $i' \in I'_{n + 1}$ is nondegenerate, say $i' = (i_0, \ldots , i_{n + 1})$, then we choose a set $J_{i'}$ and an open covering
\begin{equation} \label{hypercovering-equation-choose-covering} U_{i_0} \cap \ldots \cap U_{i_{n + 1}} = \bigcup \nolimits _{i \in J_{i'}} U_ i, \end{equation}

with $U_ i \in \mathcal{B}$ for $i \in J_{i'}$. Set

\[ I_{n + 1} = \coprod \nolimits _{i' \in I'_{n + 1}} J_{i'} \]

There is a canonical map $\pi : I_{n + 1} \to I'_{n + 1}$ which is a bijection over the set of degenerate simplices in $I'_{n + 1}$ by construction. For $i \in I_{n + 1}$ we define $d^{n + 1}_ a(i) = d^{n + 1}_ a(\pi (i))$. For $i \in I_ n$ we define $s^ n_ a(i) \in I_{n + 1}$ as the unique simplex lying over the degenerate simplex $s^ n_ a(i) \in I'_{n + 1}$. We omit the verification that this defines an $(n + 1)$-truncated hypercovering of $X$. $\square$

Lemma 25.11.4. Let $X$ be a topological space. Let $\mathcal{B}$ be a basis for the topology of $X$. Assume that

  1. $X$ is quasi-compact,

  2. each $U \in \mathcal{B}$ is quasi-compact open, and

  3. the intersection of any two quasi-compact opens in $X$ is quasi-compact.

Then there exists a hypercovering $(I, \{ U_ i\} )$ of $X$ with the following properties

  1. each $U_ i$ is an element of the basis $\mathcal{B}$,

  2. each of the $I_ n$ is a finite set, and in particular

  3. each of the coverings (, (, and ( is finite.

Proof. This follows directly from the construction in the proof of Lemma 25.11.3 if we choose finite coverings by elements of $\mathcal{B}$ in ( Details omitted. $\square$

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