Hypercovers | Higher Topos Theory notes

6.3. Hypercovers

In Section 3.4, we defined a topos $$T$$ to be hypercomplete if every $\infty$ -connected morphism is an isomorphism, and showed that the $\infty$ -connected and $\infty$ -truncated morphisms form a factorization system. In this section, we give an alternative characterization of hypercompleteness in terms of hypercovers.

Recall that an effective epimorphism $f\colon X \twoheadrightarrow Y$ in a topos $$T$$ may be thought of as a cover of $$Y$$ . The Čech nerve $\check{C}(f)$ encodes all the iterated self-intersections $X^{\times_Y^n}$ of $$X$$ over $$Y$$ , and since $$f$$ is an effective epimorphism this expresses $$Y$$ as a colimit of these iterated self-intersections.

A hypercover allows these intersections to be refined coherently in every simplicial degree. It begins with an effective epimorphism $U_0\twoheadrightarrow Y$ ; the object $$U_1$$ covers the pairwise intersections, while in degree $$n$$ the object $$U_n$$ covers the object of compatible boundary data determined by the preceding degrees. We will show that the realization of any hypercover of $$Y$$ maps $\infty$ -connectedly to $$Y$$ . Thus every hypercover is effective in a hypercomplete topos, and the converse holds as well:

Theorem 6.40. ([Lurie 2009, Theorem 6.5.3.12])

Let $$T$$ be a topos. The following conditions are equivalent:

The topos $$T$$ is hypercomplete.
For every object $X \in T$ , every hypercover of $$X$$ is effective.

We start by defining hypercovers and establishing their basic formal properties. We then prove the connectivity statement which relates their effectiveness to hypercompleteness.

6.3.1. Hypercovers and matching objects

Notation 6.41.

For each natural number $n \geq 0$ , let $\simp^{\leq n}$ denote the full subcategory of $\simp$ spanned by the objects $\{[0], [1], \dots, [n]\}$ , and set $\simp^{\leq -1}:=\emptyset$ . If $$T$$ is a topos, the restriction functor

(-)\vert_{\simp^{\leq n}} \colon \Fun(\simp\catop,T) \to \Fun((\simp^{\leq n})\catop,T)

admits a right adjoint given by right Kan extension along the inclusion $(\simp^{\leq n})\catop \hookrightarrow \simp\catop$ . We let

\cosk_n\colon \Fun(\simp\catop,T) \to \Fun(\simp\catop,T)

denote the composition of the restriction functor with its right adjoint and refer to $\cosk_n$ as the $$n$$ -coskeleton functor. In particular, $\cosk_{-1}$ is the constant simplicial object at the terminal object of $$T$$ . A simplicial object $U_\bullet$ is called $$n$$ -coskeletal if the unit map $U_\bullet \to \cosk_n(U_\bullet)$ is an isomorphism, or equivalently if $U_\bullet$ is a right Kan extension of its restriction to $(\simp^{\leq n})\catop$ .

Definition 6.42. (Hypercover)

Let $$T$$ be a topos. A simplicial object $U_{\bullet} \in \Fun(\simp\catop,T)$ is called a hypercover in $$T$$ if, for each $n \geq 0$ , the canonical map

U_n \to (\cosk_{n-1} U_{\bullet})_n

is an effective epimorphism. We say that $U_{\bullet}$ is an effective hypercover if the colimit of $U_{\bullet}$ is a terminal object of $$T$$ .

For an object $X \in T$ , a hypercover of $$X$$ is a hypercover in the slice topos $T_{/X}$ .

Remark 6.43.

The object $(\cosk_{n-1} U_{\bullet})_n$ is known as the $$n$$ -th matching object of $U_\bullet$ . By the pointwise formula for right Kan extensions, it may be computed as a limit over the category $(\simp^{\leq n-1})_{/[n]}$ . Since this category is finite, the object $(\cosk_{n-1} U_\bullet)_n$ is a finite limit of the objects $U_0, \ldots, U_{n-1}$ .

Let us spell out the hypercover condition for small $$n$$ :

For $$n = 0$$ , the $$(-1)$$ -coskeleton of any simplicial object is constant at the terminal object, so $(\cosk_{-1} U_\bullet)_0 = *$ . The condition thus says that $U_0 \to *$ is an effective epimorphism.
For $$n = 1$$ , the $$0$$ -coskeleton is determined by $$U_0$$ , and one computes $(\cosk_0 U_\bullet)_1 = U_0 \times U_0$ . The condition thus says that the map $(d_1, d_0)\colon U_1 \to U_0 \times U_0$ is an effective epimorphism.
For $$n = 2$$ , the matching object is the finite limit parametrizing compatible triples of edges which form the boundary of a $$2$$ -simplex. More explicitly, it can be written as
$(\cosk_1 U_\bullet)_2 \iso \bigl(U_1\mathbin{\mathop{\times}_{d_0,U_0,d_1}}U_1\bigr) \mathbin{\mathop{\times}_{U_0\times U_0}}U_1,$
where the first factor maps to $U_0\times U_0$ by $(d_1\pr_1,d_0\pr_2)$ and the second by $$(d_1,d_0)$$ . The hypercover condition says that $$U_2$$ covers this object of compatible boundary data.

Example 6.44.

Let $f\colon X \twoheadrightarrow Y$ be an effective epimorphism in $$T$$ . Then the Čech nerve $\check{C}(f)$ , viewed as a simplicial object in $T_{/Y}$ , is a hypercover of $$Y$$ . Its matching map in degree zero is $$f$$ , hence is an effective epimorphism. By Lemma 2.20, the simplicial object $\check{C}(f)$ is $$0$$ -coskeletal, so its matching maps in every positive degree are isomorphisms.

In preparation of the proof of Theorem 6.40, we establish various basic properties of hypercovers.

Lemma 6.45.

Let $\phi^*\colon S \to T$ be a morphism of logoi and let $U_\bullet$ be a hypercover in $$S$$ . Then $\phi^*(U_\bullet)$ is a hypercover in $$T$$ .

Proof

Since

\phi^*

is left exact and the matching object

(\cosk_{n-1} U_\bullet)_n

is a finite limit (Remark 6.43), we have

\phi^*((\cosk_{n-1} U_\bullet)_n) \simeq (\cosk_{n-1} \phi^*(U_\bullet))_n

. Since

\phi^*

preserves effective epimorphisms, the result follows.

Corollary 6.46.

Let $$T$$ be a topos with hypercompletion $L\colon T \to T^{\hyp}$ . A simplicial object $U_\bullet$ is a hypercover in $$T$$ if and only if $L(U_\bullet)$ is a hypercover in $T^{\hyp}$ .

Proof

The “only if” direction is Lemma 6.45. Conversely, suppose that

L(U_\bullet)

is a hypercover. Since

\(L\)

is left exact, it preserves the matching objects, so it sends every matching map

f_n\colon U_n\longrightarrow(\cosk_{n-1}U_\bullet)_n

to an effective epimorphism. Factor

\(f_n\)

as an effective epimorphism followed by its image monomorphism

\(m_n\)

. The functor

\(L\)

preserves this factorization. Since

\(L(f_n)\)

is an effective epimorphism, right cancellation shows that

\(L(m_n)\)

is an effective epimorphism; being monic, it is an isomorphism. Hence

\(m_n\)

belongs to the kernel of hypercompletion and is therefore

\infty

-connected. Since it is also a monomorphism, it is an isomorphism. Thus

\(f_n\)

is an effective epimorphism for every

\(n\)

, proving that

U_\bullet

is a hypercover.

Lemma 6.47.

Let $U \in T$ be an $\infty$ -connected object. Then the constant simplicial object with value $$U$$ is a hypercover in $$T$$ .

Proof

By Corollary 6.46, we may assume that

\(T\)

is hypercomplete. Then

U \simeq *

, so the constant simplicial object with value

\(U\)

is the terminal object of

\Fun(\simp\catop, T)

. Since the coskeleton functors preserve limits, the matching maps are all isomorphisms.

6.3.2. Effectiveness and hypercompleteness

The key point is that an arbitrary hypercover can be approximated by coskeletal ones. The latter are effective, and comparison with these approximations shows that the realization of any hypercover is $\infty$ -connected.

Lemma 6.48. ([Lurie 2009, Lemma 6.5.3.9])

Let $$T$$ be a topos and let $U_\bullet$ be an $$n$$ -coskeletal hypercover in $$T$$ . Then $U_\bullet$ is effective.

Proof sketch

The details of the proof are intricate; we refer to [Lurie 2009, Lemma 6.5.3.9] for a detailed argument. We only sketch the idea of Lurie's proof.We proceed by induction on

\(n\)

. If

\(n = 0\)

, then

U_\bullet

can be identified with (the underlying groupoid object of) the Čech nerve of the map

U_0 \to *

. Since

U_\bullet

is a hypercover, this map is an effective epimorphism, so the Čech nerve is a colimit diagram and

\abs{U_\bullet} \simeq *

.Now assume

\(n > 0\)

and let

V_\bullet = \cosk_{n-1} U_\bullet

. This is an

\((n-1)\)

-coskeletal hypercover, so

\abs{V_\bullet}\simeq *

by induction. The unit

f_\bullet\colon U_\bullet \to V_\bullet

is an isomorphism below degree

\(n\)

. Moreover, each

\(f_m\)

is a finite composite of pullbacks of the matching map

\(f_n\)

, and hence is an effective epimorphism.Form the degreewise Čech nerve

\(W^+\)

f_\bullet

, regarded as an augmented bisimplicial object. Realizing in the Čech direction gives

V_\bullet

, and then realizing in the other direction gives

\abs{V_\bullet}\simeq *

. By cofinality of the diagonal, the diagonal simplicial object

D_\bullet

of the underlying bisimplicial object therefore has terminal realization. Lurie constructs a retract of the underlying semisimplicial object of

D_\bullet

onto that of

U_\bullet

. Forgetting degeneracies does not change geometric realizations, so

\abs{U_\bullet}

is a retract of

\abs{D_\bullet}\simeq *

and is therefore terminal.

Lemma 6.49. ([Lurie 2009, Lemma 6.5.3.10])

Let $f_\bullet\colon U_\bullet \to V_\bullet$ be a morphism of simplicial objects in $$T$$ . Suppose that $f_k\colon U_k \to V_k$ is an isomorphism for all $k \leq n$ . Then the induced map $\abs{f_\bullet}\colon \abs{U_\bullet} \to \abs{V_\bullet}$ is $$n$$ -connected.

Proof

This is [Lurie 2009, Lemma 6.5.3.10]. Lurie reduces first to animae and then proves the claim using projectively cofibrant simplicial Kan-complex models.

Lemma 6.50.

Let $$T$$ be a topos and let $U_\bullet$ be a hypercover in $$T$$ . Then the canonical map $\abs{U_\bullet} \to *$ is $\infty$ -connected.

Proof

We show that

\abs{U_\bullet}

\(n\)

-connected for every

n \geq 0

. Let

V_\bullet = \cosk_{n+1} U_\bullet

and let

u_\bullet\colon U_\bullet \to V_\bullet

be the unit map. The matching maps of

V_\bullet

agree with those of

U_\bullet

through degree

\(n+1\)

and are isomorphisms thereafter, so

V_\bullet

is again a hypercover. Moreover,

\(u_m\)

is an isomorphism for

m \leq n+1

. By Lemma 6.49, the induced map

\abs{U_\bullet} \to \abs{V_\bullet}

\(n\)

-connected. By Lemma 6.48,

\abs{V_\bullet} \simeq *

. It follows that

\abs{U_\bullet}

\(n\)

-connected.

We are now ready to prove the main result of this section.

Proof

(1) \Rightarrow (2)

: Let

U_\bullet

be a hypercover in

T_{/X}

, or equivalently a hypercover of

\(X\)

. By Lemma 6.50, the canonical map

\abs{U_\bullet} \to X

\infty

-connected. Since

\(T\)

is hypercomplete, so is

T_{/X}

, hence this map is an isomorphism. Thus

U_\bullet

is effective.

(2) \Rightarrow (1)

: Let

f\colon U \to X

be an

\infty

-connected morphism in

\(T\)

. Consider

\(f\)

as an object of

T_{/X}

and let

f_\bullet

be the constant simplicial object with value

\(f\)

. By Lemma 6.47,

f_\bullet

is a hypercover in

T_{/X}

. By assumption,

f_\bullet

is effective, so its realization is the terminal object

\id_X

T_{/X}

. On the other hand, the realization of a constant simplicial object is its constant value, so

\abs{f_\bullet}\iso f

. Hence

\(f\)

is an isomorphism, showing that

\(T\)

is hypercomplete.

Remark 6.51.

Theorem 6.40 is often useful in practice: to show that a topos $$T$$ is hypercomplete, it suffices to verify that all hypercovers are effective. For sheaf topoi $\Shv(X)$ on a topological space $$X$$ , one may work with open hypercovers when these are sufficiently plentiful. Ordinary open covers alone do not detect hypercompleteness, since their Čech nerves are effective in every topos. The dimension criteria of Section 6.5 provide more concrete sufficient conditions.

References

Jacob Lurie. Higher topos theory. Ann. Math. Stud. 170, Princeton, NJ: Princeton University Press. 2009.