Lemma 84.17.1. Let $\mathcal{C}$ be a site with equalizers and fibre products. Let $K$ be a hypercovering. Then

1. $a^{-1} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits ((\mathcal{C}/K)_{total})$ is fully faithful with essential image the cartesian sheaves of sets,

2. $a^{-1} : \textit{Ab}(\mathcal{C}) \to \textit{Ab}((\mathcal{C}/K)_{total})$ is fully faithful with essential image the cartesian sheaves of abelian groups.

In both cases $a_*$ provides the quasi-inverse functor.

Proof. The case of abelian sheaves follows immediately from the case of sheaves of sets as the functor $a^{-1}$ commutes with products. In the rest of the proof we work with sheaves of sets. Observe that $a^{-1}\mathcal{F}$ is cartesian for $\mathcal{F}$ in $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ by Lemma 84.12.3. It suffices to show that the adjunction map $\mathcal{F} \to a_*a^{-1}\mathcal{F}$ is an isomorphism $\mathcal{F}$ in $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ and that for a cartesian sheaf $\mathcal{G}$ on $(\mathcal{C}/K)_{total}$ the adjunction map $a^{-1}a_*\mathcal{G} \to \mathcal{G}$ is an isomorphism.

Let $\mathcal{F}$ be a sheaf on $\mathcal{C}$. Recall that $a_*a^{-1}\mathcal{F}$ is the equalizer of the two maps $a_{0, *}a_0^{-1}\mathcal{F} \to a_{1, *}a_1^{-1}\mathcal{F}$, see Lemma 84.16.2. By Lemma 84.15.2

$a_{0, *}a_0^{-1}\mathcal{F} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits (F(K_0)^\# , \mathcal{F}) \quad \text{and}\quad a_{1, *}a_1^{-1}\mathcal{F} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits (F(K_1)^\# , \mathcal{F})$

On the other hand, we know that

$\xymatrix{ F(K_1)^\# \ar@<1ex>[r] \ar@<-1ex>[r] & F(K_0)^\# \ar[r] & \text{final object }*\text{ of }\mathop{\mathit{Sh}}\nolimits (\mathcal{C}) }$

is a coequalizer diagram in sheaves of sets by definition of a hypercovering. Thus it suffices to prove that $\mathop{\mathcal{H}\! \mathit{om}}\nolimits (-, \mathcal{F})$ transforms coequalizers into equalizers which is immediate from the construction in Sites, Section 7.26.

Let $\mathcal{G}$ be a cartesian sheaf on $(\mathcal{C}/K)_{total}$. We will show that $\mathcal{G} = a^{-1}\mathcal{F}$ for some sheaf $\mathcal{F}$ on $\mathcal{C}$. This will finish the proof because then $a^{-1}a_*\mathcal{G} = a^{-1}a_*a^{-1}\mathcal{F} = a^{-1}\mathcal{F} = \mathcal{G}$ by the result of the previous paragraph. Set $\mathcal{K}_ n = F(K_ n)^\#$ for $n \geq 0$. Then we have maps of sheaves

$\xymatrix{ \mathcal{K}_2 \ar@<1ex>[r] \ar@<0ex>[r] \ar@<-1ex>[r] & \mathcal{K}_1 \ar@<0.5ex>[r] \ar@<-0.5ex>[r] & \mathcal{K}_0 }$

coming from the fact that $K$ is a simplicial semi-representable object. The fact that $K$ is a hypercovering means that

$\mathcal{K}_1 \to \mathcal{K}_0 \times \mathcal{K}_0 \quad \text{and}\quad \mathcal{K}_2 \to \left(\text{cosk}_1( \xymatrix{ \mathcal{K}_1 \ar@<0.5ex>[r] \ar@<-0.5ex>[r] & \mathcal{K}_0 \ar[l] })\right)_2$

are surjective maps of sheaves. Using the description of cartesian sheaves on $(\mathcal{C}/K)_{total}$ given in Lemma 84.12.4 and using the description of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K_ n)$ in Lemma 84.15.3 we find that our problem can be entirely formulated1 in terms of

1. the topos $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$, and

2. the simplicial object $\mathcal{K}$ in $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ whose terms are $\mathcal{K}_ n$.

Thus, after replacing $\mathcal{C}$ by a different site $\mathcal{C}'$ as in Sites, Lemma 7.29.5, we may assume $\mathcal{C}$ has all finite limits, the topology on $\mathcal{C}$ is subcanonical, a family $\{ V_ j \to V\}$ of morphisms of $\mathcal{C}$ is a covering if and only if $\coprod h_{V_ j} \to V$ is surjective, and there exists a simplicial object $U$ of $\mathcal{C}$ such that $\mathcal{K}_ n = h_{U_ n}$ as simplicial sheaves. Working backwards through the equivalences we may assume $K_ n = \{ U_ n\}$ for all $n$.

Let $X$ be the final object of $\mathcal{C}$. Then $\{ U_0 \to X\}$ is a covering, $\{ U_1 \to U_0 \times U_0\}$ is a covering, and $\{ U_2 \to (\text{cosk}_1 \text{sk}_1 U)_2\}$ is a covering. Let us use $d^ n_ i : U_ n \to U_{n - 1}$ and $s^ n_ j : U_ n \to U_{n + 1}$ the morphisms corresponding to $\delta ^ n_ i$ and $\sigma ^ n_ j$ as in Simplicial, Definition 14.2.1. By abuse of notation, given a morphism $c : V \to W$ of $\mathcal{C}$ we denote the morphism of topoi $c : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/V) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/W)$ by the same letter. Now $\mathcal{G}$ is given by a sheaf $\mathcal{G}_0$ on $\mathcal{C}/U_0$ and an isomorphism $\alpha : (d^1_1)^{-1}\mathcal{G}_0 \to (d^1_0)^{-1}\mathcal{G}_0$ satisfying the cocycle condition on $\mathcal{C}/U_2$ formulated in Lemma 84.12.4. Since $\{ U_2 \to (\text{cosk}_1 \text{sk}_1 U)_2\}$ is a covering, the corresponding pullback functor on sheaves is faithful (small detail omitted). Hence we may replace $U$ by $\text{cosk}_1 \text{sk}_1 U$, because this replaces $U_2$ by $(\text{cosk}_1 \text{sk}_1 U)_2$ and leaves $U_1$ and $U_0$ unchanged. Then

$(d^2_0, d^2_1, d^2_2) : U_2 \to U_1 \times U_1 \times U_1$

is a monomorphism whose its image on $T$-valued points is described in Simplicial, Lemma 14.19.6. In particular, there is a morphism $c$ fitting into a commutative diagram

$\xymatrix{ U_1 \times _{(d^1_1, d^1_0), U_0 \times U_0, (d^1_1, d^1_0)} U_1 \ar[d] \ar[rr]_ c & & U_2 \ar[d] \\ U_1 \times U_1 \ar[rr]^{(\text{pr}_1, \text{pr}_2, s^0_0 \circ d^1_1 \circ \text{pr}_1)} & & U_1 \times U_1 \times U_1 }$

as going around the other way defines a point of $U_2$. Pulling back the cocycle condition for $\alpha$ on $U_2$ translates into the condition that the pullbacks of $\alpha$ via the projections to $U_1 \times _{(d^1_1, d^1_0), U_0 \times U_0, (d^1_1, d^1_0)} U_1$ are the same as the pullback of $\alpha$ via $s^0_0 \circ d^1_1 \circ \text{pr}_1$ is the identity map (namely, the pullback of $\alpha$ by $s^0_0$ is the identity). By Sites, Lemma 7.26.1 this means that $\alpha$ comes from an isomorphism

$\alpha ' : \text{pr}_1^{-1}\mathcal{G}_0 \to \text{pr}_2^{-1}\mathcal{G}_0$

of sheaves on $\mathcal{C}/U_0 \times U_0$. Then finally, the morphism $U_2 \to U_0 \times U_0 \times U_0$ is surjective on associated sheaves as is easily seen using the surjectivity of $U_1 \to U_0 \times U_0$ and the description of $U_2$ given above. Therefore $\alpha '$ satisfies the cocycle condition on $U_0 \times U_0 \times U_0$. The proof is finished by an application of Sites, Lemma 7.26.5 to the covering $\{ U_0 \to X\}$. $\square$

[1] Even though it does not matter what the precise formulation is, we spell it out: the problem is to show that given an object $\mathcal{G}_0/\mathcal{K}_0$ of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{K}_0$ and an isomorphism
$\alpha : \mathcal{G}_0 \times _{\mathcal{K}_0, \mathcal{K}(\delta ^1_1)} \mathcal{K}_1 \to \mathcal{G}_0 \times _{\mathcal{K}_0, \mathcal{K}(\delta ^1_0)} \mathcal{K}_1$
over $\mathcal{K}_1$ satisfying a cocycle condtion in $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/\mathcal{K}_2$, there exists $\mathcal{F}$ in $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ and an isomorphism $\mathcal{F} \times \mathcal{K}_0 \to \mathcal{G}_0$ over $\mathcal{K}_0$ compatible with $\alpha$.

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