Definition 25.4.1. Let \mathcal{C} be a site. Let K be a simplicial object of \textit{PSh}(\mathcal{C}). By the above we get a simplicial object \mathbf{Z}_ K^\# of \textit{Ab}(\mathcal{C}). We can take its associated complex of abelian presheaves s(\mathbf{Z}_ K^\# ), see Simplicial, Section 14.23. The homology of K is the homology of the complex of abelian sheaves s(\mathbf{Z}_ K^\# ).
25.4 Acyclicity
Let \mathcal{C} be a site. For a presheaf of sets \mathcal{F} we denote \mathbf{Z}_\mathcal {F} the presheaf of abelian groups defined by the rule
\mathbf{Z}_\mathcal {F}(U) = \text{free abelian group on }\mathcal{F}(U).
We will sometimes call this the free abelian presheaf on \mathcal{F}. Of course the construction \mathcal{F} \mapsto \mathbf{Z}_\mathcal {F} is a functor and it is left adjoint to the forgetful functor \textit{PAb}(\mathcal{C}) \to \textit{PSh}(\mathcal{C}). Of course the sheafification \mathbf{Z}_\mathcal {F}^\# is a sheaf of abelian groups, and the functor \mathcal{F} \mapsto \mathbf{Z}_\mathcal {F}^\# is a left adjoint as well. We sometimes call \mathbf{Z}_\mathcal {F}^\# the free abelian sheaf on \mathcal{F}.
For an object X of the site \mathcal{C} we denote \mathbf{Z}_ X the free abelian presheaf on h_ X, and we denote \mathbf{Z}_ X^\# its sheafification.
In other words, the ith homology H_ i(K) of K is the sheaf of abelian groups H_ i(K) = H_ i(s(\mathbf{Z}_ K^\# )). In this section we worry about the homology in case K is a hypercovering of an object X of \mathcal{C}.
Lemma 25.4.2. Let \mathcal{C} be a site. Let \mathcal{F} \to \mathcal{G} be a morphism of presheaves of sets. Denote K the simplicial object of \textit{PSh}(\mathcal{C}) whose nth term is the (n + 1)st fibre product of \mathcal{F} over \mathcal{G}, see Simplicial, Example 14.3.5. Then, if \mathcal{F} \to \mathcal{G} is surjective after sheafification, we have
H_ i(K) = \left\{ \begin{matrix} 0
& \text{if}
& i > 0
\\ \mathbf{Z}_\mathcal {G}^\#
& \text{if}
& i = 0
\end{matrix} \right.
The isomorphism in degree 0 is given by the morphism H_0(K) \to \mathbf{Z}_\mathcal {G}^\# coming from the map (\mathbf{Z}_ K^\# )_0 = \mathbf{Z}_\mathcal {F}^\# \to \mathbf{Z}_\mathcal {G}^\# .
Proof. Let \mathcal{G}' \subset \mathcal{G} be the image of the morphism \mathcal{F} \to \mathcal{G}. Let U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}). Set A = \mathcal{F}(U) and B = \mathcal{G}'(U). Then the simplicial set K(U) is equal to the simplicial set with n-simplices given by
A \times _ B A \times _ B \ldots \times _ B A\ (n + 1 \text{ factors)}.
By Simplicial, Lemma 14.32.3 the morphism K(U) \to B is a trivial Kan fibration. Thus it is a homotopy equivalence (Simplicial, Lemma 14.30.8). Hence applying the functor “free abelian group on” to this we deduce that
\mathbf{Z}_ K(U) \longrightarrow \mathbf{Z}_ B
is a homotopy equivalence. Note that s(\mathbf{Z}_ B) is the complex
\ldots \to \bigoplus \nolimits _{b \in B}\mathbf{Z} \xrightarrow {0} \bigoplus \nolimits _{b \in B}\mathbf{Z} \xrightarrow {1} \bigoplus \nolimits _{b \in B}\mathbf{Z} \xrightarrow {0} \bigoplus \nolimits _{b \in B}\mathbf{Z} \to 0
see Simplicial, Lemma 14.23.3. Thus we see that H_ i(s(\mathbf{Z}_ K(U))) = 0 for i > 0, and H_0(s(\mathbf{Z}_ K(U))) = \bigoplus _{b \in B}\mathbf{Z} = \bigoplus _{s \in \mathcal{G}'(U)} \mathbf{Z}. These identifications are compatible with restriction maps.
We conclude that H_ i(s(\mathbf{Z}_ K)) = 0 for i > 0 and H_0(s(\mathbf{Z}_ K)) = \mathbf{Z}_{\mathcal{G}'}, where here we compute homology groups in \textit{PAb}(\mathcal{C}). Since sheafification is an exact functor we deduce the result of the lemma. Namely, the exactness implies that H_0(s(\mathbf{Z}_ K))^\# = H_0(s(\mathbf{Z}_ K^\# )), and similarly for other indices. \square
Lemma 25.4.3. Let \mathcal{C} be a site. Let f : L \to K be a morphism of simplicial objects of \textit{PSh}(\mathcal{C}). Let n \geq 0 be an integer. Assume that
For i < n the morphism L_ i \to K_ i is an isomorphism.
The morphism L_ n \to K_ n is surjective after sheafification.
The canonical map L \to \text{cosk}_ n \text{sk}_ n L is an isomorphism.
The canonical map K \to \text{cosk}_ n \text{sk}_ n K is an isomorphism.
Then H_ i(f) : H_ i(L) \to H_ i(K) is an isomorphism.
Proof. This proof is exactly the same as the proof of Lemma 25.4.2 above. Namely, we first let K_ n' \subset K_ n be the sub presheaf which is the image of the map L_ n \to K_ n. Assumption (2) means that the sheafification of K_ n' is equal to the sheafification of K_ n. Moreover, since L_ i = K_ i for all i < n we see that get an n-truncated simplicial presheaf U by taking U_0 = L_0 = K_0, \ldots , U_{n - 1} = L_{n - 1} = K_{n - 1}, U_ n = K'_ n. Denote K' = \text{cosk}_ n U, a simplicial presheaf. Because we can construct K'_ m as a finite limit, and since sheafification is exact, we see that (K'_ m)^\# = K_ m. In other words, (K')^\# = K^\# . We conclude, by exactness of sheafification once more, that H_ i(K) = H_ i(K'). Thus it suffices to prove the lemma for the morphism L \to K', in other words, we may assume that L_ n \to K_ n is a surjective morphism of presheaves!
In this case, for any object U of \mathcal{C} we see that the morphism of simplicial sets
L(U) \longrightarrow K(U)
satisfies all the assumptions of Simplicial, Lemma 14.32.1. Hence it is a trivial Kan fibration. In particular it is a homotopy equivalence (Simplicial, Lemma 14.30.8). Thus
\mathbf{Z}_ L(U) \longrightarrow \mathbf{Z}_ K(U)
is a homotopy equivalence too. This for all U. The result follows. \square
Lemma 25.4.4. Let \mathcal{C} be a site. Let K be a simplicial presheaf. Let \mathcal{G} be a presheaf. Let K \to \mathcal{G} be an augmentation of K towards \mathcal{G}. Assume that
The morphism of presheaves K_0 \to \mathcal{G} becomes a surjection after sheafification.
The morphism
(d^1_0, d^1_1) : K_1 \longrightarrow K_0 \times _\mathcal {G} K_0becomes a surjection after sheafification.
For every n \geq 1 the morphism
K_{n + 1} \longrightarrow (\text{cosk}_ n \text{sk}_ n K)_{n + 1}turns into a surjection after sheafification.
Then H_ i(K) = 0 for i > 0 and H_0(K) = \mathbf{Z}_\mathcal {G}^\# .
Proof. Denote K^ n = \text{cosk}_ n \text{sk}_ n K for n \geq 1. Define K^0 as the simplicial object with terms (K^0)_ n equal to the (n + 1)-fold fibred product K_0 \times _\mathcal {G} \ldots \times _\mathcal {G} K_0, see Simplicial, Example 14.3.5. We have morphisms
K \longrightarrow \ldots \to K^ n \to K^{n - 1} \to \ldots \to K^1 \to K^0.
The morphisms K \to K^ i, K^ j \to K^ i for j \geq i \geq 1 come from the universal properties of the \text{cosk}_ n functors. The morphism K^1 \to K^0 is the canonical morphism from Simplicial, Remark 14.20.4. We also recall that K^0 \to \text{cosk}_1 \text{sk}_1 K^0 is an isomorphism, see Simplicial, Lemma 14.20.3.
By Lemma 25.4.2 we see that H_ i(K^0) = 0 for i > 0 and H_0(K^0) = \mathbf{Z}_\mathcal {G}^\# .
Pick n \geq 1. Consider the morphism K^ n \to K^{n - 1}. It is an isomorphism on terms of degree < n. Note that K^ n \to \text{cosk}_ n \text{sk}_ n K^ n and K^{n - 1} \to \text{cosk}_ n \text{sk}_ n K^{n - 1} are isomorphisms. Note that (K^ n)_ n = K_ n and that (K^{n - 1})_ n = (\text{cosk}_{n - 1} \text{sk}_{n - 1} K)_ n. Hence by assumption, we have that (K^ n)_ n \to (K^{n - 1})_ n is a morphism of presheaves which becomes surjective after sheafification. By Lemma 25.4.3 we conclude that H_ i(K^ n) = H_ i(K^{n - 1}). Combined with the above this proves the lemma. \square
Lemma 25.4.5. Let \mathcal{C} be a site with fibre products. Let X be an object of \mathcal{C}. Let K be a hypercovering of X. The homology of the simplicial presheaf F(K) is 0 in degrees > 0 and equal to \mathbf{Z}_ X^\# in degree 0.
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