Example 21.39.2 (Computing homology). In Example 21.39.1 we can compute the functors $H_ n(\mathcal{C}, -)$ as follows. Let $\mathcal{F} \in \mathop{\mathrm{Ob}}\nolimits (\textit{Ab}(\mathcal{C}))$. Consider the chain complex

where the transition maps are given by

and similarly in other degrees. By construction

see Categories, Lemma 4.14.12. The construction of $K_\bullet (\mathcal{F})$ is functorial in $\mathcal{F}$ and transforms short exact sequences of $\textit{Ab}(\mathcal{C})$ into short exact sequences of complexes. Thus the sequence of functors $\mathcal{F} \mapsto H_ n(K_\bullet (\mathcal{F}))$ forms a $\delta $-functor, see Homology, Definition 12.12.1 and Lemma 12.13.12. For $\mathcal{F} = j_{U!}\mathbf{Z}_ U$ the complex $K_\bullet (\mathcal{F})$ is the complex associated to the free $\mathbf{Z}$-module on the simplicial set $X_\bullet $ with terms

This simplicial set is homotopy equivalent to the constant simplicial set on a singleton $\{ *\} $. Namely, the map $X_\bullet \to \{ *\} $ is obvious, the map $\{ *\} \to X_ n$ is given by mapping $*$ to $(U \to \ldots \to U, \text{id}_ U)$, and the maps

(Simplicial, Lemma 14.26.2) defining the homotopy between the two maps $X_\bullet \to X_\bullet $ are given by the rule

for $i > 0$ and $h_{n, 0} = \text{id}$. Verifications omitted. This implies that $K_\bullet (j_{U!}\mathbf{Z}_ U)$ has trivial cohomology in negative degrees (by the functoriality of Simplicial, Remark 14.26.4 and the result of Simplicial, Lemma 14.27.1). Thus $K_\bullet (\mathcal{F})$ computes the left derived functors $H_ n(\mathcal{C}, -)$ of $H_0(\mathcal{C}, -)$ for example by (the duals of) Homology, Lemma 12.12.4 and Derived Categories, Lemma 13.16.6.

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