Example 21.39.1 (Category over point). Let $\mathcal{C}$ be a category. Endow $\mathcal{C}$ with the chaotic topology (Sites, Example 7.6.6). Thus presheaves and sheaves agree on $\mathcal{C}$. The functor $p : \mathcal{C} \to *$ where $*$ is the category with a single object and a single morphism is cocontinuous and continuous. Let $\pi : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (*)$ be the corresponding morphism of topoi. Let $B$ be a ring. We endow $*$ with the sheaf of rings $B$ and $\mathcal{C}$ with $\mathcal{O}_\mathcal {C} = \pi ^{-1}B$ which we will denote $\underline{B}$. In this way

is an example of Situation 21.38.1. By Remark 21.38.6 we do not need to distinguish between $\pi _!$ on modules or abelian sheaves. By Lemma 21.38.8 we see that $\pi _!\mathcal{F} = \mathop{\mathrm{colim}}\nolimits _{\mathcal{C}^{opp}} \mathcal{F}$. Thus $L_ n\pi _!$ is the $n$th left derived functor of taking colimits. In the following, we write

and we will name this the *$n$th homology group of $\mathcal{F}$* on $\mathcal{C}$.

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