Situation 21.30.1. With $\mathcal{C}$, $\tau$, and $\tau '$ as in Section 21.27. Assume we are given a subset $\mathcal{P} \subset \text{Arrows}(\mathcal{C})$ and for every object $X$ of $\mathcal{C}$ we are given a weak Serre subcategory $\mathcal{A}'_ X \subset \textit{Ab}(\mathcal{C}_{\tau '}/X)$. We make the following assumption:

1. given $f : X \to Y$ in $\mathcal{P}$ and $Y' \to Y$ general, then $X \times _ Y Y'$ exists and $X \times _ Y Y' \to Y'$ is in $\mathcal{P}$,

2. $f_{\tau '}^{-1}$ sends $\mathcal{A}'_ Y$ into $\mathcal{A}'_ X$ for any morphism $f : X \to Y$ of $\mathcal{C}$,

3. given $X$ in $\mathcal{C}$ and $\mathcal{F}'$ in $\mathcal{A}'_ X$, then $\mathcal{F}'$ satisfies the sheaf condition for $\tau$-coverings, i.e., $\mathcal{F}' = \epsilon _{X, *}\epsilon _ X^{-1}\mathcal{F}'$,

4. if $f : X \to Y$ in $\mathcal{P}$ and $\mathcal{F}' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A}'_ X)$, then $R^ if_{\tau ', *}\mathcal{F}' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A}'_ Y)$ for $i \geq 0$.

5. if $\{ U_ i \to U\} _{i \in I}$ is a $\tau$-covering, then there exist

1. a $\tau '$-covering $\{ V_ j \to U\} _{j \in J}$,

2. a $\tau$-covering $\{ f_ j : W_ j \to V_ j\}$ consisting of a single $f_ j \in \mathcal{P}$, and

3. a $\tau '$-covering $\{ W_{jk} \to W_ j\} _{k \in K_ j}$

such that $\{ W_{jk} \to U\} _{j \in J, k \in K_ j}$ is a refinement of $\{ U_ i \to U\} _{i \in I}$.

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