The Stacks project

Lemma 13.14.15. Assumptions and notation as in Situation 13.14.1. If there exists a subset $\mathcal{I} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$ such that

  1. for all $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$ there exists $s : X \to X'$ in $S$ with $X' \in \mathcal{I}$, and

  2. for every arrow $s : X \to X'$ in $S$ with $X, X' \in \mathcal{I}$ the map $F(s) : F(X) \to F(X')$ is an isomorphism,

then $RF$ is everywhere defined and every $X \in \mathcal{I}$ computes $RF$. Dually, if there exists a subset $\mathcal{P} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$ such that

  1. for all $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$ there exists $s : X' \to X$ in $S$ with $X' \in \mathcal{P}$, and

  2. for every arrow $s : X \to X'$ in $S$ with $X, X' \in \mathcal{P}$ the map $F(s) : F(X) \to F(X')$ is an isomorphism,

then $LF$ is everywhere defined and every $X \in \mathcal{P}$ computes $LF$.

Proof. Let $X$ be an object of $\mathcal{D}$. Assumption (1) implies that the arrows $s : X \to X'$ in $S$ with $X' \in \mathcal{I}$ are cofinal in the category $X/S$. Assumption (2) implies that $F$ is constant on this cofinal subcategory. Clearly this implies that $F : (X/S) \to \mathcal{D}'$ is essentially constant with value $F(X')$ for any $s : X \to X'$ in $S$ with $X' \in \mathcal{I}$. $\square$

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