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Tag: 04BF

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Definition 17.16.1. With $u : \mathcal{C} \to \mathcal{D}$ satisfying (a), (b) above. For $\mathcal{F} \in \textit{PAb}(\mathcal{C})$ we define {\it $g_{p!}\mathcal{F}$} as the presheaf $$V \longmapsto \mathop{\rm colim}\nolimits_{V \to u(U)} \mathcal{F}(U)$$ with colimits over $(\mathcal{I}_V^u)^{opp}$ taken in $\textit{Ab}$. For $\mathcal{F} \in \textit{PAb}(\mathcal{C})$ we set {\it $g_!\mathcal{F} = (g_{p!}\mathcal{F})^\#$}.

\begin{definition}
\label{definition-g-shriek}
With $u : \mathcal{C} \to \mathcal{D}$ satisfying (a), (b) above.
For $\mathcal{F} \in \textit{PAb}(\mathcal{C})$ we define
{\it $g_{p!}\mathcal{F}$} as the presheaf
$$V \longmapsto \colim_{V \to u(U)} \mathcal{F}(U)$$
with colimits over $(\mathcal{I}_V^u)^{opp}$ taken in $\textit{Ab}$. For
$\mathcal{F} \in \textit{PAb}(\mathcal{C})$ we set
{\it $g_!\mathcal{F} = (g_{p!}\mathcal{F})^\#$}.
\end{definition}


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