definition. filtered colimit [rosiak2022sheaf, def. 285] [tt-0050]

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A filtered colimit is a colimit of a functor \(\mathscr {F}: {\cal J} \to {\cal C}\), where \({\cal J}\) is a filtered category.

In particular, a colimit over a filtered poset \(P\) is the same as the colimit over a cofinal subset \(Q\) of that poset, where \(Q\) as a cofinal subset means that for every element \(p \in P\), there exists an element \(q \in Q\) with \(p \leq q\).