Lemma 6.4.17. Assume that the t-structure on \(C\) is left complete. Then \(C_{\geq 0}\) admits geometric realizations. If \(C'\) is another stable \(\infty \)-category with a left complete t-structure, then any exact right t-exact functor \(F\colon C \to C'\) restricts to a functor \(F\colon C_{\geq 0} \to C'_{\geq 0}\) that preserves geometric realizations.
Proof. By [Lurie (2017), Lemma 1.3.3.11(2)], the connective part of a left-complete t-structure admits geometric realizations, and a right exact functor between connective parts preserves them. The restriction of an exact right t-exact functor is right exact, which gives the second claim. □
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