Completeness and separatedness of t-structures

Definition 6.3.28. A t-structure on \(C\) is left complete if for every \(X \in C\) the canonical map \[ X \longrightarrow \lim _n \tau _{\leq n}X \] is an isomorphism. It is right complete if the canonical map \[ \colim _n \tau _{\geq -n}X \longrightarrow X \] is an isomorphism for every \(X \in C\).

We say that the t-structure is left separated if every infinitely connective object \(X \in C_{\geq \infty } := \bigcap _{n \in \Z }C_{\geq n}\) is zero. Dually, it is right separated if every infinitely coconnective object \(X \in C_{\leq -\infty } := \bigcap _{n \in \Z } C_{\leq n}\) is zero.