Bounded objects in a t-structure

Definition 6.3.33. Let \(C\) be a stable \(\infty \)-category with a t-structure. We define the subcategory of bounded objects by \[ C^{\flat } := \bigcup _{n \in \N } (C_{\geq -n} \cap C_{\leq n}) \subseteq C. \] Observe that \(C^{\flat }\) inherits a t-structure from \(C\) which by construction is bounded. Moreover, \(C\) is bounded if and only if \(C = C^{\flat }\). One may similarly define subcategories \[ C^{-} := \bigcup _{n \in \N } C_{\geq -n} \subseteq C \qquadtext { and } C^+ := \bigcup _{n \in \N } C_{\leq n} \subseteq C \] of bounded below and bounded above objects, respectively.