Truncation functors, t-cohomology, amplitude and bounded t-structures

Let $({\mathcal{T}}^{\le 0},{\mathcal{T}}^{\ge 0})$ be a t-structure on a triangulated category $\mathcal{T}$ with translation $[1]$ (Canonical t structure on a derived category, Triangulated category), and set ${\mathcal{T}}^{\le n}:={\mathcal{T}}^{\le 0}[-n]$ and ${\mathcal{T}}^{\ge n}:={\mathcal{T}}^{\ge 0}[-n]$ for $n\in \mathbb{Z}$; thus $X\in {\mathcal{T}}^{\le n}$ exactly when $X[n]\in {\mathcal{T}}^{\le 0}$, and $X\in {\mathcal{T}}^{\ge n}$ exactly when $X[n]\in {\mathcal{T}}^{\ge 0}$.

For $X\in \mathcal{T}$ and $n\in \mathbb{Z}$, the t-structure axiom applied to the translate $X[n]$ and then shifted by $[-n]$ produces a distinguished triangle ${\tau }_{\le n}X⟶X⟶{\tau }_{\ge n+1}X⟶{\tau }_{\le n}X[1]$ with ${\tau }_{\le n}X\in {\mathcal{T}}^{\le n}$ and ${\tau }_{\ge n+1}X\in {\mathcal{T}}^{\ge n+1}$; such a triangle is a truncation triangle of $X$ at $n$. Truncation triangles of the same $X$ at the same $n$ are canonically isomorphic over $X$, so the objects and maps occurring in them are determined up to a unique compatible isomorphism (Truncation triangles are essentially unique and give adjoint truncation functors ↗); one writes ${\tau }_{\le n}$ and ${\tau }_{\ge n}$ for the resulting truncation functors.

The t-cohomology of $X$ is $H_t^n(X):=({\tau }_{\le n}{\tau }_{\ge n}X)[n],$ an object of the heart $\mathcal{C}={\mathcal{T}}^{\le 0}\cap {\mathcal{T}}^{\ge 0}$ (t-cohomology is functorial, additive, shift-compatible and detects amplitude ↗). The amplitude of $X$ is the set of $n$ with $H_t^n(X)\cong ̸0$; it is contained in an interval $[a,b]$ when $H_t^n(X)=0$ for every $n\mathrm{\notin }[a,b]$. The t-structure is bounded when every object of $\mathcal{T}$ lies in ${\mathcal{T}}^{\le n}\cap {\mathcal{T}}^{\ge -n}$ for some integer $n$ depending on the object.

This item fixes notation and conventions only. Existence of the truncation triangle is the t-structure axiom itself, applied to $X[n]$ and shifted; the essential uniqueness of that triangle, the functoriality of the truncations and all algebraic properties of ${\tau }_{\le n}$, ${\tau }_{\ge n}$ and $H_t^n$ are proved in the two immediately following lemmas.