Triangulated categories

Suppose $X \xrightarrow{u} Y \xrightarrow{v} Z \xrightarrow{w} X[1]$ is distinguished. Which triangle is distinguished by the rotation axiom TR2?

Over a field $k$, let $f : k^3 \to k^5$ be a linear map of rank $2$, viewed as a morphism in $D(k)$ between complexes concentrated in degree $0$. What is $\dim H^{-1}(C(f)) + \dim H^0(C(f))$ for its cone?

In the proof that every monomorphism $u : X \to Y$ in a triangulated category splits, we deduce $w = 0$ from $u[1]\, w = 0$. Which fact justifies cancelling $u[1]$?

Apply $\operatorname{Hom}(\Z/2, -)$ to the distinguished triangle $\Z \xrightarrow{2} \Z \to \Z/2 \xrightarrow{\delta} \Z[1]$ in $D(\mathrm{Ab})$. What is the induced map $\delta_* : \operatorname{Hom}(\Z/2, \Z/2) \to \operatorname{Hom}(\Z/2, \Z[1])$?

In $D(\mathbb{F}_5)$ take $X = \mathbb{F}_5$, $Y = \mathbb{F}_5[1]$ and the zero map $X \to Y$, with cone $Z = Y \oplus X[1]$ as in the lesson. How many morphisms $c : Z \to Z$ make $(\mathrm{id}_X, \mathrm{id}_Y, c)$ a morphism of triangles?

Which of these triangulated categories are also abelian? (Choose all that apply.)

For the standard t-structure on $D(\mathcal{A})$, which subcategory is the heart?

Which Lean statement formalises “in a pretriangulated category, every monomorphism is split”? (C carries the instances of a pretriangulated category.)

A linear map $f : \mathbb{Q}^n \to \mathbb{Q}^m$, viewed in $D(\mathbb{Q})$ in degree $0$, has cone $C(f) = (\mathbb{Q}^n \xrightarrow{f} \mathbb{Q}^m)$ in degrees $-1, 0$. Write cone_cohomology(matrix, n) that returns [dim H^-1, dim H^0], where matrix is the list of the $m$ rows of $f$ (each of length n; it may be empty if $m = 0$). Use exact rational arithmetic (fractions.Fraction) to compute the rank.