@ShayBenMoshe: Thank you for the comment. I will go through these articles.

Thank you so much, Denis, for this beautiful answer.

@DmitriPavlov: Thank you so much, Dmitri. Do you think that the classification of the diagram of the form $A\langle n \rangle_\ast \stackrel{f}{\to} B\langle n \rangle_\ast \stackrel{g}{\to} C \langle n \rangle_\ast$ with $g\circ f \sim 0$, can be recovered form the $ho(Ch_\mathbb{Q})$ and the classification of the diagram of the type $E\langle n \rangle_\ast \to 0 \to \Sigma E\langle n \rangle_\ast$? Here $A\langle n \rangle_\ast$ is the chain complex concentrated in degree $n$ and $A\langle n \rangle_n=A.$

@TimCampion: Thanks. Is there any classification of the quiver representations of the form of a tree?

@DylanWilson: Yes, it sounds interesting. Can you please write down the key arguments for it?

Thank you so much for your comment. Do you think that it follows from Shimakawa's work semanticscholar.org/paper/…

@NeilStrickland: I am reading your answer and I feel I have a question, which may not directly be related to the above. I just wondering what is $[H\underline{\mathbb{Z}/2}, \Sigma H\underline{\mathbb{Z}/2}]^{C_2}$?

@TomGoodwillie: Consider the localizations maps $X \to P^n X$, $Y \to P^n Y$ and $X \wedge Y \to P^n(X \wedge Y )$. Then can we have a map from $P^n(X) \wedge P^n(Y) \to P^n(X \wedge Y)?$ Here $P^n$ is the Postnikov section.