$\DeclareMathOperator{\Ch}{\mathit{Ch}}$Let $\Ch\_\mathbb{Q}$ denote the model category of chain complexes over rational numbers. Let $T\_\ast$ be a tree in $\Ch\_{\mathbb{Q}}$ with $n$ vertices.


How to classify trees with respect to weak equivalences i.e., chain homotopies? Is it true that the classification can be recovered from the $\mathit{ho}(\Ch\_{\mathbb{Q}})$?


I think the key factor is that any chain complex $C\_\ast\cong \oplus\_n V\langle n\rangle\_\ast$, here $V\langle n\rangle\_\ast$ is the chain complex concentrated in degree $n$ and $V\langle n\rangle\_n= H\_n(C\_\ast)$


For example if we take a path with length 2, $f\_\ast : C\_\ast \to C\_\ast'$ then it is equivalent to maps $H\_n(f\_\ast): H\_n(C\_\ast) \to H\_n(C\_\ast')$, that is maps between vector spaces. We know that any map between vector spaces is completely describe by the dim(Ker). In this case, any path of length 2 is fully describe by $\dim(\mathrm{ker}(H\_n(f\_\ast)))\_n$.


I really appreciate it if someone could say something for trees.



Trees in chain complexes

> 
> Is it true that the classification can be recovered from the ho(ChQ)?
> 
> 
> 


Yes, the canonical functor Ho(Fun(T,M))→Fun(T,Ho(M))
induces a bijection on isomorphism classes if T is a tree and M is the relative category of rational chain complexes.


The canonical inclusion ι of the relative category
of graded rational vector spaces and isomorphisms
into the relative category of rational chain complexes
and quasi-isomorphisms given by equipping
graded rational vector spaces with a zero differential
preserves weak equivalences.
The functor H in the opposite direction computes homology, it also preserves weak equivalences.


These facts can be used to establish the above bijection as follows.


For surjectivity, take any functor hF: T→Ho(M)
and construct F: T→M as follows.
On objects, F(X)=hF(X).
On morphisms, choose a lift of hF(e) along M→Ho(M) for each edge e of the tree.
The values of F on other morphisms of T are now uniquely determined
because any morphism of T is a composition of a unique sequence of edges.
We bifibrantly replace the resulting diagram in the projective (respectively injective) model structure
on functors T→M, depending on which way the tree T grows.


The diagrams F and ιHF are weakly equivalent.
If T has a single object, this is established
by choosing noncanonical splittings C\_n = B'\_n ⊕ H\_n ⊕ B\_n.
If T has more than one object, we construct such a weak equivalence
by induction on the tree, starting at the root and proceeding
to the leaves.
At each step, we choose the splitting C\_n = B'\_n ⊕ H\_n ⊕ B\_n
so that it is compatible with the given map H\_n(q)→H\_n(p) (respectively H\_n(p)→H\_n(q)),
where q is the parent of p in the tree.


For injectivity, suppose that F,G: T→M are bifibrant.
We have to show that any natural isomorphism
t: H(F)→H(G) can be promoted to a natural quasi-isomorphism F→G.
This is done in the manner of the previous paragraph:
start at the root and proceed inductively to the leaves.
The transition maps F(q)→F(p) and G(q)→G(p) are injections, so we can always make a compatible choice of a quasi-isomorphism.


As pointed out in the comments, the above fact holds in a more
general setting: the canonical functor Ho(Fun(T,M))→Fun(T,Ho(M))
is essentially surjective, full, and conservative
for any model category M and tree T.
See Proposition 2.15 and 2.20 in [Catégories dérivables](https://doi.org/10.24033/bsmf.2592)
by Denis-Charles Cisinski.


