# ct.category theory – Trees in chain complexes

$$DeclareMathOperator{Ch}{mathit{Ch}}$$Let $$Ch_mathbb{Q}$$ denotes 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 $$C_astcong oplus_n Vlangle nrangle_ast$$, here $$Vlangle nrangle_ast$$ is the chain complex concentrated in degree $$n$$ and $$Vlangle nrangle_n= H_n(C_ast)$$

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.