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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.

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 by Denis-Charles Cisinski.

Yes, the canonical functor Ho(Fun(T,M))→Fun(T,Ho(M)) is an equivalence of categories if T is a tree and M is the relative category of rational chain complexes.

These facts can be used to establish the above equivalence of categories as follows.

For essential 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.

For fully faithfulness, observe that the lifts F and G constructed above are bifibrant, so the induced map Hom(F,G)/~ = RHom(F,G)/~ → RHom(ιH(F),ιH(G))/~ is an isomorphism, where Hom denotes the internal hom in chain complexes, RHom denotes the derived internal hom, and ~ denotes the chain homotopy equivalence relation on diagrams.

We have RHom(ιH(F),ιH(G))/~ ≅ Hom(ιH(F),ιH(G))/~ ≅ hom(hF,hG), the set of morphisms hF→hG in Fun(T,Ho(M)). Indeed, the RHom on the left side can be computed as the derived end of RHom(ιH(F)(t),ιH(G)(t)), where t∈T. For rational chain complexes, RHom = Hom. The derived end tells us that an equivalence class in RHom(ιH(F),ιH(G))/~ can now be written down as a collection of chain maps T_t: ιH(F)(t)→ιH(G)(t) together with chain homotopies between ιH(F)(t)→ιH(G)(t)→ιH(G)(t') and ιH(F)(t)→ιH(F)(t')→ιH(G)(t') for any edge e: t→t' of the tree T. (The chain homotopies for other morphisms in T can be recovered from this data.) The equivalence relation between two such sets of data is given by chain homotopies from T_t to T'_t for all t∈T, together with a 2-homotopy between two chain homotopies arising from the square with vertices ιH(F)(t), ιH(F)(t'), ιH(G)(t), ιH(G)(t'), for any edge e: t→t' of the tree T. (The chain 2-homotopies for other morphisms in T can be recovered from this data.) Chain homotopies of chain maps between chain complexes with vanishing differentials are trivial, so we can get rid of them, which yields the set hom(hF,hG).

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.

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.

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.

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 and is a Dwyer–Kan equivalence of relative categories. The inverse functor H computes homology, it also preserves weak equivalences.

This fact can be used to establish the above equivalence of categories as follows.

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(p)→H_n(q), where q is the parent of p in the tree.

For fully faithfulness, observe that the lifts F and G constructed above are bifibrant, so the induced map Hom(F,G)/~ → Hom(ιH(F),ιH(G))/~ is an isomorphism, where Hom denotes the internal hom in chain complexes and ~ denotes the chain homotopy equivalence relation on diagrams.

Finally, Hom(ιH(F),ιH(G))/~ = hom(hF,hG), the set of morphism hF→hG in Fun(T,Ho(M)) because chain homotopies of chain maps between chain complexes with vanishing differentials are trivial.

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 equivalence of categories as follows.

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 fully faithfulness, observe that the lifts F and G constructed above are bifibrant, so the induced map Hom(F,G)/~ = RHom(F,G)/~ → RHom(ιH(F),ιH(G))/~ is an isomorphism, where Hom denotes the internal hom in chain complexes, RHom denotes the derived internal hom, and ~ denotes the chain homotopy equivalence relation on diagrams.

We have RHom(ιH(F),ιH(G))/~ ≅ Hom(ιH(F),ιH(G))/~ ≅ hom(hF,hG), the set of morphisms hF→hG in Fun(T,Ho(M)). Indeed, the RHom on the left side can be computed as the derived end of RHom(ιH(F)(t),ιH(G)(t)), where t∈T. For rational chain complexes, RHom = Hom. The derived end tells us that an equivalence class in RHom(ιH(F),ιH(G))/~ can now be written down as a collection of chain maps T_t: ιH(F)(t)→ιH(G)(t) together with chain homotopies between ιH(F)(t)→ιH(G)(t)→ιH(G)(t') and ιH(F)(t)→ιH(F)(t')→ιH(G)(t') for any edge e: t→t' of the tree T. (The chain homotopies for other morphisms in T can be recovered from this data.) The equivalence relation between two such sets of data is given by chain homotopies from T_t to T'_t for all t∈T, together with a 2-homotopy between two chain homotopies arising from the square with vertices ιH(F)(t), ιH(F)(t'), ιH(G)(t), ιH(G)(t'), for any edge e: t→t' of the tree T. (The chain 2-homotopies for other morphisms in T can be recovered from this data.) Chain homotopies of chain maps between chain complexes with vanishing differentials are trivial, so we can get rid of them, which yields the set hom(hF,hG).