Homotopy category of chain complexes as a localization

Question

For an abelian category $\mathcal{A}$, define the homotopy category of chain complexes $\mathcal{K}(\mathcal{A})=\mathcal{C}(\mathcal{A})/\mathcal{I},$ where $\mathcal{C}(\mathcal{A})$ denotes the category of chain complexes over $A$ and $\mathcal{I}$ denotes the ideal of null-homotopic maps. Denote $\mathcal{W}$ the set of all homotopy equivalences in $\mathcal{C}(\mathcal{A})$

What I would like to know is the following:

Is $\mathcal{K}(\mathcal{A})$ canonicaly isomorphic to $\mathcal{C}(\mathcal{A})[\mathcal{W}^{-1}]$? That is, does the quotient functor $\mathcal{C}(\mathcal{A}) \rightarrow \mathcal{K}(\mathcal{A})$ satisfy the universal property of localization with respect to $\mathcal{W}$?

I am fairly certain that it is true if the localization is additive functor between additive categories. I know this is true provided that $\mathcal{W}$ is a right/left multiplicative system (i.e. satisfy a version of Ore conditions), however, I don't know whether this is the case.

Therefore, a sub-question is

Is $\mathcal{W}$ in general a right/left multiplication system?

(I should also mention that I am a little bit suspicious about this: If the claim is true, then, since $\mathcal{W}$ is a subset of quasi-isomorphisms, this would imply that the canonical functor to the derived category is just the localization with respect to quasi-isomorphisms and the step of passing to the homotopy category is there for some copmutational convenience only.)

Thanks in advance for any help.

Answer 1

The point is this: for any chain complex $A_{\bullet}$ in $\mathcal{A}$, we can form a a chain complex $(\operatorname{Cyl} A)_{\bullet}$ such that there is a natural bijection between morphisms $(\operatorname{Cyl} A)_{\bullet} \to B_{\bullet}$ and pairs of morphisms $A_{\bullet} \to B_{\bullet}$ with a chain homotopy between them. (See here.) More precisely, there are morphisms $i_0, i_1 : A_{\bullet} \to (\operatorname{Cyl} A)_{\bullet}$ such that a morphism $h : (\operatorname{Cyl} A)_{\bullet} \to B_{\bullet}$ is the same thing as a chain homotopy $h \circ i_0 \Rightarrow h \circ i_1$, and every chain homotopy of morphisms $A_{\bullet} \to B_{\bullet}$ arises like this in a unique way.

In particular, the trivial chain homotopy $\operatorname{id}_{A_{\bullet}} \Rightarrow \operatorname{id}_{A_{\bullet}}$ corresponds to a morphism $p : (\operatorname{Cyl} A)_{\bullet} \to A_{\bullet}$, and you can check that $p$ is a chain homotopy inverse for $i_0, i_1 : A_{\bullet} \to (\operatorname{Cyl} A)_{\bullet}$. Now, consider any functor $F : \mathbf{Ch} (\mathcal{A}) \to \mathcal{E}$ that sends chain homotopy equivalences in $\mathcal{A}$ to isomorphisms in $\mathcal{E}$. Then $F p$ must be an isomorphism in $\mathcal{E}$; but $p \circ i_0 = p \circ i_1 = \mathrm{id}$, so $F i_0 = F i_1$. It follows that $F$ factors through the quotient functor $\mathbf{Ch} (\mathcal{A}) \to \mathbf{K} (\mathcal{A})$. Conversely, any functor $F : \mathbf{Ch} (\mathcal{A}) \to \mathcal{E}$ that factors through the quotient functor $\mathbf{Ch} (\mathcal{A}) \to \mathbf{K} (\mathcal{A})$ must send chain homotopy equivalences to isomorphisms. We conclude that $\mathbf{K} (\mathcal{A})$ is indeed the localisation of $\mathbf{Ch} (\mathcal{A})$ with respect to chain homotopy equivalences.

It is also true that $\mathbf{D} (\mathcal{A})$ is the localisation of $\mathbf{Ch} (\mathcal{A})$ with respect to quasi-isomorphisms. There is no need to assume any Ore conditions. (In my view, this business with Ore conditions is merely a computational convenience.)