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.


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.)

  • $\begingroup$ Thank you for the answer. I tried to use cones instead of cylinders, which I was unable to make work (since they are good for describing nullhomotopies, but not so much for describing pairs of homotopic maps (without the preadditivity of all categories involved)). $\endgroup$ – Pavel Čoupek Feb 1 '15 at 13:48

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.