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If we say that two chain complexes in an abelian category are quasi-isomorphic, does this mean that there exists a quasi-isomorphism between them (in some direction) or that there exists a zigzag of quasi-isomorphisms?

Simba
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    I would be inclined to go with the zigzag definition. After all, it guarantees that "are quasi-isomorphic" becomes an equivalence relation. – Zhen Lin Jul 21 '22 at 09:25

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I suppose that the definition of quasi-isomorphic for chain complexes in an abelian category $\mathcal{A}$ depends on the preferred convention, so if using this notion, one should state explicitly what it means.

Note that the existence of a quasi-isomorphism from one chain complex to another one does not in general define an equivalence relation on the class of all chain complexes in $\mathcal{A}$. The problem is that this relation is in general neither symmetric nor transitive. However, the existence of a zigzag between two chain complexes consisting of quasi-isomorphisms does define an equivalence relation.

Simba
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