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Thomas Nikolaus
Thomas Nikolaus
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The converse is true. 

You can see this for example as follows: first since DG-categories admit a model structure each morpism $A \to B$ in Ho(dgCat) can be realized as a 3-step span like this $A \leftarrow A' \to B' \leftarrow B$ (here $A'$ is a cofibrant replacement and $B'$ is a fibrant replacement). Then such a morpism is an isomorphism in the homotopy category be the 2-out-of-3 property if and only if the middle morphism $A'\to B'$ is a weak equivalence. Thus each isomorphism in the homotopy category comes from a Zig-Zag of weak equivalences (and this argument is valid in each model category).

The converse is true. You can see this for example as follows: first since DG-categories admit a model structure each morpism $A \to B$ in Ho(dgCat) can be realized as a 3-step span like this $A \leftarrow A' \to B' \leftarrow B$ (here $A'$ is a cofibrant replacement and $B'$ is a fibrant replacement). Then such a morpism is an isomorphism in the homotopy category be the 2-out-of-3 property if and only if the middle morphism $A'\to B'$ is a weak equivalence. Thus each isomorphism in the homotopy category comes from a Zig-Zag of weak equivalences (and this argument is valid in each model category).

The converse is true. 

You can see this for example as follows: first since DG-categories admit a model structure each morpism $A \to B$ in Ho(dgCat) can be realized as a 3-step span like this $A \leftarrow A' \to B' \leftarrow B$ (here $A'$ is a cofibrant replacement and $B'$ is a fibrant replacement). Then such a morpism is an isomorphism in the homotopy category be the 2-out-of-3 property if and only if the middle morphism $A'\to B'$ is a weak equivalence. Thus each isomorphism in the homotopy category comes from a Zig-Zag of weak equivalences (and this argument is valid in each model category).

Source Link
Thomas Nikolaus
Thomas Nikolaus
  • 1.4k
  • 1
  • 13
  • 13

The converse is true. You can see this for example as follows: first since DG-categories admit a model structure each morpism $A \to B$ in Ho(dgCat) can be realized as a 3-step span like this $A \leftarrow A' \to B' \leftarrow B$ (here $A'$ is a cofibrant replacement and $B'$ is a fibrant replacement). Then such a morpism is an isomorphism in the homotopy category be the 2-out-of-3 property if and only if the middle morphism $A'\to B'$ is a weak equivalence. Thus each isomorphism in the homotopy category comes from a Zig-Zag of weak equivalences (and this argument is valid in each model category).