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Sorry for this question but I really have difficulties with model categories.

Usually a $dg$-algebra $A$ is called formal, if there exists a $dg$-algebra $B$ and quasi-isomorphisms $$A\leftarrow B\to H^*(A) $$ (Here $H^*(A)$ is the cohomology $dg$-algebra of $A$ with differential 0)

So my question is:

Is this equivalent to require, that there exists a $dg$-algebra $C$ and quasi-isomorphisms $$A\to C\leftarrow H^*(A) $$

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    $\begingroup$ The short answer is 'Yes'. The slightly longer answer is the following: Localize the category of DG-algebras at quasi-isomorphisms. Both conditions are equivalent to a third one which does not involve direction of arrows: $A$ and $H^*A$ are isomorphic in the localization. $\endgroup$ – Fernando Muro Mar 24 '14 at 11:46
  • $\begingroup$ Thanks. By the way, do you suggest any reference? $\endgroup$ – Oliver Straser Mar 24 '14 at 12:13
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    $\begingroup$ You're welcome. Gabriel-Zisman, Hovey, Hirschhorn... $\endgroup$ – Fernando Muro Mar 24 '14 at 12:17
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    $\begingroup$ Let me add that you may enjoy Gabriel-Zisman most. It is were all this business started (also in Quillen's homotopical algebra book). Those books may help you with your intuition on model categories. $\endgroup$ – Fernando Muro Mar 24 '14 at 13:57

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