Timeline for About the definition of quasi-equivalent dg-categories
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 10, 2012 at 14:29 | vote | accept | Francesco Genovese | ||
| Mar 5, 2012 at 0:22 | comment | added | Yosemite Sam | it must be a general fact about localising a class of morphisms S that if a roof $A \lefttarrow C \rightarrow B$ represents an iso in Ho then $C \to B$ is in S. anyway don't listen to me, thomas nikolaus below gave you an answer using the model structure. | |
| Mar 4, 2012 at 23:30 | answer | added | Thomas Nikolaus | timeline score: 6 | |
| Mar 4, 2012 at 17:59 | comment | added | Francesco Genovese | I'm studying that article, among others. The fact is: if $\mathcal A$ is isomorphic to $\mathcal B$ in the homotopy category, by definition we have diagrams $\mathcal A \leftarrow \mathcal C_1 \cdots \rightarrow \mathcal B$ and $\mathcal B \leftarrow \mathcal D_1 \cdots \rightarrow \mathcal A$, where the arrows "in the wrong direction" are quasi-equivalences or identities, such that the concatenation of the first and the second diagram is equivalent to the identity (and vice-versa). But this does not give immediately a diagram from $\mathcal A$ to $\mathcal B$ with quasi-equivalences only. | |
| Mar 4, 2012 at 17:41 | comment | added | Yosemite Sam | have you had a look at toen's lectures on dg-categories? anyway, I think you're right, if I'm not mistaken a way to define morphisms in a localisation goes exactly via strings $A_1 \leftarrow C_1 \cdots \to B$ as you write. | |
| Mar 4, 2012 at 16:05 | history | asked | Francesco Genovese | CC BY-SA 3.0 |