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 | |
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| Mar 10, 2012 at 14:29 | vote | accept | Francesco Genovese | ||
| Mar 10, 2012 at 14:29 | comment | added | Francesco Genovese | All right! I shall approve the answer; thanks! | |
| Mar 10, 2012 at 12:55 | comment | added | Thomas Nikolaus | On can argue more or less as you say. Given the factorization $A \leftarrow A′ \to B′ \leftarrow B$ this gives a ZigZag in the homotopy category (applying the functor $\dgCat \to Ho(\dgCat)$ where the outer two morphisms are isomorphisms. Thus the morphism in the middle is an isomorphism iff the inner is one, i.e. a weak equuivalence. | |
| Mar 10, 2012 at 12:20 | comment | added | Francesco Genovese | Yes, I found myself that the converse was indeed true: in Hovey's book on model categories it is proved in Theorem 1.2.10, part (iv) that whenever a map $f: A \rightarrow B$ in a model category is an isomorphism when viewed as a morphism in the homotopy category, then it is a weak equivalence. Could you please explain, in your argument, how to apply the 2-out-of-3 property? Thanks in advance! | |
| Mar 5, 2012 at 11:55 | history | edited | Thomas Nikolaus | CC BY-SA 3.0 |
added 4 characters in body
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| Mar 4, 2012 at 23:30 | history | answered | Thomas Nikolaus | CC BY-SA 3.0 |