Timeline for Link between homotopy equivalence of simplicial sets and categorical equivalences
Current License: CC BY-SA 4.0
8 events
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| Jan 7, 2019 at 9:10 | comment | added | Oscar P. | Actually it is the converse I need, i.e. that a weak equivalence of simplicial set is (maybe under some condtions) a categorical equivalence. He uses this in the proof of Proposition 1.2.9.3 at the end of section 2.4.5 when he wants to use Lemma 2.4.5.1. | |
| Jan 6, 2019 at 17:33 | comment | added | Harry Gindi | @OscarP. Do you know exactly where he uses that statement that a categorical equivalence of Kan complexes is a weak homotopy equivalence? I couldn't find it. | |
| Jan 6, 2019 at 10:35 | comment | added | Oscar P. | How do we know that every fibrant object in the Kan-Quillen model structure are also fibrant in the Joyal model structure? It is shown in HTT, that every $\infty$-category is fibrant in the Joyal model structure and this would be enough but I have a problem : the proof of this needs Proposition 1.2.9.3 and we would have a circular argument. Am I missing something obvious? | |
| Jan 3, 2019 at 21:33 | history | edited | Harry Gindi | CC BY-SA 4.0 |
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| Jan 3, 2019 at 21:16 | comment | added | Harry Gindi | Yes, both of you guys are right @KevinCarlson , but the definition given in the question of a categorical equivalence was the one I was aiming at. I am well-acquainted with the Joyal and Cisinski approach, but that wasn't the context of the question, for better or for worse. I was trying to be faithful to the flow of the argument in HTT without bringing in other (imo better) sources, since the other definitions are only equivalent by means of the comparison theorem itself. | |
| Jan 3, 2019 at 21:02 | comment | added | Dylan Wilson | For example, the claim is relatively straightforward using the definition of categorical equivalence given by Joyal (fully faithful and essentially surjective), or using the version of "weak categorical equivalence". In the former case it's essentially the Whitehead theorem (detecting we's by pi_* isomorphisms) and in the latter it's basically Yoneda. | |
| Jan 3, 2019 at 20:39 | comment | added | Kevin Carlson | Well, one can construct the Joyal model structure without proving the comparison theorem. Joyal did this, and so does Cisinski in his new book. But maybe I'm misreading your closing claim. | |
| Jan 3, 2019 at 19:21 | history | answered | Harry Gindi | CC BY-SA 4.0 |