Timeline for Is there a topos theoretic interpretation/proof of Quillen's Theorem A?
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9 events
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| Jun 4, 2012 at 9:15 | answer | added | Jonathan Chiche | timeline score: 4 | |
| Jun 3, 2012 at 20:51 | comment | added | Akhil Mathew | @Martin (contd): (This response is a bit dense, and I think the proof in HTT is a little more complicated than it needs to be. Feel free to email me if you'd like more details; they might also be in some of Joyal's writings.) | |
| Jun 3, 2012 at 20:49 | comment | added | Akhil Mathew | @Martin: This can be proved directly using various explicitly models for homotopy colimits, see math.harvard.edu/~eriehl/266x/lectures.pdf. Alternatively one can do this $(\infty, 1)$-categorically: this is done in HTT for instance. The whole point is that (in Lurie's/Joyal's language) cofinality is equivalent to $F$ being an equivalence in the covariant model structure. This is something that can be checked on the homotopy fibers, which turn out to be precisely the nerves of the overcategories in question. | |
| Jun 3, 2012 at 20:37 | comment | added | Benjamin Steinberg | I was hoping for something topos theoretic. I am aware of the homotopy cofinal statement. | |
| Jun 3, 2012 at 18:40 | comment | added | Martin Brandenburg | @Akhil: Isn't this reasoning circular? How do you prove Cofinality without Quillen? | |
| Jun 3, 2012 at 18:33 | comment | added | Akhil Mathew | If you believe this, then you can prove the theorem as follows: the homotopy of (the nerve of) $D$ is the homotopy colimit of $\ast$ indexed by $d \in D$. By cofinality, that is the same as the homotopy colimit of $\ast$ indexed by $c \in C$, which is the nerve of $C$. | |
| Jun 3, 2012 at 18:24 | comment | added | Akhil Mathew | This is homotopy-theoretic rather than topos-theoretic, but there is an interpretation of Quillen's theorem A in terms of cofinality: a map of categories (or quasi-categories) has contractible fiber categories if and only if it is homotopy cofinal: i.e., computing a homotopy colimit along $D$ is the same as evaluating on along $C$. | |
| Jun 3, 2012 at 13:47 | answer | added | Dany Galicer | timeline score: 0 | |
| Jun 3, 2012 at 12:15 | history | asked | Benjamin Steinberg | CC BY-SA 3.0 |