Timeline for On diagrams in model categories and rectification
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 30, 2021 at 12:18 | comment | added | D.-C. Cisinski | @SimonHenry There is still an induction hidden in the proof I sketch above: I use that commuting with small limits is the same as commuting with small sums and with pullbacks, while proving this equivalence consists in proving that any small $\infty$-category is a localization of a direct category $J$ (and that any localization is both final and cofinal) and then to do an induction on $J$. | |
| Oct 28, 2021 at 13:27 | comment | added | Simon Henry | Regarding Balzin's work there is also a more recent work by Harpaz (arxiv.org/abs/1902.04867) That proves the same result for a general base category, but at the cost of stronger assumption on the fibration ( each fiber needs to a simplicial combinatorial model structure and each transition a simplicial Quillen adjunction) | |
| Oct 28, 2021 at 13:26 | comment | added | Simon Henry | ... Then $C^I$ and $C^J$ are Quillen equivalent. | |
| Oct 28, 2021 at 13:16 | vote | accept | Simon Henry | ||
| Oct 28, 2021 at 13:16 | comment | added | Simon Henry | That's very interesting. The proofs I knew looked fairly different. In genral it is based on the idea to start with a class of categories $I$ such that the theorem can be proved by "induction" on $I$ for e.g. when $I$ is a direct or Reedy category or a Bergner-cofibrant simplicial category (in this case we can even avoid the induction and rely on the Bergner model structure, assuming C is simplicial) and then extend to a general I by showing that if $I \simeq J$ in an appropriate sense (e.g. in the Bergner model structure, or as relative categories) | |
| Oct 28, 2021 at 7:30 | history | edited | D.-C. Cisinski | CC BY-SA 4.0 |
added 121 characters in body
|
| Oct 28, 2021 at 7:22 | history | edited | D.-C. Cisinski | CC BY-SA 4.0 |
added 30 characters in body
|
| Oct 28, 2021 at 6:28 | history | edited | D.-C. Cisinski | CC BY-SA 4.0 |
added 190 characters in body
|
| Oct 28, 2021 at 6:21 | history | edited | D.-C. Cisinski | CC BY-SA 4.0 |
added 5 characters in body
|
| Oct 28, 2021 at 6:14 | history | edited | D.-C. Cisinski | CC BY-SA 4.0 |
added 170 characters in body
|
| Oct 27, 2021 at 23:23 | history | edited | D.-C. Cisinski | CC BY-SA 4.0 |
deleted 3 characters in body
|
| Oct 27, 2021 at 23:03 | history | edited | D.-C. Cisinski | CC BY-SA 4.0 |
added 33 characters in body; added 3 characters in body
|
| Oct 27, 2021 at 22:58 | history | edited | D.-C. Cisinski | CC BY-SA 4.0 |
added 66 characters in body
|
| Oct 27, 2021 at 22:53 | history | edited | D.-C. Cisinski | CC BY-SA 4.0 |
added 274 characters in body
|
| Oct 27, 2021 at 22:45 | history | edited | D.-C. Cisinski | CC BY-SA 4.0 |
added 102 characters in body
|
| Oct 27, 2021 at 22:32 | history | answered | D.-C. Cisinski | CC BY-SA 4.0 |