Timeline for Intuition for isofibrations in $\infty$-categories
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 10, 2022 at 8:59 | vote | accept | Gabriel | ||
| Nov 9, 2022 at 21:50 | comment | added | Zhen Lin | FWIW these comments also apply to isofibrations of ordinary categories. They are technically convenient and allow us to strictify. | |
| Nov 9, 2022 at 20:06 | comment | added | Kevin Carlson | @ManuelAraújo Just to clarify, the isofibrations aren't precisely the fibrations in the Joyal model structure. But every fibration is an isofibration, and every isofibration between quasicategories is a fibration, so in the $\infty$-cosmos context (where every object is fibrant) the more mysterious Joyal fibrations can be ignored. | |
| Nov 9, 2022 at 20:03 | answer | added | Kevin Carlson | timeline score: 7 | |
| Nov 9, 2022 at 9:22 | comment | added | Manuel Araújo | I am also not an expert (hence the comment instead of an answer), but here is an example. Say you want to to compute a homotopy pullback of a diagram of spaces. If one of the maps involved is a fibration, you can just take the strict pullback of the diagram. | |
| Nov 9, 2022 at 9:07 | comment | added | Gabriel | Dear @ManuelAraújo, would you mind explaining a little why this fact allow them work with 2-categories instead of bicategories? (I also know very little about model categories, but I would appreciate any explanation.) | |
| Nov 9, 2022 at 8:56 | comment | added | Manuel Araújo | They are the fibrations in the Joyal model structure on simplicial sets, whose fibrant objects are quasicategories. This means in particular they achieve what is stated in the Introduction: "To help us achieve this counterintuitive strictness, each ∞-cosmos comes with a specified class of maps between ∞-categories called isofibrations. The isofibrations have no homotopy-theoretic meaning, as any functor between ∞-categories is equivalent to an isofibration with the same codomain". | |
| Nov 9, 2022 at 8:38 | history | asked | Gabriel | CC BY-SA 4.0 |