Timeline for Do finitely presentable $\infty$-groupoids precisely correspond to the finite cell complexes?
Current License: CC BY-SA 4.0
11 events
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| Dec 9, 2023 at 18:02 | comment | added | Arshak Aivazian | @TimCampion Thanks for this comment! This is a really important clarification (and it helped me now). | |
| Nov 14, 2023 at 5:30 | comment | added | Tim Campion | A big warning is in order: there is a standard notion of a finitely presentable object in a category. Later in the book, Lurie treats the generalization of this notion to $\infty$-categories, calling it a "compact" object. Beware that not every compact object in the $\infty$-category $Spaces$ of spaces is "finitely presentable" in the sense of 1.2.14.2, by the Wall finiteness obstruction. What is true is that every compact object in $Spaces$ is a retract of a "finitely presentable" object in the sense of 1.2.14.2. | |
| Nov 11, 2023 at 20:24 | vote | accept | Arshak Aivazian | ||
| Nov 11, 2023 at 17:16 | history | edited | David White | CC BY-SA 4.0 |
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| Nov 11, 2023 at 16:43 | answer | added | David White | timeline score: 3 | |
| Nov 10, 2023 at 18:10 | comment | added | Arshak Aivazian | Really! And in fact, this is discussed further. It was a stupid question, I am sorry. | |
| Nov 10, 2023 at 18:08 | history | undeleted | Arshak Aivazian | ||
| Nov 10, 2023 at 18:08 | history | deleted | Arshak Aivazian | via Vote | |
| Nov 10, 2023 at 18:06 | comment | added | Connor Malin | I'm no expert, but I believe a simplicial set presents an $\infty$-category $C$ if they are weakly equivalent in the Joyal model structure on simplicial sets. Similarly, a simplicial set will present an $\infty$-groupoid $X$ if they are weakly equivalent in the standard model structure on simplicial sets (i.e. the one that models homotopy types). From this one can see that $\infty$-groupoids with infinite homological dimension cannot be finite presented (meaning represented by a finite simplicial set). | |
| Nov 10, 2023 at 17:20 | history | edited | Arshak Aivazian | CC BY-SA 4.0 |
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| Nov 10, 2023 at 17:06 | history | asked | Arshak Aivazian | CC BY-SA 4.0 |