Timeline for Is the inclusion functor from gaunt strict $n$-categories to weak $(\infty,n)$-categories fully faithful?
Current License: CC BY-SA 4.0
8 events
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| Mar 29, 2020 at 22:07 | comment | added | Tim Campion | @AlexanderCampbell That does sound like a reasonable guess. I wonder how it compares in general to taking the Rezk completion of the cellular nerve as Simon suggests below... I also wonder if there's a model-independently characterization of the "correct" functor... | |
| Mar 29, 2020 at 21:17 | comment | added | Alexander Campbell | A reasonable guess (which we know works for $n=2$) of a right Quillen functor from $n$-categories to $n$-quasi-categories is the nerve/singular functor induced by a Reedy cofibrant replacement of the full inclusion $\Theta_n \to \mathbf{Cat}_n$. Compose this with Ara's right Quillen functor from $n$-quasi-categories to Rezk $\Theta_n$-spaces to get a right Quillen "classifying diagram" functor. Since gaunt $n$-categories "see" weak equivalences as isomorphisms, this agrees on gaunt $n$-categories with the full inclusion I mentioned above. | |
| Mar 28, 2020 at 15:06 | comment | added | Tim Campion | @AlexanderCampbell Thanks. I think what confuses me now, as discussed with Simon below, is the fact that the cellular nerve doesn't generally produce a functor $Cat_n \to Cat_{(\infty,n)}$ unless Rezk completion is applied, which makes the situation a bit murky. Do you have any insight on how to produce the "correct" functor $Cat_n \to Cat_{(\infty,n)}$ and check that it agrees with the cellular nerve on $Gaunt_n$? | |
| Mar 28, 2020 at 14:54 | vote | accept | Tim Campion | ||
| Mar 28, 2020 at 14:59 | |||||
| Mar 27, 2020 at 22:06 | answer | added | Simon Henry | timeline score: 3 | |
| Mar 27, 2020 at 22:05 | comment | added | Alexander Campbell | This is easy to see using Rezk's $\Theta_n$-space model for $(\infty,n)$-categories. The category of gaunt n-categories is equivalent (via the cellular nerve functor) to the full (simplicially enriched) subcategory of Rezk $\Theta_n$-spaces spanned by the "discrete-valued" ones. | |
| Mar 27, 2020 at 21:43 | history | edited | Tim Campion | CC BY-SA 4.0 |
added 279 characters in body
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| Mar 27, 2020 at 21:35 | history | asked | Tim Campion | CC BY-SA 4.0 |