Timeline for On morphism spaces of $(\infty,2)$-categories failing to be $\infty$-categories
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Oct 31, 2022 at 21:31 | comment | added | Dylan Wilson | from that link it makes one wonder if the "balanced hom space" is trying to be some category of (co)spans | |
| Oct 31, 2022 at 1:17 | comment | added | Emily | There's also some funny stuff going on with morphism spaces for Duskin nerves, too: even in this very special case the morphism spaces can still fail to be $\infty$-categories, and having the hom-categories of the bicategory in question admit pushouts actually guarantees the morphism spaces to be $(\infty,\mathbf{2})$-categories! | |
| Oct 31, 2022 at 1:13 | comment | added | Emily | Lurie's Fun(-,-) is just the Hom of simplicial sets, but I'm not really sure how exactly to think about its simplices in the case of $(\infty,2)$-categories... For instance, I'm not sure if they are supposed to be lax natural transformations or pseudo ones (I guess it would be lax ones since Lurie doesn't mention thin $2$-simplices), and the same goes for the higher simplices. | |
| Oct 30, 2022 at 18:12 | comment | added | Tim Campion | Er -- It's possible I've misunderstood the notation $Fun(-,-)$, then. $[A,B]^{lax}$ in my notation should have objects the (non-lax) 2-functors $A \to B$, 1-morphisms the lax natural transformations between these, and 2-morphisms the lax modifications between those. | |
| Oct 30, 2022 at 17:22 | comment | added | Emily | (I'll leave the question open for a little longer, just in case someone has something to add about the model-dependent aspects of this issue. Thanks again for your answer! =) | |
| Oct 30, 2022 at 17:20 | comment | added | Emily | One point that still bugs me though is that the pinched morphism spaces still manage to be $\infty$-categories while the non-pinched morphism spaces fail to be so, and the only difference between them is that the non-pinched morphism spaces have "two times as many homotopies"; e.g. the $1$-simplices of Hom(A,B) look like a divided square with two $1$-simplices and those of $Hom^L(A,B)$ (resp. $Hom^R(A,B)$) look like the right-top (resp. left-bottom) triangle in this square... | |
| Oct 30, 2022 at 17:20 | comment | added | Emily | Oh, the problem was the lax hom, thanks! I guess the problem then might be that Lurie's $Fun(\Delta^1,C)$ in the case of $(\infty,2)$-categories is supposed to be the "hom of strictly unitary lax functors", as remarked here, instead of those of general lax functors. (Indeed, the $1$-simplices in the morphism spaces Hom(A,B) look like the square here, which is already a kind of "lax square" with the two $2$-simplices of $C$ there.) | |
| Oct 30, 2022 at 3:32 | history | answered | Tim Campion | CC BY-SA 4.0 |