Timeline for From the *usual* nerve of topological categories to $\infty$-categories
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Sep 3, 2023 at 0:27 | comment | added | Xin Jin | @DavidRoberts Yes, a category enriched over Top. | |
| Sep 2, 2023 at 23:54 | comment | added | David Roberts♦ | What do you mean by topological category? Enriched over Top? | |
| Sep 2, 2023 at 20:31 | vote | accept | Xin Jin | ||
| Sep 2, 2023 at 19:48 | answer | added | Dmitri Pavlov | timeline score: 5 | |
| Sep 2, 2023 at 19:17 | history | edited | Xin Jin | CC BY-SA 4.0 |
added 47 characters in body
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| Sep 2, 2023 at 19:11 | comment | added | Xin Jin | For the latter, it seems there might be a way to make a "big" canonical topological category whose objects are all points $x\in X$ and between different points $x,x'$ the morphism space is the space of paths connecting $x, x'$. But the result is equivalent to $B(\Omega_xX)$. By the way, I'm more interested in whether the construction (1) realizes $N$. | |
| Sep 2, 2023 at 19:09 | comment | added | Xin Jin | @DmitriPavlov: $Cat_\infty$ is the $\infty$-category of quasi-categories. It is either $(Set_\Delta)^{J,cf}$ as an $\infty$-category or modeled as the $\infty$-category of complete Segal spaces. I agree the construction of (2) depends on choices of basepoints, which is similar to $B(\Omega_x X)\simeq X$ for a connected space $X$. Actually, my intuition of (2) is from the "correspondence" from a connected space $X$ as a quasi-category, to the topological category $B(\Omega_x X)$ (with a single object and endomorphism space $\Omega_x X$). | |
| Sep 2, 2023 at 13:59 | comment | added | Dmitri Pavlov | What is Cat_∞? It is not defined in your post. Also, you seem to claim that (1) and (2) define adjoint functors, but (2) starts with an ∞-category (which presumably means quasicategory here), and it seems that your Cat_∞ means something else than quasicategories. Finally, the construction of (2) as it is currently stated makes noncanonical choices of basepoints, so it is not even a functor, let alone an adjoint functor. | |
| Sep 1, 2023 at 19:11 | history | asked | Xin Jin | CC BY-SA 4.0 |