Timeline for Reference request: The unit of an adjunction of $\infty$-categories in the sense of Riehl-Verity is a unit in the sense of Lurie
Current License: CC BY-SA 4.0
18 events
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| Jul 28, 2019 at 11:53 | comment | added | Robin Stoll | @DenisNardin Thanks for adding the explanation. It is quite helpful. (Sorry for the very late reply.) | |
| Jul 16, 2019 at 8:21 | history | edited | Denis Nardin | CC BY-SA 4.0 |
Cleaned up a little since the question was bumped to the front page anyway.
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| Jul 13, 2019 at 9:22 | comment | added | Denis Nardin | @RobinStoll Sorry, I had initially misunderstood your comment (I thought you were referring to the first of the three equivalences, not to the second). Hopefully the explanation I added addresses your question now. I actually have a huge commutative diagram in my notes showing that all the equivalences can be made completely natural, although I don't really want to try to render it in AMScd :) | |
| Jul 12, 2019 at 21:21 | history | edited | Denis Nardin | CC BY-SA 4.0 |
Added explanation of naturality
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| Jul 11, 2019 at 21:52 | comment | added | Robin Stoll | @DenisNardin I am not sure what you mean by "continuity of the naturality square", if not the statement I wrote. What is the difference? | |
| Jul 11, 2019 at 19:30 | comment | added | fosco | Sure, I know (but I know the result as 5.1 here; it has a different proof). I was only amused by how formal(ly formal) $\infty$-category theory has become in a few years. | |
| Jul 11, 2019 at 18:29 | comment | added | Denis Nardin | @Fosco If it was unclear, I was referring to proposition 2.3 of this article (which is of course a generalization of a standard fact in ordinary category theory) | |
| Jul 11, 2019 at 18:04 | comment | added | fosco | "by taking ends" :-) | |
| Jul 11, 2019 at 16:52 | comment | added | Robin Stoll | Ok, thanks a lot! | |
| Jul 11, 2019 at 16:40 | comment | added | Denis Nardin | @RobinStoll This starts to depend on exactly what definition you take for precomposition. I don't have time right now, but I'll try to add a short explanation today or tomorrow. Long story short: this is the map induced on fibers of the twisted arrow category fibrations, so it commutes with pushforwards and pullbacks | |
| Jul 11, 2019 at 16:33 | comment | added | Robin Stoll | Thanks for adding it. It was roughly clear to me how it goes, but I am somewhat uncomfortable with just checking it "on elements", since it needs to be continuous in $l$. It is not hard to formulate it in an element free way, but then one needs that $(e_d \circ) \circ fg \simeq (\circ e_{fc})$ as maps $\operatorname{Map}_{\mathcal D}(fc, d) \to \operatorname{Map}_{\mathcal D}(fgfc, d)$, which seems obvious, but it is unclear to me why it is true formally. (As you can tell, I am not very experienced with these things, so I am sorry if these are stupid questions.) | |
| Jul 11, 2019 at 16:27 | history | edited | Denis Nardin | CC BY-SA 4.0 |
added 21 characters in body
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| Jul 11, 2019 at 16:07 | history | edited | Denis Nardin | CC BY-SA 4.0 |
Added the "obvious" direction
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| Jul 11, 2019 at 15:51 | comment | added | Denis Nardin | @RobinStoll Sorry, I don't. But if you want I can add it to the answer. Is the level of detail in the "hard direction" ok? | |
| Jul 11, 2019 at 15:48 | comment | added | Robin Stoll | Ok, thanks a lot. Do you know of some place where at least the easy direction is written down, so that I do not have to do it myself? | |
| Jul 11, 2019 at 15:46 | vote | accept | Robin Stoll | ||
| Jul 11, 2019 at 15:01 | history | edited | Denis Nardin | CC BY-SA 4.0 |
Left & right are hard to keep apart
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| Jul 11, 2019 at 14:47 | history | answered | Denis Nardin | CC BY-SA 4.0 |