Timeline for References about "monoidal fibrations" in $\infty$-category theory
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 4, 2020 at 7:06 | vote | accept | Maxime Ramzi | ||
| Sep 29, 2020 at 11:51 | comment | added | Maxime Ramzi | Yes, that's actually what I used (using the enveloppe of an $\infty$-operad, I reduced to the case of strong symmetric monoidal functors). But I only need the statement about categories of lax symmetric monoidal functors (although in any case, I had the additional condition at my disposal) | |
| Sep 29, 2020 at 11:50 | history | edited | Yonatan Harpaz | CC BY-SA 4.0 |
added 2 characters in body
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| Sep 29, 2020 at 11:48 | comment | added | Yonatan Harpaz | Condition 3 will be needed if you wanted to prove that the induced map on categories of strong symmetric monoidal functors is also a cartesian fibration, see the last part of the proof above. | |
| Sep 29, 2020 at 11:44 | comment | added | Maxime Ramzi | That does sound very useful ! I might change the way I wrote things up then (I ended up just writing it because there were no references - but the way I wrote it, I use condition 3, and it's longer than what you wrote) | |
| Sep 29, 2020 at 11:41 | comment | added | Yonatan Harpaz | So I used the HTT reference for this, and looking at it again I see that for this one needs condition 1, i.e., that $p^{\otimes}$ preserves cocartesian edges, but not condition 3. That HTT reference (and especially the following particular case of it) is actually quite useful in practice. Applying its dual version with $K=\mathcal{E}_0 = \Delta^0$ one obtains the statement that for a map of cocartesian fibrations (which preserves cocartesian edges), an edge contained in a fiber is cartesian with respect to the induced map on fibers if and only if it is cartesian on the total spaces. | |
| Sep 28, 2020 at 10:07 | comment | added | Maxime Ramzi | It seems like you're claiming that condition 3 is not necessary, is that correct ? or am I missing the point in your argument where you use that ? (in the argument I had in mind, I used it to prove that $p^\otimes_{\langle n \rangle}$-cartesian edges were $p^\otimes$-cartesian; which it seems you're proving more generally) | |
| Sep 27, 2020 at 12:41 | history | answered | Yonatan Harpaz | CC BY-SA 4.0 |