Timeline for Recasting straightening/unstraightening equivalence as $(\infty, 2)$-adjunction
Current License: CC BY-SA 4.0
10 events
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| Aug 10, 2023 at 21:39 | answer | added | Lorenzo Riva | timeline score: 1 | |
| Aug 9, 2023 at 4:13 | history | edited | Lorenzo Riva | CC BY-SA 4.0 |
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| Aug 9, 2023 at 4:08 | comment | added | Lorenzo Riva | mistakes without realizing it. | |
| Aug 9, 2023 at 4:08 | comment | added | Lorenzo Riva | @KevinArlin I honestly do not know if I managed to avoid the size issues in these comments. I should specify that $\mathrm{OpFib}(C)$ should be the large $1$-category of opfibrations over $C$ all whose sources are small, so that it comes with a functor $\mathrm{OpFib}(C) \to \mathrm{Cat}$ which I think is integral to have in the proof of str./unstr. If we pick a universe $U$ and consider only $U$-small categories, then it should still be true that functors between them form a $U$-category, so again that's not an issue. However, I am far out of my depth so I might be making obvious... | |
| Aug 9, 2023 at 3:52 | comment | added | Lorenzo Riva | But the former is $\mathrm{Hom}_{\mathrm{CAT}}(C, \mathrm{OpFib}(D))$ and the latter is $\mathrm{Hom}_{\mathrm{CAT}^\mathrm{op}}(\mathrm{Fun}(C, \mathrm{Cat}), D)$, and so we get the desired equivalence. Now, this does not explicitly solve my problem (as it assumes straightening/unstraightening) but it at least shows that the adjunction does exist. The same arguments apply in the $(\infty, 2)$ case, since $\mathrm{Fun}(\mathcal{C}, \mathrm{Cat}_\infty)$ is still an $(\infty, 1)$-category. | |
| Aug 9, 2023 at 3:49 | comment | added | Lorenzo Riva | $G = \mathrm{OpFib} : \mathrm{CAT}^\mathrm{op} \to \mathrm{CAT}$ and $F = \mathrm{Fun}(-, \mathrm{Cat}) : \mathrm{CAT} \to \mathrm{CAT}^\mathrm{op}$. Assume we already know that there is an equivalence $\mathrm{OpFib}(D) \simeq \mathrm{Fun}(D, \mathrm{Cat})$ that is natural in $\mathrm{D}$. Then we get an equivalence $\mathrm{Fun}(C, \mathrm{OpFib}(D)) \simeq \mathrm{Fun}(C, \mathrm{Fun}(D, \mathrm{Cat})) \simeq \mathrm{Fun}(D, \mathrm{Fun}(C, \mathrm{Cat}))$ that is natural in both $C$ and $D$. ... | |
| Aug 9, 2023 at 3:39 | comment | added | Lorenzo Riva | This might be a silly answer, but let's try. All $2$-notions are weak (e.g. $2$-category = bicategory). Define a $2$-adjunction to be a pair of $2$-functors $F : C \to D$ and $G : D \to C$ such that there is an equivalence, natural in $c$ and $d$, between the hom-categories $\mathrm{Hom}_D(F(c), d)$ and $\mathrm{Hom}_C(c, G(d))$. Let $\mathrm{CAT}$ be the (very large) $2$-category of large $1$-categories and let $\mathrm{Cat}$ be the (large) $1$-category of small $1$-categories, so that $\mathrm{Cat}$ is an object of $\mathrm{CAT}$. Then there are $2$-functors... | |
| Aug 8, 2023 at 15:39 | comment | added | Kevin Carlson | There are very tricky size issues with the kind of argument you're aiming at even for 1-categories, and size issues are at the very heart of adjoint functor theorems. I'd be quite interested to see whether you think you can reproduce the equivalence between Grothendieck opfibrations and functors into Cat, or even its discrete analogue, in this way. | |
| Aug 7, 2023 at 15:33 | history | edited | Lorenzo Riva | CC BY-SA 4.0 |
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| Aug 6, 2023 at 20:56 | history | asked | Lorenzo Riva | CC BY-SA 4.0 |