Timeline for Object classifiers in 1-toposes
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 25, 2022 at 13:15 | vote | accept | Giulio Lo Monaco | ||
| May 6, 2021 at 11:05 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Apr 6, 2021 at 10:46 | answer | added | Giulio Lo Monaco | timeline score: 2 | |
| Oct 2, 2020 at 21:13 | comment | added | Jonathan Sterling | @GiulioLoMonaco This is the standard reference for the construction I referred to: www2.mathematik.tu-darmstadt.de/~streicher/NOTES/UniTop.pdf If yours turns out to be different, it might be a nice result that type theorists would appreciate. I encourage you to write it up in an answer. | |
| Oct 2, 2020 at 21:11 | comment | added | Jonathan Sterling | @GiulioLoMonaco the approach i had in mind was the classic one, where you use the equivalence between “slice of presheaf category over a representable” with “presheaves on a slice”. then, one gets a strict functorial action by composition. | |
| Oct 2, 2020 at 13:27 | comment | added | Giulio Lo Monaco | @JonathanSterling I worked out a proof myself in the meantime, which is indeed in the spirit of Ivan's comment. On presheaves, it is based on choosing a set of representatives for equivalence classes of relatively $\kappa$-compact morphisms over $y(c)$ as elements of $U(c)$. This can be transferred through a left exact localization, provided that we are not asking for uniqueness of the classifying maps. Is the strategy you have in mind similar to this? | |
| Oct 2, 2020 at 1:59 | comment | added | Jonathan Sterling | A weak classifier of k-compact families can be obtained in a 1 topos in a standard way due to hofmann and streicher. For presheaves, there is a strict enough way to give a correct definition of a presheaf in the spirit of Ivan’s attempt (which sadly did not have a functorial action). Streicher observed that by sheafifying this weakly classifying family you get a weakly classifying family for sheaves too! this follows in essence from the left exactness of the localization . If people want me to put the details into an answer, i can try to do so. | |
| Sep 24, 2020 at 10:39 | comment | added | Zhen Lin | I feel like something like this should work. Let $\mathcal{E}$ be a Grothendieck topos and let $\mathbf{s} \mathcal{E}$ have the model structure where the cofibrations are the monomorphisms and the weak equivalences are the internal weak homotopy equivalences. $\mathbf{s} \mathcal{E}$ has a classifier for small 0-truncated objects, say $\tilde{U} \to U$, which we may take to be a fibration with $U$ fibrant. Pull back along $U_0 \hookrightarrow U$ to get a fibration over a "discrete" object. If the fibres are also "discrete" then we are done, but I don't know if this happens / can be forced. | |
| Sep 23, 2020 at 15:30 | comment | added | Giulio Lo Monaco | I would need an equivalence relation on $\text{Nat}(y(c), U)$ and one on $\{ \text{relatively}\ \kappa \text{-compact morphisms over}\ c \}$ in such a way that the two quotients are naturally isomorphic, but then I could no longer use Yoneda. | |
| Sep 23, 2020 at 15:08 | comment | added | Ivan Di Liberti | Is there any chance that we define an equivalence relation on U so that the quotient is the object that you desire? | |
| Sep 23, 2020 at 11:39 | comment | added | Giulio Lo Monaco | Your second step is not applicable to my case, because there might be more than one relatively $\kappa$-compact morphism over $y(c)$ corresponding to a map $y(c) \to U$. Of course all the candidates will only differ by an isomorphism, but still it doesn't give a bijection of sets. That's the main issue in moving from $\infty$- to 1-toposes. | |
| Sep 23, 2020 at 11:03 | comment | added | Ivan Di Liberti | There is a standard strategy that always works. Say that your topos is $\mathsf{Sh}(C)$ for a site $C$. Now, if such an object $U$ exists, it means that $$U(c) \stackrel{\text{Yon}}{\cong} \text{Nat}(y(c), U) \stackrel{\text{U.P.}}{\cong} \{\kappa\text{-compact morphism over } c\}.$$ This observation gives you the only possible definition for $U$ and is the standard strategy to build the sub. classifier. Whether this is a sheaf, I don't know. | |
| Sep 23, 2020 at 10:12 | history | asked | Giulio Lo Monaco | CC BY-SA 4.0 |