Timeline for Is every petite category essentially small?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 6 at 9:24 | vote | accept | varkor | ||
| Jul 5 at 21:11 | answer | added | Roald Koudenburg | timeline score: 8 | |
| Jul 5 at 10:43 | comment | added | Peter LeFanu Lumsdaine | @TimCampion: I don’t follow your last comment; $\mathcal{D}(F-,D)$ is certainly set-valued for all $F$ as in the question regardless of $\mathcal{C}$, since $\mathcal{D}$ is assumed locally small. | |
| Jul 2 at 18:47 | comment | added | varkor | (There is another distinction, which is that Freyd and Street's characterisation is relevant to the 2-category of large categories, whereas Di Liberti–Loregian's is relevant to the 2-category of locally small categories.) | |
| Jul 2 at 18:43 | comment | added | varkor | @TimCampion: it is not clear that Freyd and Street's characterisation holds for any example other than ordinary categories (see this question, for instance). On the other hand, if Di Liberti–Loregian's definition captures smallness, I would expect it to immediately generalise to enriched and internal categories. | |
| Jul 2 at 18:30 | comment | added | Tim Campion | Ok note to self the relevant characterization is that $C$ is small iff $C$ and $Fun(C^{op},Set)$ are locally small, which apparently wasn’t proven in the literature until more than a decade after it was implied in Street and Walters. On the size of Categories Freyd and Street, TAC in 1991. And somehow I didn’t find this cited in the paper you link to. Anyway, have you thought about adapting Freyd and Street’s work to your setting? | |
| Jul 2 at 18:05 | comment | added | Tim Campion | Haven’t really thought much about this but it would surely be related to the definition of “small” from the original “Yoneda Structures” paper — if memory serves, the proof of correctness of this definition is in another standalone paper. | |
| Jul 2 at 16:29 | history | asked | varkor | CC BY-SA 4.0 |