Timeline for Example of a locally presentable locally cartesian closed category which is not a topos?
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 28, 2018 at 20:50 | comment | added | Mike Shulman | Followup question: mathoverflow.net/questions/294108/… | |
| Feb 28, 2018 at 5:55 | vote | accept | Tim Campion | ||
| Feb 28, 2018 at 5:40 | answer | added | Marc Hoyois | timeline score: 11 | |
| Feb 28, 2018 at 3:12 | comment | added | Tim Campion | Ok... so the upshot is that the statement "every 0-truncated group object is the loop space of something [its bar construction?]" is an exactness condition that holds in $\infty$-toposes but not in locally cartesian closed $\infty$-categories. That's really neat! I'm still curious if there are similar conditions that hold in the 1-category case. | |
| Feb 28, 2018 at 3:04 | comment | added | Marc Hoyois | It's not easy: one can show that the free (discrete) abelian group on the punctured affine line is not a loop space, see Remark 3.1.7 in arxiv.org/pdf/1711.05248.pdf. | |
| Feb 28, 2018 at 2:55 | comment | added | Tim Campion | Oh -- that's a perfect example! Is it easy to see that it's not an $\infty$-topos? | |
| Feb 28, 2018 at 2:54 | comment | added | Marc Hoyois | Right. If you allow ∞-categories a standard example would be motivic spaces, but I don't have another 1-categorical example at hand. | |
| Feb 28, 2018 at 2:53 | comment | added | Tim Campion | Well, a locale is a (1,0)-topos, right? Maybe I should ask my category to be large. | |
| Feb 28, 2018 at 2:49 | history | edited | Tim Campion | CC BY-SA 3.0 |
added 310 characters in body
|
| Feb 28, 2018 at 2:46 | comment | added | Marc Hoyois | Isn't a locale locally cartesian closed? | |
| Feb 28, 2018 at 2:39 | history | edited | Martin Sleziak |
there are tags for counterexamples, locally presentable categories, toposes; to me they seem (at least to some extent) relevant to the question - of course if there is a better choice of tags or if some of the tags I've added does not fit, go ahead and edit the tags further
|
|
| Feb 28, 2018 at 2:35 | history | asked | Tim Campion | CC BY-SA 3.0 |