Timeline for 1-categorical universal properties for the smash product of pointed sets
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 10 at 19:18 | comment | added | Maxime Ramzi | Of course, it's my pleasure :) I wrote an answer with what I know so far, hopefully someone else can chime in and figure out what I don't know | |
| Mar 10 at 19:18 | answer | added | Maxime Ramzi | timeline score: 1 | |
| Mar 10 at 18:46 | comment | added | Emily | As always, thank you so much, Maxime, I truly appreciate how helpful, kind, and patient you've always been to me regarding my questions here. Thank you so much! :) | |
| Mar 10 at 18:46 | comment | added | Emily | I also feel like preservation of colimits ought to follow from $(-)_+$ being strong monoidal too by applying this, but I can't seem how to conjure a proper proof either =/ | |
| Mar 10 at 18:46 | comment | added | Emily | @MaximeRamzi Ahhh that's too bad, having such a simple proof did sound too good to be true =/ | |
| Mar 10 at 18:31 | comment | added | Maxime Ramzi | Emily: that would indeed determine the functor $\wedge$, but not necessarily the monoidal structure, which is more than just that functor (by the way, the reason I haven't written an answer to your question yet is that I'm thinking about II : I've been trying to figure out whether one needs to assume that the monoidal structure preserves colimits in each variable or whether it follows from $(-)_+$ being strong monoidal; it seems like it does but I haven't been able to prove it yet. For UP I, I could already write an answer following the lines I suggested earlier) | |
| Mar 10 at 17:04 | comment | added | Emily | same as elements of $X$, and then asking that this internal Hom have a left adjoint in each variable assembling into a closed monoidal category would determine $\wedge$ up to natural isomorphism. What do you think? | |
| Mar 10 at 17:04 | comment | added | Emily | @MaximeRamzi I was thinking about UP I: is it equivalent to saying that the smash product of pointed sets and all the rest of the closed monoidal structure it comes with is the unique closed monoidal structure on $\mathsf{Sets}_*$ with unit $S^{0}$? I ask because I got the following proof idea for the later: the isomorphism $$\mathrm{Hom}_{\mathsf{Sets}_*}(S^{0},[X,Y])\cong\mathrm{Hom}_{\mathsf{Sets}_*}(X,Y),$$ valid for any closed monoidal category, seems to force $[X,Y]$ to be the internal Hom of pointed sets since morphisms of pointed sets from $S^{0}$ to a pointed set $X$ are the [...] | |
| Mar 9 at 19:02 | comment | added | Emily | @MaximeRamzi I think that would be fine, too. The only thing I want to avoid is using tools from $\infty$-category theory, so a proof using only 1- and 2-category theory would actually be lovely, even if it follows the one from $\infty$-category theory rather closely. | |
| Mar 9 at 18:20 | comment | added | Maxime Ramzi | I actually don't know how to give an enlightening proof that isn't the same as the $\infty$-categorical version (suitably contextualized), but presumably you would want to avoid that as well? | |
| Mar 8 at 16:59 | comment | added | Emily | @MaximeRamzi Yep, a $(2,1)$-categorical (or $2$-categorical) universal property for those would be lovely! | |
| Mar 8 at 4:19 | comment | added | Maxime Ramzi | Since you mention explicitly that you don't want $\infty$-categories in the picture it might help to make explicit what "unique" means. The answer to your question as I understand it is yes, in a suitable $(2,1)$-categorical sense, but I just want to make sure that you allow this | |
| Mar 7 at 19:54 | comment | added | Emily | A note on asking two questions at once: I'm asking two questions at once here since I expect there might be an easy reference or proof for them, and if someone knows the answer to one of the questions, then there's a good chance that they'll also know the answer for the other one. If this proves to be false, I'll split the question in two. | |
| Mar 7 at 19:54 | history | asked | Emily | CC BY-SA 4.0 |