Timeline for Colimit density and monads
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
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| Jan 13, 2015 at 12:38 | comment | added | Włodzimierz Holsztyński | @YemonChoi -- S. had to tag that question as a counterexample; indeed, if s/he didn't then it would be a counterexample. There is no way around it. | |
| Jan 13, 2015 at 4:32 | answer | added | Tim Campion | timeline score: 4 | |
| Jan 13, 2015 at 2:44 | comment | added | Tim Campion | @arsmath Well, the representables form a dense generating family in any presheaf category. My original intuition was that the coproduct of representables would automatically be a dense generating object, but this is totally wrong in general. Nonetheless, $\sum_n \Delta^n$ is a dense generating object in simplicial sets because the representables are all retracts of it, and it's easy to see that a full subcategory is dense if its idempotent completion is dense. All these arguments apply to $\mathsf{Cat}$, too by considering it as a subcategory of simplicial sets. | |
| Jan 12, 2015 at 7:56 | vote | accept | arsmath | ||
| Jan 12, 2015 at 7:42 | comment | added | arsmath | @ZhenLin Colimit-dense isn't a standard term? I've seen it a bunch of times, such as here or here. | |
| Jan 12, 2015 at 7:37 | comment | added | arsmath | @TimCampion: Do both Cat and Simplicial Sets have a single colimit-dense generator? That surprises me. | |
| Jan 12, 2015 at 1:39 | comment | added | Tadashi | @YemonChoi Sorry... I read the first comment thinking it was already a counterexample; Now to think about it, I guess I have misused this tag in another questions too. Thanks for the advice and sorry for the trouble. | |
| Jan 12, 2015 at 1:38 | answer | added | Todd Trimble | timeline score: 10 | |
| Jan 12, 2015 at 0:42 | comment | added | Yemon Choi | @Shamisen: are you going to tag every question that asks "is this true or false" with the counterexamples tag? | |
| S Jan 12, 2015 at 0:21 | history | suggested | Tadashi |
Added relevant tag
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| Jan 12, 2015 at 0:13 | comment | added | Todd Trimble | @TimCampion One way of seeing that $Cat$ can't be monadic (in any way, shape, or form) over $Set$ is that such categories are regular, and $Cat$ isn't. This result is true without conditions on rank. | |
| Jan 11, 2015 at 23:51 | review | Suggested edits | |||
| S Jan 12, 2015 at 0:21 | |||||
| Jan 11, 2015 at 23:34 | comment | added | Zhen Lin | @arsmath What do you mean by "colimit dense"? Do you just mean dense? | |
| Jan 11, 2015 at 23:25 | comment | added | Tim Campion | I think the category $\mathsf{Cat}$ of small categories is a counterexample. Or Simplicial Sets. More generally, if a category is locally $\lambda$-presentable but its $\lambda$-presentable objects are not generated under $\lambda$-small colimits by a single object, I would not expect it to be monadic over $\mathsf{Set}$. But I'm not that familiar with what can actually be said about categories that are monadic over $\mathsf{Set}$ without rank, so I don't have a proof. | |
| Jan 11, 2015 at 22:53 | history | asked | arsmath | CC BY-SA 3.0 |