Timeline for Colimit density and monads
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Jan 14, 2015 at 5:57 | comment | added | Tim Campion | Indeed, $\mathsf{Cat}$ has no regular projective generator. If $C$ is projective, then it lifts against the above-mentioned coequalizer, which implies that any functor $C \to \mathbf{B}\mathbb{N}$ is constant. But there are certainly functors out of $\mathbf{B}\mathbb{N}$ which can't be distinguished by where they send objects so $C$ is not a generator. | |
| Jan 14, 2015 at 0:20 | comment | added | Tim Campion | Another obvious thing that just dawned on me: none of the generators considered for $\mathsf{Cat}$ is (regular) projective. Lifting against the coequalizer $1 \overset{\to}{\to} 2 \to \mathbf{B}\mathbb{N}$ fails for all of them, and I suspect that $\mathsf{Cat}$ lacks a projective generating set. The only generator in $\mathsf{sSet}$ which I'm sure is projective is $\sum_n \Delta^n$. | |
| Jan 13, 2015 at 16:22 | comment | added | Tim Campion | All good reasons. I hope my comment didn't come across as negative. Also it occurs to me that maybe a "minimal" dense generating object of $\mathsf{sSet}$ would be the infinite unit ball: the colimit of a chain of face maps $\Delta^n \to \Delta^{n+1}$. I should also point out that basically everything I've said here is probably best viewed as an unfolding of Todd's comment about $\sum_n \Delta^n$. | |
| Jan 13, 2015 at 14:17 | comment | added | Todd Trimble | No, that's not why I chose $Pos$. I chose $Pos$ partly because it's the first example which came to my mind when I read the question, and partly for the sake of variety after I read your comment, and partly because general coequalizers in $Cat$ are messy, so that the easiest proofs of non-regularity already live in $Pos$. | |
| Jan 13, 2015 at 12:29 | history | edited | Tim Campion | CC BY-SA 3.0 |
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| Jan 13, 2015 at 12:25 | comment | added | Tim Campion | I just made an edit -- the ordinal 3 is actually already a dense generator of $\mathsf{Cat}$, which certainly seems more natural, given that it represents composition of morphisms. | |
| Jan 13, 2015 at 12:15 | history | edited | Tim Campion | CC BY-SA 3.0 |
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| Jan 13, 2015 at 12:08 | history | edited | Tim Campion | CC BY-SA 3.0 |
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| Jan 13, 2015 at 10:15 | comment | added | arsmath | Thanks, that example definitely never would have occurred to me. | |
| Jan 13, 2015 at 4:44 | history | edited | Tim Campion | CC BY-SA 3.0 |
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| Jan 13, 2015 at 4:32 | history | answered | Tim Campion | CC BY-SA 3.0 |