Timeline for Colimits of algebras for $\infty$-Monad
Current License: CC BY-SA 4.0
8 events
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| Sep 26, 2018 at 10:05 | comment | added | Simon Henry | @DylanWilson : It seems that a form of the adjoint functor theorem sufficient for your argument is in arxiv.org/abs/1803.01664 Though it still does not really answer the question: It is not really an effective construction of the colimits, and it uses a very big limits in the base category instead of iterated colimits. | |
| Sep 17, 2018 at 14:20 | comment | added | Dylan Wilson | re-added it, but the thing about being able to apply the adjoint functor theorem doesn't immediately work I think. Should still be true, but need to apply a more general version of the adjoint functor theorem than the one in HTT. In the 1-categorical case this is the "reflection theorem" in Adamek-Rosicky | |
| Sep 17, 2018 at 14:19 | comment | added | Dylan Wilson | Hmm... well are we allowed to assume that the category $\mathcal{C}$ is presentable and that the monad $T$ is accessible? Then $\mathsf{Alg}_T$ is at least accessible using the above arguments. But it has all limits, so it's automatically presentable as well (has a fully faithful functor to presheaves on compact generators (take free algebras on compact things to get those) and the functor is accessible. Since the target is presentable, the adjt functor theorem supplies a left adjoint, so we've expressed $\mathsf{Alg}_T$ as an accessible localization of a presheaf category). | |
| Sep 17, 2018 at 14:16 | comment | added | Simon Henry | Interesting ! So I'm completely fine with those assumptions, but what I'm after is an 'explicit' construction of these colimits using some transfinite construction (in the spirit of Kelly's construction). Something that can be used to prove concrete properties of the colimits in concrete situation. Your remarks does seems to qualifies as a proof that colimits exists, but I'm not really convinced yet that this give an explicit description of them... (but maybe it does) | |
| Sep 17, 2018 at 11:04 | comment | added | Simon Henry | @DylanWilson : These shows that for a monad M preserving colimits of shape K, the category of M-algebras has colimits of shape K and these are preserved by the forgetful functor to C. This case doesn't require transfinite induction. I'm interested in the case where the monad M does not preserves colimits (only filtered enough colimits) so that one needs to do some iterated colimits/application of M to gradually correct the colimits in C into a colimits in M-algebras. | |
| Sep 17, 2018 at 10:31 | comment | added | Dylan Wilson | How about Higher Algebra 4.2.3.5 and 4.2.3.7 in the case where C is the category of endofunctors for M? | |
| Sep 17, 2018 at 8:58 | history | edited | Martin Sleziak |
added the (monads) tag - feel free to rollback my edit if I missed something and the tag is not suitable here
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| Sep 17, 2018 at 8:50 | history | asked | Simon Henry | CC BY-SA 4.0 |