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Naso (Mar 31 2022 at 08:47):What are some interesting examples of monads on simplicial sets?
Zhen Lin Low (Mar 31 2022 at 10:54):There is an idempotent monad whose algebras are (strict 1-)categories. And another whose algebras are (strict 1-)groupoids.
John Baez (Mar 31 2022 at 14:21):There's also a monad on simplicial sets whose algebras are simplicial abelian groups. This is nice because simplicial groups are equivalent to chain complexes of abelian groups, thanks to the Dold-Kan theorem.
Zhen Lin Low (Mar 31 2022 at 14:43):In general if you have a monad on the category of sets then you get a monad on the category of simplicial sets by doing everything degreewise. I think such monads are automatically simplicially enriched, but I don't remember why. The idempotent monads I mentioned earlier are also simplicially enriched, but for a different reason. But simplicial enrichment is definitely not automatic in general, and I think the monad whose algebras are symmetric simplicial sets is not simplicially enriched.
Reid Barton (Mar 31 2022 at 14:47):A different sort of example is the left or right cone construction on simplicial sets
Naso (Apr 01 2022 at 05:40):Thank you all :pray: @Reid Barton are you referring to this https://math.stackexchange.com/questions/3539674/cone-monad-on-simplicial-sets ? It is defined there as a left Kan extension. Is that the same as the ordinary cone, i.e. joining with a point? If not is it easy to say what it is concretely? I would like to understand this monad. What is its multiplication and what are its algebras. I found some more information here: https://math.ucr.edu/home/baez/trimble/barconstructions.pdf section B.4 but it's a bit beyond me.
Reid Barton (Apr 01 2022 at 10:35):I was thinking of the usual cone (join with a point, on one side or the other). The functor described there is slightly different--each connected component of the original simplicial set is joined to a separate new point.
Reid Barton (Apr 01 2022 at 10:35):For the usual cone construction, the multiplication should be identifying the two new cone points (much like the multiplication on the monad on ordinary sets).
Reid Barton (Apr 01 2022 at 10:36):I am not sure if there is any useful description or name for the algebras.