Given a simplicial commutative semigroup:


(1) is it true that its underlying simplicial set is a Kan complex if and only if the simplicial semigroup was a simplicial group?


(2) is the constant simplicial set on a set, Kan fibrant?


A positive answer to (2) would give a negative answer to (1), since the constant simplicial semigroup that is degreewise the natural numbers would be Kan fibrant and not a simplicial group.



Kan complexes and semigroups
(2) is true (and so (1) is false).


To see it, note that every horn $\Lambda^n\_i\to S$ to a constant simplicial set must be constant, and so it can be filled by the constant horn $\Delta^n\to S$. Equivalently, disjoint unions of Kan complexes are Kan complexes and $\Delta^0$ is a Kan complex.


