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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.

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    $\begingroup$ Nerves of groupoids are Kan complexes, and sets are groupoids with only identity arrows. $\endgroup$
    – David Roberts ♦
    Commented Sep 30, 2017 at 12:27

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(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.

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