Let $X$ be a simplicial object in the category of commutative inverse semigroups (or monoids, if needed). Is the underlying simplicial set of $X$ always a Kan complex? If so, are there some nice extensions of this result to more general simplicial semigroups?

## 1 Answer

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By a theorem of Moore and Carboni–Kelly–Pedicchio, given a variety of algebras $A$, every simplicial object in $A$ is a Kan complex if and only if $A$ is a Malcev category.

The latter holds if and only if the algebraic theory that defines $A$ contains a Malcev operation, which is clearly false for commutative inverse monoids.

Thus, commutative inverse monoids do not form a Malcev category. Therefore, not every simplicial commutative inverse monoid is a Kan complex.