The cubical or simplicial-set construction of free cocompletions and finite n-groupoids

The paper considers an absolute free cocompletion of a point constructed from cubical sets, simplicial sets, or semisimplicial sets, together with three reflective localizations for each finite $n$, whose objects are $n$-groupoids enriched over sets, setoids, or equivalence relations at the top dimension. The free-cocompletion conjecture. One can constructively define such an absolute free cocompletion and these three reflective localizations for every finite $n$. This would provide the expected hierarchy of finite-dimensional groupoid-like relative free cocompletions and support the expectation that the distinctions among the three enrichments disappear in the infinite-dimensional limit.