Strictification of $\infty$-Groupoids is Comonadic

Abstract

We investigate the universal strictification adjunction from weak $\infty$-groupoids (modeled as simplicial sets) to strict $\infty$-groupoids (modeled as simplicial T-complexes). We prove that any simplicial set can be recovered up to weak homotopy equivalence as the totalization of its canonical cosimplicial resolution induced by this adjunction. This generalizes the fact due to Bousfield and Kan that the homotopy type of a simply connected space can be recovered as the totalization of its canonical cosimplicial resolution induced by the free simplicial abelian group adjunction. Furthermore, we leverage this result to show that this strictification adjunction induces a comonadic adjunction between the quasicategories of simplicial sets and strict $\infty$-groupoids.