Strictification of $\infty$-Groupoids is Comonadic

TL;DR

This paper demonstrates that the process of strictifying weak ∞-groupoids to strict ∞-groupoids can be understood as a comonadic adjunction, with any simplicial set recoverable via totalization of a canonical cosimplicial resolution.

Contribution

It proves the strictification adjunction is comonadic and generalizes the Bousfield-Kan result to ∞-groupoids, establishing a new categorical framework.

Findings

Any simplicial set can be recovered as the totalization of its canonical cosimplicial resolution.

The strictification adjunction induces a comonadic adjunction between ∞-groupoids and simplicial sets.

Generalization of Bousfield-Kan's result to ∞-groupoids.

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.