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Strictification of $\infty$-Groupoids is Comonadic

Kimball Strong

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.

Strictification of $\infty$-Groupoids is Comonadic

Abstract

We investigate the universal strictification adjunction from weak -groupoids (modeled as simplicial sets) to strict -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 -groupoids.
Paper Structure (19 sections, 36 theorems, 19 equations)

This paper contains 19 sections, 36 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Technical background
  3. Algebraic Kan complexes
  4. Simplicial T-complexes
  5. Crossed complexes
  6. The Dold-Kan correspondences: abelian and nonabelian
  7. The right edge: the Dold-Kan correspondence
  8. The left edge: the nonabelian Dold-Kan correspondence
  9. The top edge: the adjunction $\mathrm{sTCom} \rightleftarrows \mathrm{sAbGrp}$
  10. The bottom edge: the adjunction $\mathrm{CrCom} \rightleftarrows \text{Ch}_\mathbb{Z}^+$
  11. The Quillen Adjunction $\text{St}_{\text{Alg}} \dashv \text{U}_{\text{Alg}}$
  12. The functor $\text{AlgKan} \to \text{sTCom}$
  13. Model Structure on Simplicial T-Complexes
  14. The induced coalgebra $\text{St}(X)$ detects the homotopy type of $X$
  15. The adjunction $\text{St} \dashv \text{U}_{\text{St}}$ induces a comonadic adjunction of $\infty$-categories
  16. ...and 4 more sections

Key Result

theorem 1

The homotopy type of a space $X$ is determined by the homotopy type of its strictification $\text{Strict}(X)$, along with a natural coalgebra structure over the comonad induced by the adjunction $\text{Strict} \dashv U$.

Theorems & Definitions (83)

  • definition 1
  • theorem 1
  • theorem 2
  • definition 2: Nikolaus2011, definition 3.1
  • definition 3: Dakin_1977, Definition 1.1
  • theorem 3
  • definition 4: Brown_Higgins_Sivera_2011, Definition 7.1.9
  • definition 5
  • theorem 4
  • theorem 5
  • ...and 73 more