[Paper Review] Quillen Theorems Bn for homotopy pullbacks of (infinity, k)-categories

Clark Barwick, D Kan|arXiv (Cornell University)|Aug 8, 2012

Homotopy and Cohomology in Algebraic Topology11 references3 citations

TL;DR

This paper extends Quillen's Theorem B to homotopy pullbacks in $(∞,k)$-categories using fibrillations—fibrational structures derived from Hopkins and Rezk's sharp maps. It establishes that under property $B_n$, the $n$-arrow pullback category $f\mathbf{X} \downarrow_n g\mathbf{Y}$ is a homotopy pullback, generalizing results to relative categories and $k$-relative categories via both Reedy and Rezk model structures.

ABSTRACT

We extend the Quillen Theorem Bn for homotopy fibers of Dwyer, et al. to similar results for homotopy pullbacks and note that these results imply similar results for zigzags in the categories of relative categories and k-relative categories, not only with respect to their Reedy structures but also their Rezk structure, which turns them into models for the theories of (infinity, 1)- and (infinity, k)-categories, respectively.

Motivation & Objective

• To generalize Quillen's Theorem B to homotopy pullbacks in $(\infty,k)$-categories.
• To establish that the $n$-arrow pullback category $f\mathbf{X} \downarrow_n g\mathbf{Y}$ models the homotopy pullback when $f$ satisfies property $B_n$.
• To extend these results to relative categories and $k$-relative categories using both Reedy and Rezk model structures.
• To show that a strict $3$-arrow calculus implies property $C_3$, providing a sufficient condition for higher-order Quillen theorems.
• To develop a framework using fibrillations—fibrational analogs of Hopkins' sharp maps—for proving homotopy pullback characterizations.

Proposed method

• Utilizes fibrillations, a reinterpreted version of Hopkins and Rezk's sharp maps, which behave like fibrations in right proper model categories.
• Applies the Fibrillation Lifting Lemma (Lemma 9.4) to transfer fibrillation properties across functors like $w_*$ and product constructions.
• Employs the $k$-simplicial nerve functor $\mathrm{s}^k\mathrm{N}$ to relate $k$-relative categories to $k$-simplicial spaces.
• Applies the Quillen fibrillation lemma (Lemma 9.7) to construct fibrillations from overcategories, especially in $\widehat{\mathbf{Cat}}$.
• Uses the Rezk structure on $\mathrm{s}^k\widehat{\mathbf{Cat}}$ and $\mathbf{Rel}^k\mathbf{Cat}$ to model $(\infty,1)$- and $(\infty,k)$-categories.
• Applies the Global Equivalence Lemma (Lemma 5.13) to transfer homotopy pullback properties through homotopy equivalences such as $w_*$.

Experimental results

Research questions

• RQ1 Under what conditions is the $n$-arrow pullback category $f\mathbf{X} \downarrow_n g\mathbf{Y}$ a homotopy pullback for a zigzag $f: \mathbf{X} \to \mathbf{Z} \leftarrow \mathbf{Y}: g$?
• RQ2 How can Quillen's Theorem $B_n$ be extended from individual functors to homotopy pullbacks of zigzags in $(\infty,k)$-categories?
• RQ3 What role do fibrillations—derived from sharp maps—play in characterizing homotopy pullbacks in relative categories and $k$-relative categories?
• RQ4 Can a strict $3$-arrow calculus serve as a sufficient condition for property $C_3$, enabling higher-order Quillen theorems?
• RQ5 In what ways do the Reedy and Rezk model structures on relative categories and $k$-relative categories support the same homotopy pullback characterizations?

Key findings

• The $n$-arrow pullback category $f\mathbf{X} \downarrow_n g\mathbf{Y}$ is a homotopy pullback if $f$ satisfies property $B_n$, with a monomorphism into the actual homotopy pullback that is a weak equivalence when the pullback is homotopy-invariant.
• For $k$-relative categories ($k > 1$), the same result holds under $B_n$ or $C_n$ conditions, valid in both Reedy and Rezk model structures.
• The map $\pi: f\mathbf{X} \downarrow_n \mathbf{Z} \to \mathbf{Z}$ is a fibrillation when $f$ has property $B_n$, and this structure lifts through $w_*$ and product functors.
• The $k$-simplicial nerve functor $\mathrm{s}^k\mathrm{N}$ preserves the fibrillation structure, enabling transfer of results from $\mathrm{s}^k\widehat{\mathbf{Cat}}$ to $\mathbf{Rel}^k\mathbf{Cat}$.
• A strict $3$-arrow calculus implies property $C_3$, providing a concrete sufficient condition for the $B_3$-type theorems to apply.
• The $w_*$ functor from $\mathbf{RelCat}$ to $\mathrm{s}^k\widehat{\mathbf{Cat}}$ is a homotopy equivalence, allowing transfer of homotopy pullback results between categories via the Global Equivalence Lemma.