[Paper Review] A Quillen theorem Bn for homotopy pullbacks

Clark Barwick, D Kan|arXiv (Cornell University)|Jan 25, 2011

Homotopy and Cohomology in Algebraic Topology4 references3 citations

TL;DR

This paper extends Quillen's Theorem B to homotopy pullbacks by introducing a new construction, $(foldsymbol{X} abla_n goldsymbol{Z})$, which serves as a homotopy pullback for a zigzag $oldsymbol{X} o oldsymbol{Y} rom oldsymbol{Z}$ when $f$ satisfies property $B_n$—particularly if $oldsymbol{Y}$ has property $C_n$. The key result is that the ordinary pullback $oldsymbol{X} imes_{oldsymbol{Y}} oldsymbol{Z}$ is a homotopy pullback if the canonical monomorphism into $(foldsymbol{X} abla_n goldsymbol{Z})$ is a weak equivalence.

ABSTRACT

We prove an extension of the Quillen Theorem Bn for homotopy fibres to a similar result for homotopy pullbacks and use this to obtain sufficient conditions on a pullback diagram of categories to guarantee that it be a homotopy pullback.

Motivation & Objective

• To generalize Quillen's Theorem B from homotopy fibres to homotopy pullbacks in the context of zigzag diagrams between categories.
• To establish sufficient conditions under which the ordinary pullback of a zigzag $\boldsymbol{X} \to \boldsymbol{Y} \from \boldsymbol{Z}$ is a homotopy pullback.
• To provide a non-inductive proof of the homotopy pullback result by leveraging Quillen's foundational lemma on Grothendieck constructions.
• To apply the result to simplify verification of the Segal condition in complete Segal spaces, particularly in relative categories with the two-out-of-six property and a 3-arrow calculus.
• To show that the category of weak equivalences in such relative categories has property $C_3$, enabling the use of the main theorem for $n=3$.

Proposed method

• Introduce a new construction, $(foldsymbol{X} abla_n goldsymbol{Z})$, as a candidate for the homotopy pullback of a zigzag $f: \boldsymbol{X} \to \boldsymbol{Y} \from \boldsymbol{Z}: g$.
• Use the Grothendieck construction on the functor $(f\boldsymbol{X}\nabla_n -): \boldsymbol{Y} \to \mathbf{Cat}$ to define the homotopy pullback category.
• Leverage Quillen's key lemma: if a functor $F: \boldsymbol{D} \to \mathbf{Cat}$ sends all maps in $\boldsymbol{D}$ to weak equivalences (property $Q$), then the fibre of the projection $\pi: \operatorname{Gr}F \to \boldsymbol{D}$ is a homotopy fibre.
• Apply this lemma to the Grothendieck construction of $(f\boldsymbol{X}\nabla_n g-)$ over $\boldsymbol{Z}$, showing that $(f\boldsymbol{X}\nabla_n gZ)$ is a homotopy fibre over $Z$.
• Construct a pullback square involving $\pi: (f\boldsymbol{X}\nabla_n \boldsymbol{Y}) \to \boldsymbol{Y}$ and $g: \boldsymbol{Z} \to \boldsymbol{Y}$, and show it is a homotopy pullback using fibre isomorphism and homotopy fibre properties.
• Establish that the canonical monomorphism $k: \boldsymbol{X} \times_{\boldsymbol{Y}} \boldsymbol{Z} \to (f\boldsymbol{X}\nabla_n g\boldsymbol{Z})$ is a weak equivalence if the pullback is to be a homotopy pullback.

Experimental results

Research questions

• RQ1 Under what conditions is the ordinary pullback $\boldsymbol{X} \times_{\boldsymbol{Y}} \boldsymbol{Z}$ of a zigzag $\boldsymbol{X} \to \boldsymbol{Y} \from \boldsymbol{Z}$ a homotopy pullback?
• RQ2 How can Quillen's Theorem B be extended from homotopy fibres to homotopy pullbacks in the context of category-valued functors?
• RQ3 What role does the property $B_n$ (or $C_n$) play in ensuring that the construction $(f\boldsymbol{X}\nabla_n g\boldsymbol{Z})$ is a homotopy pullback?
• RQ4 Can the Segal condition in complete Segal spaces be verified more simply using this extended theorem, particularly in relative categories without a simplicial structure?
• RQ5 How does the Grothendieck construction of the functor $(f\boldsymbol{X}\nabla_n g-)$ facilitate the proof of homotopy pullback properties?

Key findings

• The construction $(f\boldsymbol{X}\nabla_n g\boldsymbol{Z})$ is a homotopy pullback of the zigzag $f: \boldsymbol{X} \to \boldsymbol{Y} \from \boldsymbol{Z}: g$ whenever $f$ has property $B_n$.
• If $\boldsymbol{Y}$ has property $C_n$, then $f$ automatically has property $B_n$, so the homotopy pullback condition is satisfied under this weaker assumption.
• The ordinary pullback $\boldsymbol{X} \times_{\boldsymbol{Y}} \boldsymbol{Z}$ is a homotopy pullback if the canonical monomorphism $k: \boldsymbol{X} \times_{\boldsymbol{Y}} \boldsymbol{Z} \to (f\boldsymbol{X}\nabla_n g\boldsymbol{Z})$ is a weak equivalence.
• The proof relies on Quillen’s foundational lemma: if a functor $F: \boldsymbol{D} \to \mathbf{Cat}$ has property $Q$ (sends all maps to weak equivalences), then the fibre of the projection $\pi: \operatorname{Gr}F \to \boldsymbol{D}$ is a homotopy fibre.
• The result provides a non-inductive proof of the homotopy fibre version of Theorem $B_n$, which then extends naturally to the homotopy pullback case.
• The theorem simplifies the verification of the Segal condition in complete Segas spaces, as shown in the authors’ follow-up work, by reducing it to a weak equivalence check in a category with property $C_3$.