Dice Question Streamline Icon: https://streamlinehq.com

Equivalence of the two zigzag filtrations of the pullback B ×_D C

Establish that the two zigzag-based sequential filtrations of the pullback B ×_D C produced by starting the zigzag construction from the spans B ← 0 → 0 (yielding the sequence R_1 → R_3 → …) and from 0 ← 0 → C (yielding the sequence Q_0 → Q_2 → …) are canonically identical stage-by-stage, thereby showing that the apparent asymmetry of the zigzag construction disappears.

Information Square Streamline Icon: https://streamlinehq.com

Background

The paper introduces a zigzag construction that builds a sequence of spaces approximating the pullback B ×_D C associated to a pushout square B ← A → C with pushout D. Depending on whether one starts the construction from the span B ← 0 → 0 or from 0 ← 0 → C, one obtains two a priori different filtrations of the same object B ×_D C (via the R-sequence versus the Q-sequence).

Although the authors provide structural tools (Construction 3.6 and subsequent analysis) suggesting a relationship between these filtrations, they explicitly state that they have not proved that the two filtrations coincide. Resolving this would clarify the symmetry of the construction and potentially simplify applications that depend on a canonical filtration.

References

The zigzag construction is a priori asymmetric, in the sense that a priori one gets different filtrations of B \times_D C depending on whether we start from B \leftarrow 0 \to 0 and consider S \times_D C, or start from 0 \leftarrow 0 \to C and consider S \times_D B. We expect, but do not prove, that these two filtrations are actually the same.

— Path spaces of pushouts (2402.12339 - Wärn, 19 Feb 2024) in Section 3 (The zigzag construction), paragraph beginning “The zigzag construction is a priori asymmetric”