Meet obstructions and saturation for the constant window convolution on graded posets

@misc{yokoyama2026meet,
  title = {{Meet obstructions and saturation for the constant window convolution on graded posets}},
  author = {Shinobu Yokoyama},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.19904},
  eprint = {2608.19904},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.19904},
}

TY  - UNPB
TI  - Meet obstructions and saturation for the constant window convolution on graded posets
AU  - Shinobu Yokoyama
PY  - 2026
DA  - 2026/08/31
PB  - arXiv
DO  - 10.48550/arxiv.2608.19904
UR  - https://arxiv.org/abs/2608.19904
AB  - Let P be a finite graded poset and Δ_a^P the height-a thickening of its diagonal, with projections q_1,q_2 to P. We study the \emph{window convolution} C_a=Lan_q_1∘ q_2^∗ on Shv(P;k), a discrete analogue of convolution against a thickening kernel. An interleaving distance needs the homotopy window convolution ℂ_a to compose as a flow, ℂ_aℂ_b≃ℂ_a+b; where the meet assignment Φ is total it is a functor and carries a comparison map. Finality is sufficient, and necessary at every minimal apex and wherever the finality defect of Φ is essential; where Φ is total at a minimal apex with unit windows, it is the failure of a length-two interval above the apex to have a single interior element. The flow fails at every branching length-two interval with minimal bottom element, and with it on the face poset of every finite regular cell complex of dimension ≥2. It survives on tame posets, where id⇒ℂ_a gives a canonical extended interleaving pseudometric on D^b(Shv(P;k)); in the saturation cases computed here it takes no finite value above the length of P, and is finite if and only if the derived colimits agree.
ER  -