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

Abstract: Let $\mathsf{P}$ be a finite graded poset and $Δ<em>a<sup>{\mathsf{P}}$ the height-$a$ thickening of its diagonal. We study the \emph{window convolution} $$C_a=q</em>{1\sharp}(k_{Δ<em>a^{{\mathsf{P}}}\otimes^{\mathbf} L}q_2<sup>\ast(-))$$ on $\mathrm{Shv}(\mathsf{P};k)$. An interleaving distance needs the left derived $\mathbb{L}C_a$ to compose as a flow, $$\mathbb{L}C_a\mathbb{L}C_b\simeq\mathbb{L}C</em>{a+b}$$; the total meet functor $Φ$ gives rise to the canonical comparison. Finality is sufficient, and necessary where 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 to have a single interior element. The flow fails at every branching length-two interval, and with it on the face poset of every finite regular cell complex of dimension $\ge2$. It survives on tame posets, where $\mathrm{id}\Rightarrow\mathbb{L}C_a$ gives a canonical extended interleaving pseudometric on $$\operatorname{D<sup>{b}}(\mathrm{Shv}(\mathsf{P};k))$$; in the saturation cases computed here it takes no finite value above the length of $\mathsf{P}$, and is finite if and only if the derived colimits agree.