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

Abstract

Let $\mathsf{P}$ be a finite graded poset and $\Delta_a^{\mathsf{P}}$ the height-$a$ thickening of its diagonal, with projections $q_1,q_2$ to $\mathsf{P}$. We study the \emph{window convolution} $C_a=\operatorname{Lan}_{q_1}\circ q_2^\ast$ on $\mathrm{Shv}(\mathsf{P};k)$, a discrete analogue of convolution against a thickening kernel. An interleaving distance needs the homotopy window convolution $\mathbb{C}_a$ to compose as a flow, $\mathbb{C}_a\mathbb{C}_b\simeq\mathbb{C}_{a+b}$; where the meet assignment $\Phi$ 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 $\Phi$ is essential; where $\Phi$ 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 $\ge2$. It survives on tame posets, where $\mathrm{id}\Rightarrow\mathbb{C}_a$ gives a canonical extended interleaving pseudometric on $\operatorname{D^{b}}(\mathrm{Shv}(\mathsf{P};k))$; 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.