AI 中文总结
本文研究分次偏序集上的窗口卷积，分析同伦窗口卷积满足流复合律的条件，刻画其失效的偏序结构特征，并在驯顺偏序集上构造典范扩展交错伪度量，探讨其饱和性质与有限性条件。
AI 中文摘要
设$\u2a06$为有限分次偏序集，$Δ_a^{\u2a06}$为其对角线的高度为$a$的加厚，带有到$\u2a06$的投影$q_1,q_2$。我们研究$\text{Shv}(\u2a06;k)$上的“窗口卷积”$C_a=\text{Lan}_{q_1}∘q_2^∗$，它是加厚核卷积的离散类似物。交错距离需要同伦窗口卷积$\u2102_a$能像流一样复合，即$\u2102_a\u2102_b≃\u2102_{a+b}$；在相遇赋值$Φ$处处有定义的区域，它是一个函子且带有一个比较映射。终结性是充分条件，且在每个极小顶点以及$Φ$的终结性缺陷为本质的位置都是必要的；当$Φ$在带单位窗口的极小顶点处有定义时，流不成立等价于该顶点上方的长度为2的区间不具备唯一内部元素。流在每个以极小元为底的分支型长度2区间处失效，因此在维度≥2的所有有限正则胞腔复形的面偏序集上均失效。它在驯顺偏序集上成立，此时$\text{id}⇒\u2102_a$在$\text{D}^{\text{b}}(\text{Shv}(\u2a06;k))$上给出一个典范的扩展交错伪度量；在本文计算的饱和情形中，该伪度量在$\u2a06$的长度之上不取有限值，且它有限当且仅当导出余极限一致。
英文摘要
Let $\mathsf{P}$ be a finite graded poset and $Δ_a^{\mathsf{P}}$ 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 $Φ$ 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 $\ge2$. 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.
Comments54 pages. Substantial revision. v2 corrects the derived construction: C_a is defined as a Kan extension, and its homotopy version is the homotopy left Kan extension, not the total left derived functor of C_a as v1 had it. Necessity of finality is proved outright at every minimal apex via a new detection criterion. The facet-fibre computations and the degeneration branch of v1 are removed
