Generalized Rank via Minimal Subposet
Thomas Brüstle, Justin Desrochers, Samuel Leblanc
TL;DR
This work develops a categorical framework for the generalized rank invariant of persistence modules indexed by small connected categories. It proves that final and initial subcategory embeddings preserve limits and colimits, yielding multiplicity equalities for interval modules when the embedding is both final and initial, and extends this to all $\mathcal{C}$-modules. The authors construct minimal final and initial subposets under mild assumptions, and assemble them into a connected subposet $\mathcal{M}^{\mathrm{rk}}$ that preserves generalized rank, enabling practical reductions in computing multiplicities and ranks. They also provide a concrete strategy (Construction D) and discuss representation-type consequences, showing that the reduced subposet often has finite type even when the ambient poset does not. The results unify and extend prior work on rank invariants (e.g., Kinser, Dey–Lesnick, Dey–Kim–Mémoli) and offer a principled method to identify minimal, rank-preserving substructures in multiparameter persistence.
Abstract
Let $\mathcal{C}$ be a small, connected category with finite hom-sets. We show that if the embedding of a connected subcategory $\mathcal{J}$ is both initial and final, then the restriction of any $\mathcal{C}$-module along $\mathcal{J}$ preserves the generalized rank-or equivalently, the multiplicity of the ``entire" interval modules for $\mathcal{C}$ and $\mathcal{J}$. Conversely, we prove that this property characterizes initial and final embeddings when both $\mathcal{C}$ and $\mathcal{J}$ are posets satisfying certain mild constraints and the embedding is full. For $\mathcal{C}$ a poset under these conditions, we describe the minimal full subposet whose embedding is initial or final. We also extend a result of Kinser on the generalized rank invariant to small categories.