Limit Computation Over Posets via Minimal Initial Functors
Tamal K. Dey, Michael Lesnick
TL;DR
This work studies how restricting diagrams along initial functors can drastically simplify limit computations when the index category is a finite poset or an interval in N^d. It introduces and analyzes minimal initial functors and initial scaffolds, proving that, in many cases, minimal initial functors are inclusions of initial scaffolds and providing concrete algorithms to compute them. The authors derive tight size bounds for initial scaffolds in N^d intervals, connect these bounds to Betti numbers of monomial ideals, and deliver practical algorithms with provable complexity for computing limits and generalized ranks. The results yield new, near-optimal bounds on the cost of limit computations and generalized rank calculations, with direct relevance to topological data analysis and multiparameter persistence. Overall, the paper provides a cohesive framework linking category-theoretic initiality to efficient linear-algebraic limit computation and invariant extraction in TDA.
Abstract
It is well known that limits can be computed by restricting along an initial functor, and that this often simplifies limit computation. We systematically study the algorithmic implications of this idea for diagrams indexed by a finite poset. We say an initial functor $F\colon C\to D$ with $C$ small is \emph{minimal} if the sets of objects and morphisms of $C$ each have minimum cardinality, among the sources of all initial functors with target $D$. For $Q$ a finite poset or $Q\subseteq \mathbb N^d$ an interval (i.e., a convex, connected subposet), we describe all minimal initial functors $F\colon P\to Q$ and in particular, show that $F$ is always a subposet inclusion. We give efficient algorithms to compute a choice of minimal initial functor. In the case that $Q\subseteq \mathbb N^d$ is an interval, we give asymptotically optimal bounds on $|P|$, the number of relations in $P$ (including identities), in terms of the number $n$ of minima of $Q$: We show that $|P|=Θ(n)$ for $d\leq 3$, and $|P|=Θ(n^2)$ for $d>3$. We apply these results to give new bounds on the cost of computing $\lim G$ for a functor $G \colon Q\to \mathbf{Vec}$ valued in vector spaces. For $Q$ connected, we also give new bounds on the cost of computing the \emph{generalized rank} of $G$ (i.e., the rank of the induced map $\lim G\to \mathop{\mathrm{colim}} G$), which is of interest in topological data analysis.