Limit Computation Over Posets via Minimal Initial Functors

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 $|P|=\Theta(n)$ for $d\leq 3$, and $|P|=\Theta(n^2)$ 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.