Poset functor cocalculus and applications to topological data analysis

Abstract

We introduce a new flavor of functor cocalculus, called \emph{poset cocalculus}, as a tool for studying approximations in topological data analysis. Given a functor from a distributive lattice to a model category, poset cocalculus produces a Taylor telescope of codegree $n$ approximations of the functor, where a codegree $n$ functor takes strongly bicartesian $(n+1)$--cubes to homotopy cocartesian $(n+1)$--cubes. We give several applications of this new functor cocalculus. We prove that the codegree $n$ approximation of a multipersistence module is stable under an appropriate notion of interleaving distance. We draw connections to filtrations of simplicial complexes, and show that the Vietoris-Rips filtration is precisely the codegree 2 approximation of the Čech filtration. We demonstrate that the codegree 1 approximation of the space of simplicial maps between two simplicial complexes is in some sense the space of continuous maps between their realizations, and that this statement can be made precise.

Figures (4)

• Figure 1: The Hasse diagrams of two non-distributive lattices.
• Figure 2: An illustration of the poset $\mathbf{I}^{\Gamma}$, where $P = \mathbb{N}$, and $\Gamma \in \mathbf{Trans}_P$ is the translation given by $\Gamma(n) = n+1$.
• Figure 3: A graph.
• Figure 4: Illustrated is a 3--simplex $\sigma$. Suppose that $X$ is a 1--coskeletal simplicial complex and $\sigma \in X$. If $\{a,b,d\}, \{a,c,d\}$ and {b,c} are all in $X$, then $\{a,b,c,d\}$ is in $X$.

Theorems & Definitions (118)

• Theorem A
• Theorem B
• Proposition
• Lemma 2.1
• proof
• Definition 2.2
• Proposition 2.3
• Definition 2.4
• Lemma 2.5
• proof