nLab persistence object

Idea

A persistence object of a category $C$ is a functor from a poset, often a Cartesian product of linear orders, to $C$. This is a concept with an attitude: One calls such functors “persistence objects” when one is interested in determining their persistence diagrams or other measures of “persistence” as used in topological data analysis.

The main example in this context arises when $C$ is a category of vector spaces or more generally a category of modules, in which case one speaks of persistence modules as used in persistent homology. Alternatively, $C$ could be a category of groups, such as homotopy groups, or even of full homotopy types, which is the case of interest in persistent homotopy.

When the indexing poset is the poset of real numbers or, more generally, a product of copies of the poset of real numbers, the collection of persistent objects of a fixed category $C$ admits a distance called the interleaving distance, which, informally, measures how isomorphic any two persistence objects are.

The key property which one will typically demand of a good theory of persistence objects is a notion of persistence diagrams (measuring “how persistent” a given persistence object is) which is stable with respect to interleaving distance.

Definitions

Let $P$ be a partially ordered set, seen as a category, and let $C$ be a category. A $P$-persistent object of $C$ is a functor $P \to C$. The category of $P$-persistent objects of $C$ is the functor category $C^P$.

Let $(\mathbb{R}, \leq)$ be the poset of real numbers with its standard order, and let $C$ be a fixed category. Let $X : \mathbb{R} \to C$ be a persistent object of $C$, and let $\epsilon \geq 0 \in \mathbb{R}$. The $\epsilon$-shift of $X$ is the persistent object $X[\epsilon] : \mathbb{R} \to C$ such that, for all $r \in \mathbb{R}$ we have $X[\epsilon](r) = X(r+\epsilon)$, and for all $r \leq s \in \mathbb{R}$, the structure morphism $X[\epsilon](r) \to X[\epsilon](s)$ is equal to the structure morphism $X(r+\epsilon) \to X(s+\epsilon)$. The $\epsilon$-shift construction gives a functor $(-)[\epsilon] : C^\mathbb{R} \to C^\mathbb{R}$. Note that the structure morphisms of $X$ give a natural transformation $\eta^X_\epsilon : X \to X[\epsilon]$.

An $\epsilon$-interleaving between $X$ and $Y$ consists of natural transformations $f : X \to Y[\epsilon]$ and $g : Y \to X[\epsilon]$ such that $g[\epsilon] \circ f = \eta^X_{2\epsilon}$ and $f[\epsilon] \circ g = \eta^Y_{2\epsilon}$.

As an example, note that a $0$-interleaving is precisely an isomorphism in the category $C^\mathbb{R}$.

The interleaving distance between persistent objects $X, Y \in C^{\mathbb{R}}$ is

$$d_I(X,Y) = inf\; \big( \big\{\epsilon \geq 0 \;:\; X \;\text{and}\; Y\;\text{are}\; \epsilon\text{-interleaved}\;\big\} \cup \{ \infty \} \big).$$

Examples

Let $(\mathbb{R}, \leq)$ denote the poset of real numbers with its standard order, and let $Vect_k$ denote the category of vector spaces over a fixed field $k$. The $\mathbb{R}$-persistent objects of $Vect_k$ are known as persistence modules, and are a central object of study in topological data analysis.

Let $n \geq 2$ be a natural number. Persistent objects of the form $\mathbb{R}^n \to Vect_k$ are known as multiparameter persistence modules.

Let $\mathbb{R}_{\geq 0}$ denote the poset of non-negative real numbers with its standard order, and let $Set$ denote the category of sets. The $\mathbb{R}_{\geq 0}$-persistent objects of $Set$ are known as persistent sets.

Related concepts

• persistence module

References

The terminology “persistence object” is used for instance in

• Donald Pinckney, Section 3.1 Persistence objects in Topological Data Analysis and Persistent Homology (2019)

General category theoretic discussion of persistence objects (not using that terminology, though):

• Peter Bubenik, Jonathan Scott Categorification of Persistent Homology, Discrete & Computational Geometry 51 (2014) 600-627 [doi:10.1007/s00454-014-9573-x]

• Francesca Cagliari, Massimo Ferri & Paola Pozzi. Size Functions from a Categorical Viewpoint, Acta Applicandae Mathematica, 2001.

Original references on multiparameter persistent homotopy groups and multiparameter persistence modules:

• P. Frosini, M. Mulazzani. Size homotopy groups for computation of natural size distances, Bulletin of the Belgian Mathematical Society Simon Stevin, 1999.

• G. Carlsson, A. Zomorodian. The theory of multidimensional persistence, Discrete and Computational Geometry, 2009.

Survey on multiparameter persistent homology:

• Magnus Bakke Botnan, Michael Lesnick. An Introduction to Multiparameter Persistence. (arXiv:2203.14289)