# of multiparameter persistence modules: Additivity -theory

## Abstract

Persistence modules stratify their underlying parameter space, a quality that makes persistence modules amenable to study via invariants of stratified spaces. In this article, we extend a result previously known only for one-parameter persistence modules to grid multiparameter persistence modules. Namely, we show the of grid multiparameter persistence modules is additive over strata. This is true for both standard monotone multi-parameter persistence as well as multiparameter notions of zig-zag persistence. We compare our calculations for the specific group -theory with the recent work of Botnan, Oppermann, and Oudot, highlighting and explaining the differences between our results through an explicit projection map between computed groups.

## 1. Introduction

Persistent homology has become a central tool in *topological data analysis* (TDA). It has also been added to the toolbox of symplectic geometers, see for instance Reference PRSZ20. The typical setting for persistent homology is that of a filtered topological space, and a field of coefficients, , The sequence of vector spaces and linear maps encoded by . are then used to analyze the space and/or a dataset from which has been constructed. Persistence modules are a generalization of persistent homology in that they are simply a functorial assignmentтАФsay from filtered spacesтАФto a тАЬreasonableтАЭ category Typical examples of тАЬreasonableтАЭ categories include no condition, Abelian, or exact categories. .

The filtered spaces, of persistence theory often arise by considering a dataset, fixing a parameter space, and deciding on a scheme which associates a space to each parameter value, e.g., the ─Мech or VietorisтАУRips complexes associated to a point cloud and a real number. In practice, our parameter space, , is a manifold. If our parameter space is equipped with an embedding , we are in the setting of one-parameter persistence. If our parameter space is embedded in , then we are in the setting of multiparameter persistence. Although , Euclidean space is perhaps the typical embedding target, in general, an embedding into any manifold of dimension -dimensional is a setting for persistence. -parameter

One-parameter persistence is well understood and under reasonable hypotheses, there are complete, discrete, computable invariants. Multiparameter persistence is more subtle, and it is a charge of the community to find computable, descriptive invariants. See the original work of Carlsson and Zomorodian Reference CZ09 or the more recent survey Reference BL22.

Inspired by the works of Botnan, Oppermann, and Oudot Reference BOO21, and our previous work Reference GS21, we study the universal additive invariant of persistence modules: their algebraic In the present, we use the same setup as in -theory.Reference GS21, defining persistence modules as constructible cosheaves over the parameter space, which itself has been stratified by an тАЬevent stratification.тАЭ By imposing some mild hypotheses on our stratified parameter spaces, the category of such constructible cosheaves is equivalent to the category representations of the (combinatorial) entrance path category of our space. (See Reference CP20 for a nice overview of this correspondence.) This category of representationsтАФa functor categoryтАФhas well-defined algebraic for nice target categories, e.g., modules over a commutative ring on pointed sets. -theory

Note that, while persistence modules are most typically defined as representations of partially ordered sets (posets), we define stratified parameter spaces in a direction-free way. Roughly speaking, we keep track of тАЬevent timesтАЭ as zero strata, but the data of how event times relate to one another is not explicitly stored in the stratification of a parameter space. Instead, we incorporate the poset structure of a persistence module into the assignments of morphisms out of the stratified parameter space; see Reference GS21 for further insights into this perspective. One utility of this formalism is that it puts monotone persistence, zig-zag persistence, and their multiparameter generalizations on common footing.

Our main contributions in the present article are to:

- (1)
Prove that the of multiparameter persistence (grid) modules is additive over the strata of the parameter space. This main result is Theorem -theory3.0.7. As an immediate corollary, we obtain the groups and for persistence modules valued in finite dimensional vector spaces over a field. Our result is a generalization of Theorem 4.1.6 of Reference GS21 to the multiparameter setting.

- (2)
For the groups compare our results with those of ,Reference BOO21. In particular, while our is not isomorphic to the Grothendieck group of rank-exact persistence modules of Botnan, Oppermann, and Oudot, we do obtain an explicit comparison homomorphism. This comparison is the content of Section 4.1.

## 2. Background and conventions

### 2.1. Stratified spaces and entrance paths

Note that any topological space is stratified by the terminal poset consisting of a singleton set. Moreover, the simplices of a simplicial complex, come equipped with the structure of a poset, and we call the resulting stratification of , the *face stratification*, which we denote by Reference Sta91.

Analogous to simplicial complexes/combinatorial manifolds, cubical manifolds are naturally stratified by the face poset of the cubulation and we denote the resulting stratification by .

### 2.2. Grid modules

As noted above, we define our persistence modules as representations of the (combinatorial) entrance path category associated to a stratified parameter space.

In what follows, we are interested in *grid modules*, i.e., those persistence modules parametrized by subspaces of cubulated To that end, we need a few preliminary definitions. .

Given a stratified space , and an embedding , there is a stratification of , extending that on called the *connected ambient stratification* and the resulting stratified space is denoted See Definition 2.1.11 of .Reference GS21 for details.

Note that any closed and bounded substratified space of is naturally cubulated.

See Figure 2.1 for an instance of a cubical grid manifold.

### 2.3. Waldhausen -theory

Dan Quillen, justifiably, received much recognition for defining higher algebraic via Abelian and exact categories. Some twenty years later, Friedhelm Waldhausen found a further generalization of QuillenтАЩs setting in his work on the algebraic -theory of spaces -theoryReference Wal85. Today, this setting is that of Waldhausen categories and exact functors between them. Here we recall, tersely, some key notions leading to WaldhausenтАЩs Additivity Theorem. A modern introduction to this material can be found in Reference FP19 or the encyclopedic Reference Wei13.

The following is one of the fundamental theorems of algebraic It is known as -theory.*Waldhausen Additivity*.

## 3. of grid modules -theory

In this section we prove our main theorem, which is the multiparameter analog of Theorem 4.1.6 of Reference GS21.

It is standard that the category of finitely generated, projective modules for a commutative ring defines a Waldhausen category. So too does the category of functors from a small category into this category of modules. (Some details are provided in Appendix A of Reference GS21.)

The argument for Lemma 3.0.1 is, *mutatis mutandis*, as for Lemma 4.1.1 of Reference GS21.

Since the sequence in Lemma 3.0.1 is a standard split short exact sequence of Waldhausen categories, Waldhausen Additivity (Theorem 2.3.3) immediately gives us the following key corollary, which is the main tool we will use in our argument toтАЬbreak apart and glue togetherтАЭ modules from submodules.

The following is a main result of Reference GS21 (Lemma 4.1.4), and serves as a base case for our induction into multiparameter modules.

In proving the preceding result, we inducted on the number of zero-strata of In our multiparameter setting, we induct on an analogous notion: .*height*.

It is clear from the definition that in the one-parameter case, the height of is indeed just the number of zero-strata of In the general . case, height is a measure of the longest axis-aligned тАЬslice,тАЭ although note that the height of -parameter may be realized in more than one parameter.

We are now ready to state our main result, the of multiparameter grid modules. The proof uses a double induction on the number of parameters and on the height of the module. -theory

We end this section by discussing how Theorem 3.0.6 translates to the specific case of multiparameter persistence modules. -valued

The first two of a field are well known. The following isomorphisms are induced by the dimension and determinant maps, respectively. -groups

See Chapter IV of Reference Wei13 for an in-depth description of the higher of fields. -theory

## 4. Connections: Euler manifolds and rank exact -theory

As we previously observed in the one-dimensional case Reference GS21, given a persistence module, over a parameter space, , the class of , in is the Euler curve of the persistence module. The same conclusion holds in the multiparameter setting with the exception that we no longer consider the Euler curve, but rather the Euler surface or manifold depending on the number of parameters. Indeed, the description of in terms of constructible functions goes back to Kashiwara and Schapira Reference KS94, and the construction of their isomorphism uses a local Euler index.

### 4.1. Rank exact -theory

Finally, we briefly discuss the relationship between our present results and the recent work of Botnan, Oppermann, and Oudot on Grothendieck groups, in multiparameter persistence via rank-exact structures ,Reference BOO21.

Let be a field and be an arbitrary poset. Let denote the category of functors with finite total rank. Note, we use rank, instead of dimension (over , as there are many other choices of target category to which the work of ),Reference BOO21 applies. Of course, in our simplified setting .

Theorem 4.4 of Reference BOO21 states that equipped with rank-exact short exact sequences is an exact category, which we denote Hence, . is well-defined. The higher also exist, but following the authors of -groups*loc. cit.* we restrict our discussion to .

For a poset, let be the collection of *segments*, i.e., pairs defining the partial order on the underlying set of i.e., ,

Consider the linear order, Following .Reference GS21, the poset defines a cubical one-manifold with three zero-strata and two one-strata, which we denote by By our previous additivity result, or the one-parameter specialization of the results above, . However, note that . so , Of course, . so we seek an explanation/explicit comparison. ,

Let be a finite poset. The *order complex*, is the abstract simplicial complex whose faces are chains in , Let . denote the geometric realization of which is a (geometric) simplicial complex. It is a standard exercise that , i.e., the realization of the linear order , is the standard Returning to the previous paragraph, note that the cubical one-manifold -simplex. is a subcomplex of it sits inside as the ;*spine* of the 2-simplex. Note further that there is a natural bijection between and the set of simplices of the one skeleton of Indeed, this bijection is determined by considering the chains of length at most one in . i.e., the one skeleton of the order complex, , We conclude that the inclusion .

induces a projection map This argument immediately generalizes to any finite linear order, and we have the following. .

Similarly, we can analyze the case of a grid as well. Let be a finite grid equipped with the (categorical) product poset structure. Again, we have a natural bijection Next, note that strata of . are indexed by the Cartesian product of sets where , is the cubical grid manifold associated to -dimensional Now, for each . and each edge or vertex there is a ,*source* and a *target* There is an injection of sets .

given by

Geometrically, the map exhibits as a coarsening of a subcomplex of As before we obtain a projection map at the level of . -groups.

Thus, when is of the form or more generally is a finite grid, we define a projection map that allows for direct comparison between the group , of Botnan, Oppermann, and Oudot, and the group of the present work.

## Acknowledgments

The authors wish to thank Steve Oudot for helpful correspondence as well as Brittany Fasy for thoughtful feedback.