In this article we aim to demonstrate the utility of viewing persistent phenomena from the perspective of constructible (co)sheaves. In particular, we demonstrate how cosheaves provide a convenient interpretation of augmented descriptors of persistence modules and how cosheaves are a convenient setting for constructing invariants via algebraic K-theory. The present is in the same spirit of the program we first employed in , namely, applying stratified mathematics and higher algebra to topological data analysis (TDA) .
), serves as an inspiration for own perspectives. The key idea interpolating between persistence modules and constructible cosheaves is that of a stratified space. A persistence moduleis obtained by sampling (or otherwise selecting a discrete subset) of a larger parameter space. For concreteness, consider as a selection of “instances” in our one-dimensional ray of “time.” As our persistence module only changes at elements of , it is locally constant on , which is the defining property of a constructible cosheaf.
Constructible cosheaves are particularly nice mathematical objects for several reasons, chief among them is their equivalence to representations of the so called entrance path category; this is known as “the” Exodromy Theorem. Any stratified space has an associated entrance path category and in good cases (e.g., when the space is a combinatorial manifold), the entrance path category is a straightforward combinatorial object – in many cases it’s simply a poset. The idea of exodromy goes back to MacPherson and proofs in different settings appear in work of Curry and Patel , Treumann , Lurie , and Barwick with Glasman and Haine .
Given a parameter space (and a choice of sampling instances), exodromy allows us to consider all persistence modules/constructible sheaves on that space as a category of functors. Such categories of functors then inherit desirable properties from the target category. For instance, if we consider modules valued in vector spaces, the category of persistence modules is naturally an Abelian category. Abelian categories are the home of homological and homotopical algebra, so we are free to apply the tools of algebraic topology/homotopy theory, e.g., algebraic K-theory. The combinatorial nature of the entrance path category makes K-theory computations tractable and allows us to consider connections with other persistent invariants such as Euler curves and persistence diagrams.
In the present article, we are mainly concerned with one-dimensional parameter spaces. The resulting persistence modules are the zig-zag persistence modules of Carlsson and de Silva , which includes the more typically seen monotone (standard) modules.
Readers familiar with the persistent homology transform (PHT) may be interested to note that the PHT is a special type of persistence module itself. Our computation of -theory for zig-zag persistence modules has an interpretation in the setting of the PHT where the sphere of directions is . Thus, the results of this paper may be useful for future work in the computation of other invariants of the PHT. See  for further background on the PHT.
1.1. Why K-theory?
In the later part of this article, we compute the K-theory of the category of zig-zag modules. Here, we briefly overview why K-theory is a useful invariant.
K-theory began as simply as group completion of a monoid. Indeed, let be a commutative monoid and define to be the unique (up to isomorphism) Abelian group, equipped with a monoid homomorphism from , satisfying the universal property: for any Abelian group and homomorphism (of monoids) , there exists a unique group homomorphism factorization through . This universal property is described as the universal Euler characteristic and conveyed diagrammatically is as follows:
For instance, let be the isomorphism classes of finite dimensional vector spaces over the field (with direct sum) and the rank function, then we have an induced map , which happens to be an isomorphism. Expanding this example to complexes, let denote isomorphism classes of bounded complexes of -vector spaces. The natural extension of the rank function is the Euler characteristic, which again factors uniquely through . (In the topological setting, the Chern character of a vector bundle is an example of such an additive map.) The universal property of extends to categories equipped with a symmetric monoidal structure, as isomorphism classes of objects in such a category form a commutative monoid.
K-theory is more than just a single Abelian group, but rather a spectrum, , associated to a category (equipped with additional structure). Recall that spectra are the central objects of homotopy theory. The homotopy groups of define the K-groups of , i.e., . To first approximation, spectra can be thought of as the objects that parametrize cohomology theories. As such, they, so K-theory in particular, admit a wealth of computational tools, refined structures, and interpretations from algebraic topology. Cohomology theories are also the natural home for obstruction/anomaly theory and in this way, K-theory has become a central tool in topology (index theory, finiteness obstructions) and algebraic number theory (class field theory).
When refined to the level of spectra, K-theory has a remarkable additive structure with respect to split short exact sequences of categories. (We discuss this in Appendix A, see also .) Combined with its property as the universal Euler characteristic, K-theory is the universal additive invariant of (Waldhausen) categories.
1.1.1. Flavors and History of K-theory
There are several flavors and constructions of K-theory, we trace here the history to the two we use in the present article: Waldhausen’s construction and Zakharevich’s theory of assemblers. See the canonical texts of Rosenberg  and Weibel  for historical references and more details on the development of K-theory.
The genesis of K-theory came in the late 1950’s and early 1960’s through the work of Grothendieck in complex (algebraic) geometry and Atiyah and Hirzebruch in topology. Algebraic K-theory—the kind relevant to the present work—is an extension of Grothendieck’s ideas to build a family of functors from rings to Abelian groups . While Grothendieck only defined , suitable definitions for and where found by the mid 1960’s; the contributions of Bass, Schanuel, and Milnor are most notable. (Bass and Karoubi also gave definitions of negative K-theory, .) Definitions of higher K-groups was a major open problem in the early 1970’s, which was first solved by Dan Quillen: the -construction. (Milnor had given a definition of higher K-groups as well, though this Milnor K-theory is only a summand of the now accepted definition of higher K-theory.) Given a ring, , Quillen produced a space, , whose homotopy groups recovered/defined the K-theory of .
Quillen quickly followed his -construction with the -construction. The -construction takes as input an exact category, , e.g., the category of finitely generated projective modules for a ring, and outputs a space, , whose homotopy groups define K-theory. Quillen used the -construction to prove many fundamental results in algebraic K-theory that restricted to those for rings, as he also proved that , that is, the -construction is a strict generalization of -construction for rings.
The next revolution in algebraic K-theory came through Waldhausen’s work in manifold topology . Published in 1985, Waldhausen gave a construction that takes as input categories with structure that generalizes that of exact categories—nowadays called Waldhausen categories—and outputs a spectrum (the basic building block of homotopy theory) whose homotopy groups define the corresponding K-groups. (Segal some 15 years earlier used his -objects to produce a K-theory spectrum in certain cases.) Waldhausen’s construction is often referred to as the -construction and we give a brief overview in Appendix A. The -construction is a strict extension of the -construction. Perhaps most significantly, the -construction satisfies an additivity result for split short exact sequences; this result has become a central tool in algebraic K-theory.
Motivated by so called scissor crongruences and algebraic varieties, Zakharevich  found a simple construction of K-theory using her theory of assemblers. Zakharevich’s construction works well even in the absence of linear structure (Abelian and exact categories are in some way categorified linear algebra), e.g., the category of finite sets. One interpretation of this construction is that it builds a Waldhausen category from the elementary structure of the assembler and then applies the -construction and hence satisfies analogues of the foundational structural results: localization, additivity, etc.
Finally, we note that there has been an extension of K-theory to the higher categorical/homotopical algebraic setting as well. The work of Blumberg, Gepner, and Tabuada  proves that K-theory satisfies certain universal properties, such as additivity, (and hence is essentially uniquely defined by such properties) in this setting.
1.1.2. K-theory and persistence
Through the work of Patel, Bubenik and collaborators, K-theoretic considerations have started to appear in the TDA literature. Patel considered the Grothendick, i.e., , of one-dimensional persistence modules valued in symmetric monoidal categories .
Subsequently, in , Bubenick and Milićević show that the category of persistence modules over any preorder is Abelian. The key idea—which we use below as well—is that functor categories inherit many of the properties of the target category, so if the target is Abelian or Grothendieck, i.e., AB5 with a generator, then the functor category with domain a preorder (or any small category) is Abelian or Grothendieck. It would be interesting to apply Quillen’s -construction to these categories of persistence modules and compare the resulting K-theories to our computations below. (We note that  contains much more than we just outlined, e.g., the authors prove an embedding theorem in the vain of the Gabriel–Popescu Theorem.)
More relevant for us is the recent article  by Bubenik and Elchesen. In this work, the group completion of the monoid of persistence diagrams is described, i.e., is defined (semi-)explicitly. Points in diagrams are counted with multiplicity, so the binary operation is simply induced by . The input data for the construction of Bubenik and Elchesen is pretty flexible, so one can talk about diagrams (and their group completions) indexed by the entire first quadrant, the integers, etc. We make contact with this work in Section 5.2 below.
1.2. What we do
We have aimed to illustrate the connection between persistence modules and cosheaves and the utility of this interplay. To this end, we accomplish the following.
1.2.1. coSheaves from filtrations
The relevance of cosheaves in TDA has been advocated by Curry and others for a number of years. In Section 3, we give explicit constructions of constructible cosheaves associated to persistence modules. We are particularly interested in persistence modules arising from index filtrations of spaces. In Section 3.2.3, we describe the augmented filtration cosheaf, which records both non-instantaneous and instantaneous events. (We flag the recent work of Berkouk, Ginot, and Oudot  where level-set persistence is recast in terms of sheaves over .)
1.2.2. Equivalence Theorem
We prove an equivalence of categories between a localization of the category of zig-zag modules a la Carlsson and de Silva and constructible cosheaves on . The explicit statement of the result is Theorem 3.3. This result is stated in passing (Example 6.3) in the recent work of Curry and Patel , and we make it explicit with proof. We hope the proof is as interesting to the reader as the result, though it uses techniques that are different from the rest of the paper so we relegate it to Appendix B.
1.2.3. K-theory of Zig-Zag Modules
In Section 4, we define and compute K-theory of persistence modules. To describe the K-theory of persistence modules valued in the category of sets, we use the theory of Zakharevich’s assemblers. For modules valued in , we use Waldhausen’s construction of K-theory. A key input is additivity, in this case with respect to strata. For instance, in the case that our parameter space is one-dimensional, e.g., monotone or zig-zag persistence, the group is the free abelian group on the strata of parameter space (Theorems 4.1 and 4.2).
The higher K-groups do not vanish but rather are given by the algebraic K-theory of fields and/or the sphere spectrum. In forthcoming work, we aim to interpret these higher K-groups as arising from data.
The constructions and techniques we present apply to parameter spaces of arbitrary dimension.
1.2.4. Euler Curves and Virtual Diagrams
We conclude the body of the paper by connecting our K-theoretic work back to some recent work in TDA. First, we show how the Euler curve of a persistence module has a natural interpretation as a class in K-theory. (This is as expected, e.g., Kashiwara and Schapira  prove that is isomorphic to constructible functions via a local Euler index.) With this observation, we define an Euler class for arbitrary persistence modules irregardless of dimension; this is Definition 5.1. Lastly, Section 5.2 builds a group homomorphism from of persistence modules to Bubenik and Elchesen’s Abelian group of virtual persistence diagrams.
We assume the reader has some familiarity with algebraic topology, and freely use concepts from Hatcher’s standard text .
Throughout, we will let be the category of finite dimensional vector spaces over the field and linear maps. Much of our work doesn’t depend on making a choice of field and we simply use the notation .
Unless otherwise noted, we will assume all stratified spaces are combinatorial manifolds equipped with their native stratification, notions we define in the next section.
We thank David Ayala for many discussions related to the content of this and other articles. We also thank Peter Bubenik for several discussions related to zig-zag persistence and other mathematical topics in TDA.
2. Constructible coSheaves
2.1. Stratified/Constructible Basics
A stratified topological space is a triple consisting of
a paracompact, Hausdorff topological space, ,
a poset , equipped with the upward closed topology, and
a continuous map .
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 native stratification.
A map of stratified topological spaces to is a pair of continuous maps making the following diagram commute.
Let be a stratified space and a point. The space is conically stratified at if there exists an open neighborhood, , of and a stratified homeomorphism where is a topological space and is the cone on a space stratified by . A space is conically stratified if it is conically stratified at all of its points.
A piecewise linear (PL) manifold is a topological manifold along with an equivalence class of triangulations. A combinatorial manifold is a triangulated PL manifold. That is, a combinatorial manifold is a PL manifold along with a simplicial complex and PL homeomorphism . The manifold inherits a native stratification from the simplicial complex .
[] Let be a stratified space, a topological embedding, and . Define the poset, , as the set , subject to the following generating relations:
The relations of ;
For and , if and only if , i.e., the -stratum is in the closure of the connected component indexed by .
There is an obvious extension of the map , and we call this stratification the connected ambient stratification.
A typical (easy) example of the preceding is considering a discrete subset . The resulting stratified space, , is a combinatorial manifold.
Let be a topological space, the poset of open sets in , and a category. A precosheaf on valued in is a functor . A precosheaf is a cosheaf if for each open and any open cover of , , there is an equivalence (in )
For what remains, we will assume is a nice category, so that cosheafification exists. (Cosheafification is quite subtle, even compared to its dual notion of sheafification.) In particular, we will later focus on the case that or .
Let be a basis for the topology of the space and let be a cosheaf on the poset determined by . There is a unique (up to isomorphism) extension of to a cosheaf on .
The idea of the lemma can be thought of in terms of a Kan extension picture:
Let be a stratified space (not nec. conical or simplicial) and a cosheaf on . The cosheaf, , is constructible if it is locally constant when restricted to any stratum of .
Let be a (pre)cosheaf on and . The costalk of at is defined by
2.2. Operations on coSheaves
Given a continuous map , there is in an induced functor on the posets of opens given by preimages with respect to .
Let be a continuous map and a (pre)cosheaf on . The pushforward of , , is the (pre)cosheaf on given by .
There is a contravariant functor as well associated to a map : the pullback . As a continuous map is not necessarily an open map, is (slightly) more involved to define: it is the limit over opens containing for an open. Only pushforwards appear below.
Let be a point in the topological space and the inclusion map. Let be a cosheaf on , then is the costalk at of , . Let be a cosheaf on , then is a skyscraper cosheaf on .
Let be a stratified map and a constructible cosheaf on . Although the pullback of a constructible cosheaf is always constructible, it is not necessarily the case that is constructible on .
Consider the inclusion and be a nonzero constant cosheaf on . Further stratify and with zero-stratum and one-stratum (resp. ). The cosheaf is not constant on as
Constructibility is preserved by pushforwards in certain cases. Let be the “elementary collapse” of the interval , i.e.,
The map is constructible with respect to the (connected) ambient stratifications induced by and Let be any constructible cosheaf on it is straightforward to verify that is constructible on .
2.3. Entrance Paths and Their Representations
Given a stratified space, , an entrance path is a continuous path in such that it for all time it stays in a stratum or enters into a deeper (with respect to ) stratum.
Let be a stratified space. The entrance path category of , has objects the points of and morphisms (elementary) homotopy classes of entrance paths.
[Theorem 6.1 of ] Let be a conically stratified space and a category. There is an equivalence of categories
between constructible cosheaves on and representations of its entrance path category.
Let be a combinatorial manifold. The combinatorial entrance path category, has as objects the strata of and a morphism whenever is a face of .
Let be a combinatorial manifold. There is an equivalence of categories .
Define a functor , where the image of a point is unique simplex containing , and the image a morphism is the combinatorial entrance path from (well-defined since the simplex containing must be a face of the simplex containing in order to be an entrance path). We claim that is fully faithful and essentially surjective. Again, let , , so that is a face of . Since and are face-coface pairs of a non-degenerate triangluation, the subspace deformation retracts onto , meaning there is a unique homotopy class of entrance paths . Furthermore, there is a unique morphism in , i.e., is fully faithful. Next, we observe that, for any simplex , we can always find a point so that (for example, let be the barycenter of ). That means we have shown is also essentially surjective, and thus gives an equivalence of categories. ∎
One might hope that there is an equivalence of entrance and combinatorial entrance path categories for a larger class of stratifications. However, even when a space is stratified by a “degenerate” triangulation, this equivalence does not generally hold. For instance, consider , stratified by , a single vertex, and , a single edge. Let . Then there are two distinct homotopy classes of entrance paths from in , but only one combinatorial entrance path given by the face relation in .
One useful interpretation of the preceding proposition is that the data of a constructible cosheaf on a combinatorial manifold is just a specification of costalks on the stratifying poset and linear maps between them.
3. Persistence Modules, Persistence Cosheaves, and Filtrations
We now introduce our main actors: persistence modules and persistent cosheaves. To start, we consider constructible cosheaves that arise from common types of persistence modules and/or filtrations. We construct these cosheaves in a way that is compatible with traditional models of the specific filtration or module in question. We finish the section with an equivalence result relating zig-zag modules to one-dimensional constructible cosheaves.
3.1. Persistent Definitions
A persistence module is a functor , where is some poset category. Specifically, we may refer to these as -indexed persistence modules. -indexed persistence modules define a category: the functor category, whose morphisms are natural transformations between the functors.
Hereafter, we take to be , the category of vector spaces over a field , and by , we mean a vector space of dimension in .
The previous definition is general – in what follows, we will mostly restrict our attention to single-parameter persistence modules. There are two flavors of such modules common in the literature: zig-zag  and monotone (standard) persistence modules . We note that monotone persistence modules are most commonly called simply ‘persistence modules;’ we have added the word ‘monotone’ to emphasize their distinction from more general modules.
To any poset there is an associated undirected graph: its Hasse diagram. Properties of the Hasse diagram, e.g., planarity, are used in order theory as they are often more accessible than the abstract poset. We will find it useful to consider the Hasse diagram of a poset as a one-dimensional simplicial complex.
Let be a poset.
The poset is zig-zag if its Hasse diagram is homeomorphic to the closed interval, half-closed interval, or .
A representation of a zig-zag poset is a zig-zag persistence module.
If is a linear order, then a representation is a monotone persistence module.
Consider a zig-zag persistence module , where the objects of are a discrete subset of real numbers (with potentially not-standard ordering). This, in turn, defines a stratification of , the connected ambient stratification (where we have a zero-stratum for every object of and a one-stratum for every connected component of , as in ). To define a cosheaf on , it suffices to define its values on a basis of the topology on .
First, we give a cosheaf theoretic interpretation of the notion of zig-zag modules found in . We call this cosheaf propagated because the functor is entirely determined by the ordering of and assignments to zero strata; the value over a one-stratum is propagated from either endpoint depending on the ordering of the relevant poset.
[The Propagated Persistence Cosheaf on ] Given with discrete, we define the the propagated persistence cosheaf as follows. Let be a metric -ball so that is less than the distance between any pair of zero-strata. Then either contains a single zero-stratum or no zero-stratum, and we assign values for as follows:
Next, we describe the assignment of morphisms. If contains a zero-strata, , or if is entirely contained in some one-strata , then . Suppose instead that contains the vertex but . Then if (or if ). See cosheaf.
The cosheaf is locally constant on strata, so it defines a constructible cosheaf on the stratified space .
Suppose that is the poset , where and are ordered with the standard ordering on as in the figure above. Then , , and .
3.2. Filtered Spaces and Cosheaves
Next, we discuss how persistence-modules and persistence module cosheaves relate to filtrations of spaces.
[Filtration] Let be a simplicial complex. A filtration of is a sequence of subcomplexes such that, for every , there is an inclusion of spaces and so that and .
If we take to be the indexing set of a filtration, then there is a natural way to view as a poset with the standard ordering of . Passing to homology in degree defines an associated monotone persistence module via the assignment . The propagated persistence cosheaf on (see pmcosheaf) is easy to describe. Indeed, for a single one-stratum, we have and that the costalk of at a zero-stratum is .
3.2.1. Monotone and Index Filtrations
Let be a monotone function on simplices. That is, whenever is a face of , we have . Let be the ordered set of minimum values in for which is a distinct non-empty simplicial complex. Setting and , we define the (monotone) filtration of by as
Notice that, by construction, all inclusion maps are in the direction of increasing index. Furthermore, if has non-empty simplices, .
Next, suppose that is a total order of the simplices of so that if either , or if is a face of , then . Letting , the increasing sequence of subcomplexes is an index filtration compatible with the monotone filtration.
In what follows, we will use to denote the number of non-empty simplices in a simplicial complex and to denote the number of steps in a monotone filtration.
Index filtrations are themselves monotone. Index filtrations are compatible with themselves, but to no other index filtrations.
3.2.2. From Index to Monotone
Suppose that is a monotone filtration with filter function and is a compatible index filtration. Then for every , there is some maximum interval for such that (where, whenever or , we define and , respectively). These intervals cover , and every interval corresponds to a unique . Then we define a map of stratified spaces, that maps intervals with a particular value under the filter function to intervals with that same value under as follows.
Suppose that , where is the associated interval for some . Three cases arise: if , we assign
if , we assign
and if , we assign
is a stratified map.
3.2.3. Augmented Descriptors via Index Filtrations
Let , , and be a monotone filtration and compatible index filtration, respectively (as in the previous section).
Given a monotone filtration, we are perhaps interested in the so-called instantaneous events that are captured in augmented topological descriptors, a remnant of the fact that many standard algorithms to produce descriptors for monotone filtrations are often actually employing compatible index filtrations. For example, an instantaneous -dimensional homology event at time records the presence of an -boundary in the (closed) space .
Note that many applications of TDA, such as the classic application of manifold learning through a Vietoris-Rips filtration, discard events with a short lifespan because they may be attributed to noise, so non-augmented persistence diagrams are the traditional tool of choice (see ). However, recent developments in areas such as shape comparison and inverse TDA problems (see, e.g., ) rely on the instantaneous events of augmented descriptors for efficient representation of simplicial or cubical complexes, particularly when the filtration used is directional (e.g., height filtration, lower-star filtration, etc.)
We aim to track both instantaneous and non-instantaneous events at every step of a monotone filtration. We introduce to account for instantaneous events (the extra data of an augmented module). Let denote the -dimensional boundaries of and let denote the kernel of the map on -dimensional homology induced by the inclusion . Although we may think of as generated by simplices in , the fact that includes into means that is also naturally viewed as a subgroup of . Then, we define:
Note that the rank of is the number of -boundaries that appear in , the th step of the filtration, but that were not previously boundaries or did not kill a cycle in the inclusion. Equivalently, the rank of is the number of points (counting multiplicity) on the diagonal in the standard -dimensional augmented persistence diagram corresponding to a monotone filtration. Thus, we can view as a repackaging of the ‘entire’ information in index filtrations, independent of the choice of compatible index filtration. The connection to compatible index filtrations is made explicit in the following lemma.
Suppose that is a monotone filtration corresponding to a filter function and is any compatible index filtration. Let denote the kernel of the map induced on homology in degree by the inclusion . Furthermore, let denote the kernel of the map induced on homology in degree by the composition of inclusions , where is as in cmap. Then:
Since each step in an index filtration adds a single simplex, either (if the simplex added does not fill in any -cycle) or (if the simplex added in fills in an -cycle). Thus, the direct sum in the equation above has nontrivial terms only for values of such that witnesses the death of -cycles in the index filtration. Recall that is the maximum interval whose image under the filter is . This means that and . Thus, every boundary of that was not present as a boundary in is introduced or becomes a boundary in some step of the index filtration between the values and . That is, terms of the direct sum above are nontrivial when boundaries are created (and otherwise). This is exactly the count of boundaries added in the inclusion , i.e., it is , using the previously introduced in the paragraph above. However, recall that should not account for boundaries that fill in a cycle from a previous step in the filtration. Thus, we quotient out by the kernel of the composition of maps between and . This kernel is generated by boundaries that include into themselves and by cycles in the initial filtration step that are eventually filled, i.e., cycles of that are filled in . Since and , and since the index filtration is compatible with the monotone filtration, we see that . Thus, the value in indexaug is exactly the value in monotoneaug, , as desired. ∎
Suppose that and are monotone and index filtrations as in the bottom and top of cmap. Then . Computed using the identification of coker, we see that this is the same as .
The following cosheaf organizes the information of both instantaneous and non-instantaneous events.
[Augmented Filtration Cosheaf on ] Let be a monotone filtration of a simplicial complex , and suppose is stratified by . We define the augmented filtration cosheaf on , , on metric -balls as follows.
Observe that the above definition implies that the costalk at a zero-stratum of is .
For an index filtration , any new -cycles introduced through the map are not -boundaries, since the boundaries and interiors of simplices are added at distinct filtration events. Thus, is trivial, i.e., the augmented filtration cosheaf that arises from an index filtration is equivalent to its (non-augmented) filtration cosheaf.
An instance of the previous remark is illustrated by following example.
Let be the index filtration in the top of cmap. Notice that the one-dimensional costalk of the non-augmented filtration cosheaf at is , which is isomorphic to .
The stratified map define above provides a clean interpolation between the augmented, non-augmented, and index cosheaves associated to a filtration.
Let and be the non-augmented and augmented filtration cosheaves for some monotone filtration and let be the filtration cosheaf for a compatible index filtration . Let be the map of stratified spaces as above. Then,
We have an isomorphism of cosheaves ;
Let be open such that , then .
That and agree on one-strata follows directly from their definitions (they can differ at zero-strata). In claim (i), there are two parts: that is constructible and that is isomorphic to . To prove the first, note that is a composition of “elementary collapses” as described in Example 2.2, so by functoriality is constructible. The second part of (i) is an explicit unwinding of the definition of the pushforward. ∎
3.3. An Equivalence Result
Let be the category of posets with Hasse diagrams homeomorphic to the interval, half-closed interval or and whose underlying set is at most countable. Morphisms in are surjective maps of posets. So from above, the category of zig-zag modules is the category of pairs with and a representation of .
Define the poset to have objects with non-identity morphisms
The poset arises naturally when considering stratified (ambiently) by the natural numbers.
There is a canonical isomorphism of categories
We wish to “mark” our posets by passing to the under category of .
Define the category of marked zig-zag modules, , to be the category of pairs with and a representation. A morphism is a pair with defining a morphism in the under category and a natural transformation.
It turns out that isomorphism in is too strong to capture our preferred notion of “sameness,” so we introduce a notion of weak equivalence. An example of an operation that creates a weakly equivalent module is “subdividing” a vertex in a poset into several vertices provided that all of the new maps in the corresponding representation are isomorphisms.
A morphism in is a weak equivalence if is a natural isomorphism. Let denote the collection of weak equivalences.
We caution the data-analytically oriented reader here; notice that weakly equivalent objects of do not generally have the same indices of “events,” i.e., vertices at which the corresponding image of the representation changes. That is, the standard map from to persistence diagrams (as described in ) does not factor through . However, the order and number of events is preserved.
The category of (marked) zig-zag modules localized at weak equivalences is equivalent to the category of constructible cosheaves on stratified by the natural numbers. That is, we have an equivalence of categories
The second equivalence is just an example of the exodromy equivalence. The first equivalence, which is actually an isomorphism of categories, is proved in Appendix B. There are some technicalities in proving the previous theorem, but the main idea of the equivalence is as follows. Let be a representation of . Pullback along the map to obtain a representation of so that, via Lemma 3.3, we have a representation of the corresponding entrance path category, i.e., a constructible cosheaf.
4. K-theory of Zig-Zag Modules
We now shift gears and compute the K-theory of the category of zig-zag modules. The category in which our modules take values plays a central role and we consider two different constructions: one for set-valued modules and one for modules valued in vector spaces.
Let be a stratified space, its combinatorial entrance path category, and any category. By way of the Exodromy Theorem, we will study persistence modules through the functor category . Therefore, we define the K-theory of zig-zag modules as follows.
Let be a combinatorial manifold, its combinatorial entrance path category, and any category. The K-theory of valued zig–zag modules (parametrized by ) is given by
One issue with the preceding definition is that for different , we may need to use different constructions of -theory. For instance, when is a category of vector spaces we use Waldhausen’s construction. As noted previously, when multiple constructions of K-theory exist for the same category the resulting K-groups are all isomorphic.
4.1. Assemblers: Modules
Let be a category with pullbacks (the existence of pullbacks is not universally assumed in the literature, so condition (FP) below is not automatic).
A Grothendieck pretopology on is a collection of covering families in such that
For all , any morphism contained in a covering family of , and all morphisms , the pullback exists;
For all , any covering family , and all morphisms , the family is covering;
If is a covering family and for each , is a covering family, then the composite family is covering;
If is an isomorphism in , then is a covering family.
A category (with pullbacks) equipped with a pretopology is called a Grothendieck site.
A closed assembler is a Grothendieck site such that
has an initial object and the empty family is a covering of ;
All morphisms in are monomorphisms.
We will let be the category of finite sets and injective functions. Given a (finite) set declare a family to be covering if the images of the is the entirety of ; we call such families jointly surjective. We verify that, so equipped, is indeed a closed assembler.
The category equipped with the jointly surjective topology is a closed assembler.
That pullbacks exist and that conditions (I) and (M) are satisfied is immediate. It only remains to show that collection of jointly surjective families satisfy the axioms of a pretopology. Similarly, since all pullbacks exist, condition (FP) is clear. In the category , an isomorphism is a bijection, so condition (Iso) is satisfied. The locality of covering families—condition (L)—follows (essentially) from the fact that the composition of surjective functions is itself surjective.
We now consider (BC). Let be a morphism, a covering family, and . By definition, there is a set and a morphism such that is in the image of . Explicitly, let be an element such that . Then maps to , i.e., is indeed a jointly surjective family. ∎
Let be a small category and a closed assembler. The category of functors is a Grothendieck site where a family is a covering family if and only if for all is a covering family in . So equipped, is a closed assembler.
The preceding Lemma can easily be verified directly. In our case, it specializes to the following result.
Let be a stratified space and its combinatorial entrance path category. The category of representations of , , is a closed assembler.
The following is the central construction of  (and the content of Theorem B contained therein).
Let be a closed assembler. There exists a spectrum such that is the free abelian group on the isomorphism classes of indecomposables of .
One way to construct the spectrum is to associate a Waldhausen category to the closed assembler and apply Waldhausen’s construction.
A zig-zag module is indecomposable if and only if it is an interval module.
Let be a combinatorial one-manifold and the category of injective set valued zig–zag modules parametrized by . Then we have an isomorphism of abelian groups
where is the set of -dimensional strata of .
In fact, this decomposition lifts to the level of spectra as an application of Theorem D of . We do not give a proof as the ideas are completely analogous to the proof of Theorem 4.2 below. In loc. cit., that the K-theory of finite sets recovers the sphere spectrum via the Barratt–Priddy–Quillen–Segal Theorem is explained.
Let be a combinatorial one-manifold, we have an equivalence of spectra
where is the set of -strata of and denotes the sphere spectrum.
For a general assembler, it can be difficult to identify the spectrum and its homotopy groups. In , Zakharevich gives an explicit description of arising from a closed assembler.
4.2. Abelian Categories: coSheaves
Linear categories don’t have great gluing constructions, e.g., a vector space is not the union of proper subspaces! So we wouldn’t expect (even after restricting to injective linear maps) to be an assembler. Fortunately, the category of finitely generated modules for a commutative ring is an Abelian category, so we can define/compute K-theory using the work of Quillen and Waldhausen. (If our ring is a field, we recover our old friend ). In this section we will freely use the material of Appendix A.
Let be a commutative ring, the associated Waldhausen category of finitely generated modules, a (finitely, simplicially) stratified space, and a connected zero-dimensional stratum, i.e., a point that is a stratum. The following sequence is split short exact sequence of Waldhausen categories
where and are the inclusion maps.
The content of Lemma A is precisely that the three categories appearing are Waldhausen. We next observe that the inverse and direct image functors (in this setting) are compatible with the equivalences and cofibrations, so indeed we have a sequence of exact functors.
It is standard that is right adjoint to and in this case that counit of the adjunction is a natural isomorphism. Because our domain categories are discrete (finite even), is indeed left adjoint to and the unit is a natural isomorphism; is the extension by zero map. The composition is manifestly the zero functor and presents as the cokernel of . In summary, the sequence is short exact and the adjointness properties we observed further show it is split.
The preceding lemma is straightforward as we are considering constructible cosheaves on the complement of a point (which is open). One could try to prove a version of the lemma above where is replaced by an arbitrary stratum and at the level of non-combinatorial entrance path categories, but—in general—this fails as will not have the appropriate adjointness properties. There is, however, a corresponding lemma for an arbitrary closed/open complement decomposition that is compatible with the stratification.
The split short exact sequence of Lemma 4.2 is standard.
Let . Each component of the natural transformation is an isomorphism, except for the component corresponding to . The component corresponding to is the inclusion of zero, which is a cofibration. Therefore, is a cofibration in the functor category.
Finally, let be a cofibration in . We need to check that unique map
is a cofibration. By definition, we must check this condition componentwise. For a component corresponding to , the kernel of is exactly the submodule of by which we quotient when constructing pushouts in categories of modules; that is, the component of is a monomorphism. For the component, the pushout is identified with and , so by hypothesis it is a monomorphism. ∎
We require one final observation/Lemma before assembling the proof of bigOplus. From the definition of entrance paths and the fact that K-theory preserves colimits, it immediately follows that K-theory is addivitive with respect to connected components of our parameter space. That is:
Let be a stratified space, then there is an equivalence of spectra
Although the strata of a one-dimensional stratified space are not generally disjoint, we still have an additivity result similar to the previous lemma, as we will now show.
Let be a one-dimensional combinatorial manifold. There is an equivalence of spectra
where is the set of -strata of .
We proceed by induction over the number of zero-strata. As our base case, note that when there are no zero-strata, we have and , so the claim holds. Now suppose that the claim holds whenever contains zero-strata, for all . Then consider the case that contains zero-strata. For an arbitrary zero-stratum , we know by key that
is a split short exact sequence of Waldhausen categories. Then by additivity, we see that there is an equivalence of spectra