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Sheaves, Cosheaves and Applications

Justin Curry

arXiv:1303.3255v2math.ATmath.RT

TL;DR

The thesis addresses how to make sheaf and cosheaf theory usable for science and engineering while retaining mathematically rich constructions. It develops computable cellular models, extends them across topology, persistence, networks, and sensing, and establishes equivalences, homological tools, and metric structure. The thesis also identifies communication, higher-order computation, and statistical theory as ongoing boundaries.

  • Problem

    Sheaf theory’s abstract language and derived-functor constructions are difficult to communicate and apply outside pure mathematics, motivating simpler and computable descriptions.

  • Method

    The thesis develops cellular sheaves and cosheaves as finite vector-space assignments on cell complexes, alongside constructible-cosheaf, categorical, homological, and metric frameworks.

  • Results

    The thesis applies cellular (co)sheaves to topological data analysis, network coding, and sensor networks, and develops equivalences and constructions supporting persistence, duality, and sheaf homology.

  • Takeaways & Limitations

    Sheaves and cosheaves are presented as frameworks for tethering spatially distributed data and for using homology in applications where it is easier to visualize.

  • Takeaways & Limitations

    A broader theory of statistics for sheaves remains an open problem.

Abstract

from arXiv · show

This thesis develops the theory of sheaves and cosheaves with an eye towards applications in science and engineering. To provide a theory that is computable, we focus on a combinatorial version of sheaves and cosheaves called cellular sheaves and cosheaves, which are finite families of vector spaces and maps parametrized by a cell complex. We develop cellular (co)sheaves as a new tool for topological data analysis, network coding and sensor networks. A foundation for multi-dimensional level-set persistent homology is laid via constructible cosheaves, which are equivalent to representations of MacPherson's entrance path category. By proving a van Kampen theorem, we give a direct proof of this equivalence. A cosheaf version of the i'th derived pushforward of the constant sheaf along a definable map is constructed directly as a representation of this category. We go on to clarify the relationship of cellular sheaves to cosheaves by providing a formula that defines a derived equivalence, which in turn recovers Verdier duality. Compactly-supported sheaf cohomology is expressed as the coend with the image of the constant sheaf through this equivalence. The equivalence is further used to establish relations between sheaf cohomology and a herein newly introduced theory of cellular sheaf homology. Inspired to provide fast algorithms for persistence, we prove that the derived category of cellular sheaves over a 1D cell complex is equivalent to a category of graded sheaves. Finally, we introduce the interleaving distance as an extended pseudo-metric on the category of sheaves. We prove that global sections partition the space of sheaves into connected components. We conclude with an investigation into the geometry of the space of constructible sheaves over the real line, which we relate to the bottleneck distance in persistence.

A MATHEMATICAL INTRODUCTION

This part is organized for different audiences, offering distinct reading paths for category-theory beginners, general sheaf readers, and readers focused on exact-sequence or cosheaf perspectives.

  • Category-theory beginners are advised to read Chapter 1 before moving to Part II, especially Chapters 4 and 6.
  • Readers seeking general topological definitions should study Chapter 2 and then proceed to Chapter 3 for examples.
  • Section 2.2 addresses readers who find the sheaf axiom as an exact sequence opaque and Čech homology notation frustrating.
  • Sections 2.5 and 2.5.4 target readers who view cosheaves simply as sheaves valued in the opposite category.

1 A PRIMER ON CATEGORY THEORY

The primer presents category theory as a language of objects and composable maps, then develops open-set categories and related constructions needed for sheaves and cosheaves.

  • Categories provide a language for identifying formal similarities throughout mathematics and originated in part from studying functors assigning algebraic invariants to spaces.
  • The primer prepares readers to regard open sets as a category and to summarize functor behavior using limits and colimits.
  • Category theory formalizes objects, morphisms, composition, associativity, and identity morphisms.
  • The open-set category Open(X) has open sets as objects and a unique morphism U → V whenever U ⊆ V.
  • Pointed open-set categories attach a point to each open set, while the fundamental groupoid uses points and homotopy classes of paths as objects and morphisms.
  • The primer introduces Set, Ab, Vect, vect, and Top as data categories used throughout the paper.

2 THE THEORY OF SHEAVES AND COSHEAVES

The theory section develops sheaves and cosheaves as functorial assignments satisfying gluing conditions, then makes those conditions computable through equalizers, coequalizers, and homological constructions.

  • Sheaves and cosheaves assign data to subsets, with sheaves restricting from larger to smaller spaces and cosheaves extending from smaller to larger ones.
  • The abstract functorial perspective captures sheaves and cosheaves through gluing axioms expressed by limits and colimits.
  • For vector spaces, cover conditions reduce to equalizers and coequalizers, and Čech homology yields complexes whose zeroth homology computes pre-cosheaf values on covers.
  • Any functor satisfying Mayer–Vietoris is a cosheaf.
  • If a sheaf or cosheaf axiom holds for a cover, it holds for every refinement, so checking a finest cover can guarantee the axiom for all covers.
  • Cosheafification exists for vector-space pre-cosheaves because cosheaves form a coreflective subcategory of Fun(Open(X), Vect).

3 PRELIMINARY EXAMPLES

This section develops sheaves and cosheaves through examples involving local and global sections, fiber bundles, and local systems. It shows how locally constant sheaves and cosheaves correspond to representations of fundamental groupoids under suitable topological assumptions.

  • Sheaves model sections: Local sections may exist everywhere without combining into a global section.The restricted time projection provides an example where local continuous sections exist but no globally defined section does.
  • Sheaves model sections: The square-root map on the complex plane has two local sections near each nonzero point but no global section.The obstruction is that a global square root would necessarily be multivalued over the whole plane.
  • Local systems: A fiber bundle produces local systems from the homology or cohomology of its fibers.Paths induce fiber homology isomorphisms, while homotopies preserve these maps.
  • Local systems: Under local path-connectedness, local simple connectivity, paracompactness, and Hausdorffness, locally constant sheaves and cosheaves correspond to fundamental-groupoid representations.Representations determine both types of locally constant objects, yielding an equivalence of their categories.

LINEAR ALGEBRA OVER CELL COMPLEXES

Cellular sheaves and cosheaves are presented as linear algebra parametrized by cell complexes, with their sheaf-theoretic status justified through the Alexandrov topology. The section also introduces computational treatments and barcodes for interpreting their homology and cohomology.

  • Cellular sheaves and cosheaves are linear algebra parametrized by a cell complex.
  • The Alexandrov topology justifies treating functors from posets as sheaves or cosheaves.
  • Cellular sheaf cohomology and cosheaf homology are developed computationally before being grounded in derived categories.
  • Barcodes are introduced to interpret cellular sheaf cohomology and cosheaf homology.

4 CELLULAR SHEAVES AND COSHEAVES

This section formalizes cellular sheaves and cosheaves as computable diagrams over cell complexes and relates them to sheaves on Alexandrov-topologized posets. It also develops explicit extension and pushforward constructions.

  • Cell complexes provide a computationally convenient setting that is more general than regular cell complexes and need not be simplicial after subdivision.
  • A cell complex determines a face-relation poset whose category indexes cellular sheaves and cosheaves.
  • Cellular sheaves assign objects to cells and restriction maps to incident face-coface pairs, while cellular cosheaves reverse the map direction.
  • For a torus height function, taking homology of star pre-images produces a cellular cosheaf; limits and colimits recover open-set sheaf and cosheaf values.
  • Cellular sheaves and cosheaves are respectively sheaves and cosheaves on the Alexandrov topology, uniquely determined by functors on the associated poset.
  • On Alexandrov spaces, pushforwards can be defined directly through inverse images of opens or pointwise Kan extensions, avoiding explicit sheafification.

6 HOMOLOGY AND COHOMOLOGY

This section extends cellular homology and cohomology by placing vector spaces and compatible linear maps on cells. The resulting theory is combinatorial, computable, and interpretable through indecomposable representations.

  • Cellular (co)homology augments classical coefficients with vector spaces on cells and linear maps between incident cells, capturing data that varies over a cell complex.
  • When every cell carries k and every incidence map is the identity, cellular (co)homology recovers classical cellular (co)homology.
  • The theory is combinatorial and computable using linear algebra, while representation theory decomposes sheaves and cosheaves into indecomposable building blocks.
  • The construction uses signed incidence relations to define cellular chain or cochain differentials whose consecutive compositions vanish.
  • Generalized barcodes use indecomposable representations to expose how data travels through a space.
  • Finite-poset sheaf and cosheaf categories satisfy Krull-Schmidt decompositions, and constant (co)sheaves on connected cell complexes are indecomposable.

7 THE DERIVED PERSPECTIVE

This section introduces derived methods for cellular sheaves and cosheaves to retain information that ordinary fiber homology can miss. It constructs resolutions and relates derived functors to cellular homology, duality, and subdivision.

  • Ordinary homology of fibers cannot distinguish two maps to S2, motivating the derived category as a framework that retains the missing structure.
  • The derived perspective replaces isolated sheaves with complexes of sheaves or sheaves of complexes, using fiberwise cochains and differentials.
  • Injective and projective resolutions are explicit for cellular sheaves and cosheaves, and every sheaf has an injective resolution while every cosheaf has a projective resolution.
  • For a cellular cosheaf, the left derived functors of the constant-map pushforward agree with the computational cellular homology groups.
  • On triangulated closed manifolds, cellular sheaves on a triangulation correspond to cellular cosheaves on the dual triangulation with matching homologies and cohomologies.
  • Sheaf homology is invariant under subdivision, while the newly developed sheaf homology and cosheaf cohomology theories can depend on target subdivision and embedding.

APPLICATIONS TO SCIENCE AND ENGINEERING

The thesis applies cellular sheaves and cosheaves to persistent homology, network coding, and sensor networks. These applications support distributed computation, barcode visualization, duality results, and new sensing models.

  • Persistent homology is reformulated with sheaves and cosheaves to distribute homology computations and aggregate results efficiently.
  • The persistence chapter connects level-set and sub-level-set persistence using spectral sequences and introduces generalized barcodes for multi-dimensional persistence.
  • Cellular sheaves and barcode methods model network coding and visualize data flow, alongside two proofs of a network-coding duality theorem.
  • In sensor networks, cosheaf machinery proves a no-go result for using level-set persistence on the intruder problem.
  • The sensor-network chapter introduces a linearized model for multi-modal sensing.

8 TOPOLOGICAL DATA ANALYSIS

This section presents persistent homology as a computable, barcode-based approach to describing shape from point clouds and level sets, then extends the perspective using sheaves and cosheaves. It also develops barcode homology and connections between level-set and sublevel-set persistence.

  • Point Cloud Persistence: Persistent homology replaces a point cloud with unions of growing balls, applies homology, and represents the resulting persistence module by a barcode.Long bars indicate robust topological signals; for the example point cloud, long H0 and H1 bars indicate connectivity and an apparent circle.
  • Point Cloud Persistence: The prototypical point-cloud pipeline consists of defining the cloud, forming a filtration of unions of balls or Vietoris–Rips complexes, applying homology, and decomposing into interval modules.The same construction extends to sublevel sets of sufficiently well-behaved functions.
  • Level Set and Zigzag Persistence: Zigzag persistence handles filtrations built from alternating maps, and Gabriel’s theorem preserves interval-module decompositions and barcodes for these representations.This supports level-set persistence when the usual persistence-module structure theorem does not directly apply.
  • Level Set and Zigzag Persistence: Sheaf-theoretic level-set persistence provides locality, while sheaf cohomology preserves the information of the domain space that the degree-one level-set sheaf alone can lose.The section explicitly identifies locality and information preservation as complementary roles of sheaves and sheaf cohomology.
  • Cellular Maps and Absolute Homology Cosheaves: Absolute homology cosheaves over compact subsets of the real line decompose into four barcode types, whose cosheaf homology agrees with Borel–Moore homology of the underlying barcode.For these barcodes, H0 counts closed bars and H1 counts open bars; finite direct sums permit computation from the barcode decomposition.

A Unifying Perspective

This section uses the generalized nerve and Mayer–Vietoris blowup to unify local sheaf-cohomology constructions and to derive persistence from level-set data. It also frames sheaves as a foundation for multidimensional persistence through Leray sheaves.

  • Generalized Nerve and Mayer–Vietoris Blowup: The Mayer–Vietoris blowup of an open cover is homotopic to the covered paracompact Hausdorff space.The construction lies in the product of the space and the nerve, with projection maps used to establish the homotopy equivalence.
  • Generalized Nerve and Mayer–Vietoris Blowup: When the nerve is at most one-dimensional, the Leray cellular sheaves associated to the blowup are isomorphic to the Čech cellular sheaves of the cover.This identifies the two local cohomological strategies within the stated nerve-dimension setting.
  • Generalized Nerve and Mayer–Vietoris Blowup: Both Čech and Leray approaches support distributed cohomology computation because their defining data require only local cohomology calculations.The computations can in principle be assigned to different processors, while dualization converts the corresponding results into homology computations with cosheaves.
  • Level Set to Sublevel Set Persistence: For a proper map to the real line, the Leray sheaf’s stalks record level-set cohomology, while restrictions and sheaf cohomology over intervals produce sublevel-set persistence.The resulting functor combines cohomology groups of restricted Leray sheaves to record cohomology of entire sublevel sets.
  • Multidimensional Persistence: The section presents sheaves and cosheaves as a response to the challenge of constructing multidimensional persistence while retaining locality.Its multidimensional persistence module is defined as a Leray sheaf, with level-set persistence treated as the primary object.

9 NETWORK CODING AND ROUTING SHEAVES

This section models network coding and routing with cellular sheaves, using linear maps to represent capacities and coding, and studies their decompositions, duality, and limitations for representing information paths. Routing sheaves admit a particularly tractable interval-and-circle structure.

  • Network Coding Sheaves: A network coding sheaf assigns each edge a vector space whose dimension equals capacity, uses projections for incoming edges, and allows arbitrary linear coding maps on outgoing edges.The total coding map at a vertex combines the maps from the vertex space to its outgoing edges.
  • Routing Sheaves: Routing sheaves are the capacity-one specialization in which coding maps are binary matrices with at most one 1 in each row and column.The remaining incoming edges are matched bijectively to a subset of outgoing edges, while other incoming data may be sent to zero.
  • Routing Sheaves: Every routing sheaf can be realized as the compactly supported pushforward of a disjoint union of half-open intervals or circles whose images intersect only at vertices.This decomposition makes routing sheaves easier to visualize and supports their barcode perspective.
  • Duality and Routing Sheaves: For routing sheaves, H0 represents independent information flows, whereas H1 counts closed trajectories of information flow.The latter interpretation motivates a sheaf-theoretic analogue of a cut-equals-flow relationship.
  • Counting Paths Cohomologically: A pseudo network-coding sheaf with independent source capacity cannot encode source-to-target paths cohomologically.The failure occurs even when the intended network has no flow from source to target, because the sheaf may still decompose into intervals with nontrivial behavior.
  • Counting Paths Cohomologically: Adding decoding edges changes the decomposition in the example from two half-open and two closed intervals to only two half-open intervals.The example uses decoding edges to alter the barcode representation in a network with no source-to-target flow.

10 SHEAVES AND COSHEAVES IN SENSOR NETWORKS

The chapter develops sheaf- and cosheaf-based tools for sensor-network coverage and evasion, showing both their computational promise and limits as path-detection criteria. It introduces sensing and evasion constructions whose homology and cohomology yield structural results about detectable properties and evasion regions.

  • Motivation: Cellular sheaf theory formalizes global-section questions and makes them programmable, addressing the rapid complexity of intuition-based reasoning in sensor networks.The chapter frames sheaf theory as a way to turn section-existence questions into computable mathematics.
  • Linearized sheaf and cosheaf models: A compact connected evasion region projecting onto every time has a barcode spanning the full interval [0, 1].This establishes the existence of a persistent homological feature when some evasion location exists at every time.
  • Linearized sheaf and cosheaf models: A long barcode or nonzero H0([0, 1]; F) does not imply an evasion path, because homology is insensitive to how the evasion region is embedded.The linearized sheaf model therefore produces false positives as an if-and-only-if detector.
  • Multi-modal sensing: The sensing sheaf assigns sensor-property subspaces on the nerve of the sensor supports, while the evasion cosheaf is identified with the dual of the embedding cokernel.These constructions connect sensor capabilities, detected properties, and undetected regions through linear algebra.
  • Multi-modal sensing: A short exact sequence induces a long exact cohomology sequence relating the topology of the covered region to sensing and evasion sheaves.The sensing-evasion decomposition identifies cokernel cohomology with evasion-cosheaf homology.
  • Multi-modal sensing: When sensor properties come from an orthonormal basis, the evasion cosheaf splits into constant cosheaves supported on individual evasion sets.Under overlapping detection sets, the sensing sheaf has no global sections; the framework also yields forcing results such as disconnected red evasion sets.

NOVEL MATHEMATICAL CONTRIBUTIONS

The thesis contributes new mathematical foundations connecting constructible cosheaves, cellular sheaves, cohomology, and persistence. It proves categorical equivalences and metric results that clarify duality, simplify one-dimensional computations, and relate sheaf geometry to persistence.

  • Constructible cosheaves: Constructible cosheaves are proved equivalent to representations of MacPherson’s entrance path category via a van Kampen theorem.The chapter also constructs representations from stratified definable maps using stratification theory.
  • Sheaf-cosheaf duality: An explicit derived equivalence between cellular sheaves and cosheaves presents Verdier duality as an exchange between the two theories.This formula provides the bridge used in later cohomological constructions.
  • Sheaf-cosheaf duality: Compactly supported cellular sheaf cohomology is expressed as a derived coend with the constant sheaf’s image under the equivalence.The result applies the sheaf-cosheaf correspondence to a concrete cohomological operation.
  • One-dimensional cellular sheaves: Over a one-dimensional base space, the derived category of cellular sheaves is equivalent to a graded category.This formalizes why spectral sequences over graphs collapse on the E2 page.
  • Sheaf geometry and persistence: The interleaving distance is an extended metric on sheaves, and global sections place sheaves in distinct connected components.The thesis further describes constructible sheaves over the real line and relates that geometry to persistence.

11 THE DEFINABLE ENTRANCE PATH CATEGORY

This chapter models fiber homology of definable maps using the definable entrance path category and establishes its relationship with constructible cosheaves. It proves local and stratified-map results that support this correspondence, including a van Kampen theorem and cellular cosheaf constructions.

  • Multi-dimensional persistence studies how fiber homology changes over multi-parameter families, a problem whose complexity increases sharply beyond one parameter.
  • Definable maps yield representations of the definable entrance path category, while its opposite, the exit path category, yields constructible sheaves.
  • Thom-Mather stratifications: Whitney stratified spaces support strong local structure, including tubular-neighborhood local triviality along each stratum.
  • Thom-Mather stratifications: Finite cell decompositions and regular-neighborhood deformation retractions provide computable stratified models and homotopy equivalences.
  • Thom mappings: For codomain strata differing by one dimension, the restricted stratified map is Thom; every C1 stratified map to R is therefore Thom.
  • Entrance paths and cosheaves: The van Kampen theorem builds entrance path categories locally, despite the difficulty that restricted homotopies preserve entrance paths but need not preserve endpoints.
  • Entrance paths and cosheaves: Representations of the entrance path category define constructible cosheaves, and constructible cosheaves with finite-dimensional costalks determine such representations.
  • Representations from stratified maps: Proper definable stratified maps produce representations of definable entrance path categories, and stratified maps to R produce cellular cosheaves degreewise.

12 DUALITY: EXCHANGE OF SHEAVES AND COSHEAVES

This section constructs a derived equivalence between cellular sheaves and cosheaves by extending cell data over closures and arranging it into chain complexes. The equivalence recovers compactly supported cohomology, sheaf homology, Poincaré duality, and Verdier duality.

  • 12.1 The Poincaré-Verdier equivalence functor: The construction extends each cell’s sheaf value across its closure while storing pre-existing face values in separate complex degrees.This realizes duality as an exchange between open and closed cells.
  • 12.1 The Poincaré-Verdier equivalence functor: The functor bP sends a cellular sheaf to a cosheaf of chain complexes, placing F(σ) in degree dim |σ| or homological degree −dim |σ|.Its differentials use signed cell-incidence relations and the sheaf restriction maps.
  • 12.1 Computational consequences: Applying bP and taking colimits term by term yields the computational formula for compactly supported sheaf cohomology.The section presents this as a computational consequence of the equivalence.
  • 12.1 Computational consequences: For a compact manifold, the construction relates compactly supported sheaf cohomology to Borel–Moore homology and the newly introduced cellular sheaf homology.The proof uses the dual cell structure and identifies sheaf homology with cosheaf homology.
  • 12.2 Derived equivalence: Theorem 12.2.1 proves that bP : Db(Shv(X)) → Db(CoShv(X)) is an equivalence of categories.The proof establishes quasi-isomorphic inverse composites and concludes that the functor is an adjoint equivalence.
  • 12.2 Linear duality and Verdier duality: Composing bP with linear duality gives the Verdier dual anti-equivalence on constructible cellular sheaves.The result is stated as D ∼= VbP.

13 COSHEAVES AS VALUATIONS ON SHEAVES

This section interprets cosheaves as valuations acting on sheaves through a coend or tensor product. The framework expresses compactly supported cohomology and related computations using cosheaves, while also yielding a one-dimensional graded-sheaf equivalence and higher sheaf homology.

  • 13.1 Tensoring sheaves with cosheaves: A coend generalizes the tensor product and defines a pairing between sheaves and cosheaves using categorical colimit operations.The pairing applies even to pre-sheaves and pre-cosheaves because neither sheaf axiom is required.
  • 13.1 Tensoring sheaves with cosheaves: Tensoring a sheaf with a skyscraper cosheaf recovers the operation of taking its stalk at a point.The observation extends the notion of cosheaves acting as valuations on sheaves.
  • 13.1 Tensoring sheaves with cosheaves: Every colimit-preserving functor on sheaves arises by tensoring with a cosheaf.This provides the categorical basis for viewing cosheaves as valuations on sheaves.
  • 13.2 Compactly-supported cohomology: The section gives an explicit complex of cosheaves that realizes derived pushforward with compact supports.This construction is motivated by the adjunction for derived pushforward and the derived sheaf–cosheaf equivalence.
  • 13.2 Compactly-supported cohomology: For cellular sheaves, derived pushforward to a point is equivalent to tensoring with the image of the constant sheaf under the derived equivalence.The resulting perspective couples the topology of X with a sheaf’s cohomology.
  • 13.3 Sheaf homology and future directions: A projective resolution of the constant cosheaf provides a way to compute higher sheaf homology.The passage identifies this theory as recently applied to max-flow min-cut.
  • 13.3 Sheaf homology and future directions: For a one-dimensional cell complex, every bounded complex of cellular sheaves is quasi-isomorphic to its graded cohomology sheaf.Thus the derived category is equivalent to a category of graded sheaves.

15 A METRIC ON THE CATEGORY OF SHEAVES

The chapter introduces interleaving distance for sheaves, establishes its metric properties, and analyzes how sheafification and global sections constrain interleavings. It then gives explicit distance formulas and geometric coordinates for constructible sheaves on the real line.

  • Interleavings and stability: Interleavings of sheaves provide a derived stability result for pushforward sheaves of maps that are close in the supremum norm.Higher fiber invariants such as H_i for i ≥ 1 can be unstable under large perturbations.
  • Global sections and metric structure: Global sections obstruct interleavings: presheaves with non-isomorphic global sections cannot be ε-interleaved for any ε.For sheaves, thickening preserves global sections under the stated closed-map condition.
  • The effect of sheafification: Thickening a sheaf need not preserve the sheaf axiom because it can create intersections absent from the original space.The chapter therefore studies sheafification as a separate operation on interleavings.
  • The effect of sheafification: Sheafification can either increase or decrease interleaving distance: finite-distance presheaves may become infinitely separated, while infinitely separated presheaves may become isomorphic.In the displayed example, sheafifications produce radically different sheaves; in another, sheafification sends a presheaf to the zero sheaf.
  • Global sections and metric structure: The interleaving distance is an extended metric on the skeleton of the sheaf category, because distance zero implies isomorphic sheaves.The value ∞ is allowed when no interleaving exists.
  • Indecomposable sheaves and coordinates: For indecomposable sheaves on the real line, explicit thickening and distance formulas yield geometric coordinates and taxicab or sup-norm descriptions.For example, k(b,d) lies at distance (d − b)/2 from the zero sheaf, while k[b,d] lies at that distance from the midpoint skyscraper sheaf.
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