Sheaves, Cosheaves and Applications Justin M. Curry March 13, 2013 Abstract This note advertises the theory of cellular sheaves and cosheaves, which are de- vices for conducting linear algebra parametrized by a cell complex. The theory is presented in a way that is meant to be read and appreciated by a broad audience, including those who hope to use the theory in applications across science and engineer- ing disciplines. We relay two approaches to cellular cosheaves. One relies on heavy stratification theory and MacPherson's entrance path category. The other uses the Alexandrov topology on posets. We develop applications to persistent homology, net- work coding, and sensor networks to illustrate the utility of the theory. The driving computational force is cellular cosheaf homology and sheaf cohomology. However, to interpret this computational theory, we make use of the Remak decomposition into indecomposable representations of the cell category. The computational formula for cellular cosheaf homology is put on the firm ground of derived categories. This leads to an internal development of the derived perspective for cell complexes. We prove a con- jecture of MacPherson that says cellular sheaves and cosheaves are derived equivalent. Although it turns out to be an old result, our proof is more explicit than other proofs and we make clear that Poincar´e-Verdier duality should be viewed as an exchange of sheaves and cosheaves. The existence of enough projectives allows us to define a new homology theory for sheaves on posets and we establish some classical duality results in this setting. Finally, we make use of coends as a generalized tensor product to phrase compactly supported sheaf cohomology as the pairing with the image of the constant sheaf through the derived equivalence. 1 Contents 1 Introduction 5 1.1 Outline of the Document . .8 1.2 How to Get Straight to the Applications . 11 1.3 Acknowledgements . 12 2 Categories: Limits and Colimits 13 2.1 Categories . 13 2.2 Diagrams and Representations . 17 2.3 Cones and Limits . 18 2.4 Co-Cones and Colimits . 21 3 Sheaves and Cosheaves 24 3.1 The Abstract Definition . 25 3.2 Limits and Colimits over Covers: a Structure Theorem . 30 3.2.1 Rephrased as Equalizers or Co-equalizers . 31 3.2.2 Rephrased as Exactness . 31 3.3 Cechˇ Homology and Cosheaves . 32 3.4 Refinement of Covers . 36 3.5 Generalities on Sheaves and Cosheaves . 37 4 Preliminary Examples 42 4.1 Sheaves Model Sections . 42 4.2 Cosheaves Model Topology . 45 4.3 Taming of the Sheaf... and Cosheaf . 48 5 Cellular Sheaves and Cosheaves 50 5.1 Stratifications: A Forge of Theory and Examples . 52 5.1.1 Whitney and Thom-Mather Stratified Spaces . 54 5.1.2 Stratified Maps: Glued Together Fiber Bundles . 60 5.1.3 Persistence and Tame Topology . 64 5.1.4 Local Systems and Constructibility . 70 5.1.5 Representations of the Entrance Path Category . 74 5.2 Partially Ordered Sets: Finite Spaces and Functors . 93 5.2.1 The Alexandrov Topology . 93 5.2.2 Functors on Posets . 95 6 Functoriality Under Maps 99 6.1 Maps of Posets and Associated Functors . 100 6.1.1 Pullback or Inverse Image . 101 6.1.2 Application: Subdivision . 102 6.1.3 Pushforward or Direct Image . 102 6.1.4 fy, Pushforwards and Closed Sets . 103 2 6.1.5 f!: Pushforward with Compact Supports on Cell Complexes . 105 6.2 Calculated Examples . 107 6.2.1 Projection to a point . 107 6.2.2 Inclusion into a Closed Interval . 107 6.2.3 Map to a Circle . 108 6.3 Adjunctions . 108 7 Homology and Cohomology 111 7.1 Computational Sheaf Cohomology and Cosheaf Homology . 111 7.1.1 Cellular Sheaf Cohomology . 112 7.1.2 Cellular Cosheaf Homology . 114 7.2 Explaining Homology and Cohomology via Indecomposables . 115 7.2.1 Representation Theory of Categories and the Abelian Structure . 115 7.2.2 Quiver Representations and Cellular Sheaves . 119 8 Barcodes: Persistent Homology Cosheaves 121 8.1 Barcodes and One-Dimensional Persistence . 121 8.2 Barcode Homology and Multi-Dimensional Persistence . 125 9 Network Coding and Routing Sheaves 130 9.1 Duality and Routing Sheaves . 132 9.2 Counting Paths Cohomologically, or Failures thereof . 133 10 Sheaves and Cosheaves in Sensor Networks 135 10.1 A Brief Introduction to Sensors . 135 10.2 The Coverage Problem: Static and Mobile . 136 10.3 Intruders and Barcodes . 138 10.3.1 Tracking the Topology over Time . 139 10.3.2 Linearizing the Sheaf of Sections . 141 10.4 Multi-Modal Sensing . 144 10.4.1 A Deeper Look at Sensing . 146 10.4.2 Indecomposables, Evasion Sets, Generalized Barcodes . 149 11 The Derived Perspective 152 11.1 Taylor Series for Sheaves . 152 11.1.1 Elementary Injectives and Projectives . 153 11.1.2 Injective and Projective Resolutions . 155 11.2 The Derived Category and Homotopy Theory of Chain Complexes . 157 11.3 The Derived Definition of Cosheaf Homology and Sheaf Cohomology . 161 11.3.1 Borel-Moore Cosheaf Homology . 163 11.3.2 Invariance under Subdivision . 164 11.4 Sheaf Homology and Cosheaf Cohomology . 165 11.4.1 Invariance under Subdivision . 167 3 11.4.2 Examples of Sheaf Homology over Graphs . 168 12 Duality: Exchange of Sheaves and Cosheaves 170 12.1 Taking Closures and Classical Dualities Re-Obtained . 170 12.2 Derived Equivalence of Sheaves and Cosheaves . 173 12.2.1 Linear Duality . 175 12.2.2 Verdier Dual Anti-Involution . 176 13 Cosheaves as Valuations on Sheaves 177 13.1 Left and Right Modules and Tensor Products . 177 13.2 Compactly-Supported Cohomology . 179 13.3 Sheaf Homology and Future Directions . 181 4 1 Introduction The utility of linear algebra in mathematics, science and engineering is manifest. Linear algebra exhibits a surprising fidelity for applications with discrete structures possessing con- tinuous qualities. When wedded with geometry, it provides true local approximations to highly complex phenomena. The linear reflects nonlinear properties that can be effectively computed and, today, put on a computer. The embedding of the continuous world into the discrete matrix of the computable world is a remarkable force of technological innovation. Classical theorems of 17th, 18th and 19th century mathematics describing the flow and flux of heat, fluid and abstract electromagnetic fields were framed continuously via The Cal- culus. This form usually came after first thinking in terms of discrete particles continuously moving through small boxes. The passage from discrete to continuous and again to dis- crete { or model to theory to applications { by programming these renowned equations on computers, represents a cycle of innovation of unparalleled productivity. The theorist and aesthete may lament the degradation of her ideal models, but the engineer and pragmatic judiciously apprehends the differences and the similarities. To say that for algebraic topologists the situation described is most familiar is an un- derstatement more than mild. The whole enterprise of algebraic topology consists of the principled study of pushing the continuous through the mesh of algebra. However, this perspective betrays the history of the subject, since up until a hundred years ago, the proto- type of this study was called \combinatorial topology." Without algebra, numbers were the receptacle for discretization. To perform an alternating count of vertices, edges and faces and discover that for all the Platonic solids, the result is two, would usher in history's first topological invariant in the 1750s { the Euler characteristic. The complexity of the objects studied by topology rose, the linear giving way to the subtle twists of holomorphic maps and geometry, the grounding in discrete computable invariants continued. It was Gauss { that father figure of so much mathematics { who then understood that averaging the curvature of a surface yielded Euler characteristic. One of the many disciples of Gauss, Johann Benedict Listing, coined the very word \topology" in 1836 and in attempting to define its subtle nature said: \By topology we mean the doctrine of the modal features of objects, or of the laws of connection, of relative position and of succession of points, lines, surfaces, bodies and their parts, or aggregates in space, always without regard to matters of measure or quantity." [50] Listing, together with Gauss, would influence that next great hero-figure of mathematics { Bernhard Riemann { by communicating some notions of topology. Alas, that coarse art of counting required serious strengthening in order to organize the inchoate world of high- dimensional manifolds that Riemann summoned from the heavens. Riemann, who lived a short life constantly in poor health, spent his last years in Italy where he returned a visit to Enrico Betti. Shortly after Riemann's death, Betti produced { without knowing it { the first lifting of Euler characteristic in 1870, by defining a sequence of numbers whose purpose was 5 to count \holes" in a higher dimensional manifold. To do so, Betti introduced the notion of a boundary, thus anticipating the algebraization of combinatorial topology. The person credited with inventing algebraic topology is Henri Poincar´e.In addition to Riemann and Betti, he was influenced by his own investigations in differential equations, celestial mechanics and discontinuous group actions. In his famous treatise \On Analysis Situs" (the preferred name for topology before Lefschetz revived Listing's term), Poincar´e tried to assuage an oddly prescient concern of modern times: \Persons who recoil from geometry of more than three dimensions may believe this result to be useless and view it as a futile game, if they have not been informed of their error by the use made of Betti numbers by our colleague M. Picard in pure analysis and ordinary geometry." [69] Having cursorily addressed the criticism of a lack of applications of analysis situs, Poincar´e went on to push Betti's notion of a boundary and defined what is now known as homology and the fundamental group.