Cellular Sheaves of Lattices and the Tarski Laplacian
aa r X i v : . [ m a t h . A T ] J u l Cellular Sheaves of Lattices and the Tarski Laplacian
Robert Ghrist and Hans Riess A BSTRACT . This paper initiates a discrete Hodge theory for cellular sheaves taking values in a category of latticesand Galois connections. The key development is the
Tarski Laplacian , an endomorphism on the cochain complexwhose fixed points yield a cohomology that agrees with the global section functor in degree zero. This has im-mediate applications in consensus and distributed optimization problems over networks and broader potentialapplications.
1. Introduction
The goal of this paper is to initiate a theory of sheaf cohomology for cellular sheaves valued in a cat-egory of lattices. Lattices are algebraic structures with a rich history [ ] and a wide array of applica-tions [ , , , , , ]. Cellular sheaves are data structure that stitch together algebraic entities accordingto the pattern of a cell complex [ ]. Sheaf cohomology is a compression that collapses all the data over atopological space — or cell complex — to a minimal collection that entwines with the homological featuresof the base space [ ]. Our approach is to set up a Hodge-style theory, developing analogues of the com-binatorial Laplacian adapted to sheaves of lattices. Specific contributions of this work include the following.(1) In §
2, we review posets, lattices, lattice connections, cellular sheaves, and Hodge theory. It isnontrivial to define cohomology for sheaves valued in the (nonabelian) category of lattices andconnections.(2) In § Tarski Laplacian , L , is a reasonable candidate for a diffusion operator.(3) In § ( id ∧ L ) has fixed point set equal to the quasi-sublattice ofglobal sections of the sheaf.(4) In § harmonic flow projects arbitrary 0-cochains toglobal sections.(5) Interpreting global sections as zeroth cohomology of the sheaf, in § Tarskicohomology in terms of fixed points on higher cochains.(6) In § § § Readers who need no motivation for Hodge-theoretic sheaf cohomology valued inlattices may skip this subsection.The authors are motivated by certain problems in data science of a local-to-global nature, in which mul-tiple instances of local data are required to satisfy constraints based on a notion of proximity. Such problemsoften can be formalized in the language of sheaf theory, where the local data are stored in stalks, and local
Key words and phrases.
Cellular sheaves, lattice theory, non-abelian homological algebra, fixed points, discrete Hodge theory.The authors were supported by Office of Naval Research (Grant No. N00014-1442-16-1-2010). constraints are encoded in restriction maps. The cohomology of a sheaf collates global information, such asglobal sections and obstructions to such.Certain sheaves prominent in applications take values in sets. For example, Reeb graphs can be viewedin terms of global sections of (co)sheaves valued in finite sets [ , ]. Other examples arising from problemsin quantum computation can be found in the works of Abramsky et al. [ ]. Applications closer to engineer-ing are present in the pioneering work of Goguen [ ] and in more recent work of others [ , ]. The lackof a full sheaf cohomology theory in these settings limits applicability.Recently, there has been substantial activity in cellular sheaves (and dual cosheaves), prompted by thethesis of Curry [ ]. The theory of cellular sheaves valued in vector spaces is especially well-developed, andtheir cohomology is not difficult to define or compute [ , ]. Cellular sheaves of vector spaces and theircohomology have been used in network flow and coding problems [ ], persistent homology computation[ ], signal processing [ ], distributed optimization [ ], opinion dynamics [ ], and more.Our motivations for working with sheaves of lattices stems from both their generality (ranging fromlattices of subgroups to Boolean algebras) and broad applicability in logic, topology, and discrete mathemat-ics. The reader need look no further than computational topology for recent work harnessing lattice theory(e.g. in computational Conley theory [ ], classifying embeddings factoring through a Morse function [ ],and computing interleavings of generalized persistence modules [ ]). The desire for a Hodge theory comesfrom more than mere computation of sheaf cohomology. For a sheaf of vector spaces, the Hodge Laplacianhas numerous applications, as detailed in the thesis of Hansen [ ]. As a generalization of the graph Lapla-cian, the Hodge Laplacian for sheaves of vector spaces has a rich spectral theory [ ] and leads to notionsof harmonic extension that are useful in several contexts [ , ].Among the many contingent avenues for applications, one is of special note. Graph signal processing —the extension of signal processing methods from signals over the reals to signals over graphs — has of latebeen a vibrant hive of activity, all based on the graph Laplacian, and all amenable to the Hodge-theoreticapproach to sheaves over networks. Lifting graph signal processing from real-valued to lattice-valueddata is an intriguing concept, with only the first steps being imagined by P ¨uschel et al for functionals onsemi-lattices [ ] and powersets [ ]. This paper provides the technical background for establishing graphsignal processing valued in lattices. Other potentialities, especially those concerning deep learning andconvolutional neural nets (at this moment valued almost exclusively in vectorized data, with expception[ ]) wait in the wings.
2. Background
The following is terse but sufficient for the remainder of the paper. The reader may find more detailedreferences for lattice theory [ , , ], cellular sheaves [ ], and their Hodge theory [ ]. There is somevariation in terminology among references in lattice theory. Caveat lector. A preordered set is a set P with a binary relation, (cid:22) , satisfying reflexivity , x (cid:22) x , and transitivity : x (cid:22) y and y (cid:22) z implies x (cid:22) z . A preordered set is a partial ordered set, or poset , if (cid:22) satisfies anti-symmetry : x (cid:22) y and y (cid:22) x implies x = y . Denote a strict partial order, x ≺ y , if x (cid:22) y , but y (cid:14) x .Given two elements x and y of a poset P, define the meet , x ∧ y , and join , x ∨ y , to be the greatest lowerbound and least upper bound respectively. That is, x ∨ y = min { z : z (cid:23) x , z (cid:23) y } : x ∧ y = max { z : z (cid:22) x , z (cid:22) y } .Likewise, for any subset S ⊆ P, we may define V S and W S whenever they exist.A lattice is a poset X closed under all finite (possibly empty) meets and joins. A lattice is complete if all arbitrary meets and joins exist (finite lattices thus being complete). By this definition, 0 = ∨ ∅ and1 = ∧ ∅ exist, so that all lattices have maximal (1) and minimal (0) elements. Such lattices are sometimescalled bounded lattices in the literature: our convention is such that all lattices in this paper are bounded.An element x ∈ X is join irreducible if x = x = y ∨ z implies x = y or x = z . An element x ∈ X is meet irreducible if x = x = y ∧ z implies x = y or x = z . A lattice can be defined eitheralgebraicly or by its partial order. From the binary operations join and meet, we can recover the partialorder by x (cid:22) y ⇔ x ∨ y = y ⇔ x (cid:22) y ⇔ x ∧ y = x . ELLULAR SHEAVES OF LATTICES AND THE TARSKI LAPLACIAN 3
For x ∈ P, a poset, define the principal downset of x, ↓ x = { y ∈ P : y (cid:22) x } , and the principal upset of x , ↑ x = { y ∈ L : y (cid:23) x } . A subset D ⊆ P is a downset if for all x ∈ D , if y (cid:22) x , then y ∈ D . A subset U ⊆ P is an upset if for all x ∈ U , if y (cid:23) x , then y ∈ U .An interval in a poset is a set, [ x , y ] = ↓ y ∩ ↑ x . A sublattice S ⊆ X is a subset closed under meets andjoins, including 0 and 1. It is more common to use a weaker notion of a sublattice. Define a quasi-sublatticeQ ⊆ X to be a subset closed under meets and joins, not necessarily containing 0 or 1. The product of afamily of lattices { X α } α ∈ J is the cartesian product ∏ α ∈ J X α with meets and joins defined coordinate-wise.We denote an element of the product x ∈ ∏ α ∈ J X α and and the bottom and top elements of the productlattice.In a poset P, we say y covers x , denoted x ⊏ y , if x ≺ y and there does not exist z ∈ P such that x ≺ z ≺ y . A lattice (or poset) X is graded if there exist a ranking r : X → N such that r ( x ) < r ( y ) whenever x ≺ y and r ( y ) = r ( x ) + x ⊏ y . For example, given a vector space V , let Gr ( V ) be the(Grassmanian) poset of subspaces of V with the order U (cid:22) U ′ if U is a subspace of U ′ . One may check thatthis is a lattice with U ∨ W = U + W the subspace sum, and U ∧ W = U ∩ W the intersection. For V finitedimensional, Gr ( V ) is graded via dimension: r ( U ) = dim ( U ) . For another example, let 2 S be the powersetlattice of a set, S , ordered by inclusion. For S finite, this is graded by cardinality: r ( U ⊆ S ) = U .A subset I ⊆ P is a chain if I is totally-ordered. A finite chain I = { x ≺ x ≺ · · · ≺ x ℓ } is said to have length ℓ . A poset P satisfies the descending chain condition (DCC) if no infinite strictly descending chainexists.L EMMA Any graded poset satisfies the descending chain condition. P ROOF . The grading of any strictly descending chain is a strictly decreasing sequence of natural num-bers and thus finite. (cid:3)
The height of a poset, denoted h ( P ) , is the length of the maximal chain if it exists; otherwise, h ( P ) = ∞ .The distance d P ( x , y ) between x , y ∈ P is defined as the interval height h ([ x , y ]) . Height is additive under(finite) products.L EMMA For { P i } Ni = be a finite family of posets,h N ∏ i = P i ! = N ∑ i = h ( P i ) (1)P ROOF . It suffices by finite induction to show h ( P × P ′ ) = h ( P ) + h ( P ′ ) . A maximal chain of P × P ′ projects to maximal chains I and I ′ in P and P ′ respectively and lies in the product I × I ′ . By projection andmaximality, its length is ℓ ( I ) + ℓ ( I ′ ) . (cid:3) A poset P is
Jordan-Dedekind if for every x ≺ y , then all maximal chains between x and y have thesame finite length. In particular, the lattices, Gr ( V ) and 2 S are Jordan-Dedkind for V finite dimensional and S finite. These lattices are graded by height by the following classical result ( [ , p. 16]):T HEOREM
A poset with and no infinite chains is Jordan-Dedekind if and only itis graded by height. We are especially concerned with maps of lattices f : X → Y .A (poset or) lattice map f is order-preserving if x (cid:22) x ′ implies f ( x ) (cid:22) f ( x ′ ) . A lattice map f is join-preserving if f ( x ∨ x ′ ) = f ( x ) ∨ f ( x ′ ) and f ( ) =
0. Similarly, f is meet-preserving if f ( x ∧ x ′ ) = f ( x ) ∧ f ( x ′ ) and f ( ) =
1. If f : X → Y is join preserving, then it is automatically order preserving since, if x (cid:22) x ′ ,then x ∨ x ′ = x ′ and f ( x ) ∨ f ( x ′ ) = f ( x ∨ x ′ ) = f ( x ′ ) which holds if and only if f ( x ) (cid:22) f ( x ′ ) . The same holds for meet-preserving maps.An order-preserving map Φ : X → X is expanding if Φ ( x ) (cid:23) x and contracting if Ψ ( x ) (cid:22) x . The fixed point set of Φ : X → X is the set Fix ( Φ ) = { x ∈ X : Φ ( x ) = x } . The fixed point set can be realizedas the intersection of the prefix points of Φ , Pre ( Φ ) = { x ∈ X : Φ ( x ) (cid:22) x } and the suffix points of Φ ,Post ( Φ ) = { x ∈ X : Φ ( x ) (cid:23) x } . ROBERT GHRIST AND HANS RIESS
The critical result about the fixed point set of a lattice endomorphism concerns its subobject structurewithin the lattice. The classical
Tarski Fixed Point Theorem is well-known [ ], but, for completeness andfor use of perspectives in the sequel, we give a brief proof.T HEOREM
The fixed point set of an order-preserving endomorphism of acomplete lattice is a complete lattice. P ROOF . Let Φ : X → X be order-preserving for X complete. For S ⊂ Pre ( Φ ) nonempty and x ∈ S , Φ (cid:16) ^ S (cid:17) (cid:22) Φ ( x ) (cid:22) x .As this holds for every x ∈ S , V S ∈ Pre ( Φ ) . Since _ S = ^ \ x ∈ S ↑ x ! ,it follows that Pre ( Φ ) is complete. By duality, Post ( Φ ) is complete also. The fixed point set of Φ can beexpressed as Fix ( Φ ) = Post (cid:16) Φ | Pre ( Φ ) (cid:17) .Hence, Fix ( Φ ) is complete. (cid:3) The computational complexity of finding a fixed point via querries to Φ is well-understood [ ]. There is a type of lattice map which interfaces well with categorical methodsand homological algebra. A (Galois) connection between a pair of lattices ( X , Y ) is an order-preservingpair, f = ( f q , f q ) , X −−−→←−−− f q f q Y such that f q ( x ) (cid:22) y ⇔ x (cid:22) f q ( y ) . (2)One calls f q the lower or left adjoint , and f q the upper or right adjoint . One interpretation of a connection is abest approximation to an order inverse, as the following critical proposition indicates.P ROPOSITION The following are equivalent: (1) f q ( x ) (cid:22) y ⇔ x (cid:22) f q ( y ) ; (2) f q f q (cid:23) id X and f q f q (cid:22) id Y . P ROOF . Suppose f q ( x ) (cid:22) y if and only if x (cid:22) f q ( y ) . In particular, by reflexivity, f q ( x ) (cid:22) f q ( x ) . Hence, f q f q ( x ) (cid:23) x for all x ∈ X . Similarly, by reflexivity, f q ( y ) (cid:22) f q ( y ) . Hence, f q f q ( y ) (cid:22) y .Conversely, suppose f q f q ( x ) (cid:23) x and f q f q ( y ) (cid:22) y . If f q ( x ) (cid:22) y , then, since f q is order-preserving, f q f q ( x ) (cid:22) f q ( y ) . But, f q f q ( x ) (cid:23) x which implies x (cid:22) f q ( y ) by transitivity. Similarly, if x (cid:22) f q ( y ) , then,since f q is order-preserving, f q ( x ) (cid:22) f q f q ( y ) which implies f q ( x ) (cid:22) y since f q f q ( y ) (cid:22) y . (cid:3) A category-theoretic interpretation is in order. Viewing a lattice X as a category with the underlyingposet structure defining morphisms, the meet and join operations are the product and coproduct respec-tively, 0 is initial, and 1 is terminal. Functors between lattices are simply order-preserving maps and aconnection between lattices is precisely an adjunction. This leads to the following well-known useful char-acterization of connections.P ROPOSITION If f = ( f q , f q ) , then f q preserves joins and f q preserves meets. Conversely, if f q : X → Y preserves joins, then there exist a map f q : Y → X that preserves meets such that ( f q , f q ) is a connection. Dually,if f q : Y → X preserves meets, then there exist a map f q : X → Y that preserves joins such that ( f q , f q ) is aconnection. Explicitly, these are given asf q ( y ) = _ f − q ( ↓ y ) and f q ( x ) = ^ f q − ( ↑ x ) . (3)P ROOF . The first statement follows from the (co)limit-preserving properties of adjuctions. The secondstatement is a consequence of the Adjoint Functor Theorem. For completeness, we give a constructiveproof in one direction (the other following from duality). Given f q , it suffices to show by Proposition 5 that f q f q ( x ) (cid:23) x and f q f q ( y ) (cid:22) y . We have f q f q ( x ) = _ f − q ( ↓ f q ( x )) . ELLULAR SHEAVES OF LATTICES AND THE TARSKI LAPLACIAN 5
But f − q f q ( x ) = { x } ⊆ f − q ( ↓ f q ( x )) implies f q f q ( x ) = _ f − q ( ↓ f q ( x )) (cid:23) _ { x } = x .For the other inequality, we have f q f q ( y ) = f q (cid:16) _ f − q ( ↓ y ) (cid:17) = _ f q f − q ( ↓ y ) since, by the first part, f q preserves joins. Notice, f q f − q ( ↓ y ) ⊆ ↓ y . Hence, f q f q ( y ) = _ f q f − q ( ↓ y ) (cid:22) _ ↓ y = y . (cid:3) Equation 3 is important for computational purposes as it gives an explicit formula for computing theupper adjoint of a join-preserving map. The complexity of computing adjoints is a question of interest.
Grandis [ ] gives several definitions of categories of lattices that we willfind useful.D EFINITION
1. Let
Ltc be the category of lattices and connections . The objects, ob ( Ltc ) , are lattices.Morphisms are connections, X −−−→←−−− f q f q Y denoted as a pair, f = ( f q , f q ) . The identity morphism is the connection, X −−→←−− idid X . Composition of arrows X −−−→←−−− f q f q Y −−−→←−−− g q g q Z is given by g ◦ f = ( g q ◦ f q , f q ◦ g q ) .Often, we desire to work with maps that are not bi-directional in order to form limits and colimits.Consider the category, Sup , of lattices and join-preserving maps . Likewise, consider the category,
Inf , of lattices and meet-preserving maps . We can view these categories as the images of forgetful functors, U : Ltc → Sup : V : Ltc op → Inf (4)given by the identity on objects and U : ( f q , f q ) f q and V : ( f q , f q ) f q on morphisms. For completelattices, there are corresponding full subcategories, cLtc , cSup , and cInf .There are two other important subcategories of Ltc : the distributive and the modular lattices,
Dlc ⊂ Mlc ⊂ Ltc .A lattice, X , is modular if for x , y ∈ X , x (cid:22) y ⇒ x ∨ ( z ∧ y ) = ( x ∨ z ) ∧ y for all z ∈ X , whereas it is distributive if for x , y , z ∈ X , x ∧ ( y ∨ z ) = ( x ∧ y ) ∨ ( x ∧ z ) or dually, x ∨ ( y ∧ z ) = ( x ∨ y ) ∧ ( x ∨ z ) .A distributive lattice satisfies the modular identity. A connection f = ( f q , f q ) is left exact if f q f q ( x ) = x ∨ f q right exact if f q f q ( y ) = y ∧ f q
1. An exact connection is one which is both left and right exact.The category
Mlc consists of modular lattices with exact connections, while
Dlc is given by distributivelattices and exact connections.
ROBERT GHRIST AND HANS RIESS
Our goal is to set up and work with cellular sheaves of lattices. A sheafis a type of data structure, built for the aggregation of local data and constraints into global solutions. Thesubject of sheaf theory is rich and technically intricate [ – ], but in recent years, a discrete version adaptedto posets from cell complexes has been shown to be useful in a number of applications [ ]. We thereforepresent a simple overview of the cellular theory.It is assumed that the reader is familiar with the definition of a cell complex : these are slightly moregeneral than simplicial complexes, and not quite as general as CW-complexes [ ]. A cell complex X isfiltered by its k -skeleta X ( k ) , for k ∈ N , where X ( ) is the vertex set. We write X k for the set of all k -cells in X,where we identify each cell σ with its image in X ( k ) .Let Fc ( X ) denote the face poset of a cell complex X given by the transitive-reflexive closure of therelation: σ P τ if σ is a face of τ . Every cell σ of X has:(1) boundary , ∂σ = { ρ : ρ P σ } ; and(2) coboundary , δσ = { τ : τ Q σ } .It is helpful to regard the face poset as a poset category for what follows.Cellular sheaves attach data to cells, glued together according to the face poset. A cellular sheaf takingvalues in a complete category D is simply a functor F : Fc ( X ) → D . Explicitly, F attaches to each cell σ ofX an object, F σ , called the stalk over σ . For pairs σ P τ , F prescribes restriction maps , F σ P τ : F σ → F τ , sothat for ρ P σ P τ ,(1) F σ P σ = id F σ (2) F σ P τ ◦ F ρ P σ = F ρ P τ The reader familiar with sheaves over topological spaces should think of a cellular sheaf as a discrete ver-sion, using the nerve of a locally-finite collection of open sets as the cell complex.Sheaves describe consistenty or consensus relationships between data, programmed via the restrictionmaps: this perspective has generated applications in flow networks [ ], sensing [ ], opinion networks[ ], and distributed optimization [ ]. The category of cellular sheaves over a cell complex X valued in D , denoted Sh X ( D ) , has as objects sheaves, F , and as morphisms, natural transformations η : F → G in D Fc ( X ) . Pullbacks, direct images, (oftentimes) tensor products, and other operations can be defined [ , ].In a given sheaf, the transition from local restrictions to global satisfaction is coordinated via the globalsection functor. The (global) sections of F , denoted Γ ( X; F ) , is the limit Γ ( X; F ) = lim ( F : Fc ( X ) → D ) .There is an explicit description in terms of assignments of data to 0-cells that agree over 1-cells: Γ ( X; F ) = ( x ∈ ∏ v ∈ X F v : ∀ v P e Q w , F v P e x v = F w P e x w ) . (5)For cellular sheaves taking values in an abelian category, sheaf cohomology is straightforward. Oneforms the cochain complex ( C • , δ • ) , where C k ( X; F ) = ∏ dim σ = k F σ ,are the k -cochains and the sheaf coboundary map δ : C k ( X; F ) → C k + ( X; F ) is given by ( δ x ) τ = ∑ σ P τ [ σ : τ ] F σ P τ x σ , (6)where [ σ : τ ] = ± H k ( X; F ) = Ker δ /Im δ . In degree zero, this computes the global sections via a naturalisomorphism. Cellular sheaf cohomology has proven useful in a number of settings of late [ – , ], andit is our goal to extend such to the setting of sheaves of lattices. The Hodge Theorem for Riemannian manifolds is a well-known exampleof the topological content of the Laplacian on differential forms: the kernel of the Laplacian gives thede Rham cohomology, which is isomorphic to the singular compactly supported cohomology with realcoefficients [ ]. There are simple combinatorial versions of Hodge theory for cell complexes [ ], which ELLULAR SHEAVES OF LATTICES AND THE TARSKI LAPLACIAN 7 in its most stripped-down cartoon form implies the widely-known fact that the kernel of the combinatorialgraph Laplacian has dimension equal to the number of connected components of the graph [ ].There is a richer variant of combinatorial Hodge theory for cellular sheaves of inner-product spaces [ ].Given a cell complex X and a cellular sheaf F on X taking values in the category of inner product spacesand linear transformations, one has a well-defined sheaf cohomology as per the previous subsection. Thecoboundary map δ from (6) has an adjoint δ ∗ given by taking the linear adjoint of each restriction map F σ P τ . The resulting Hodge Laplacian of F is given by L = δ ∗ δ + δδ ∗ . (7)This specializes to the combinatorial Laplacian in the case of a constant sheaf over a cell complex.There is a very nice Hodge theory for sheaves of finite-dimensional real vector spaces, beginning withthe observation that the kernel of L is isomorphic to the sheaf cohomology. The Hodge Laplacian is sym-metric and positive semidefinite, endowing the cochain complex with a quadratic form h· , L ·i which, e.g.,gives a measure of how close a cochain in C is to being a global section. More is possible, including ageneralization of spectral graph theory to the setting of sheaves [ ] via the spectral data of L , as well as theapplication of diffusion dynamics via L to compute sheaf cohomology [ ]. It is the latter that we wish togeneralize to sheaves of lattices.
3. The Tarski Laplacian3.1. Defintion and Properties.
Let F : Fc ( X ) → Ltc be a lattice-valued sheaf on a cell complex, X. Therestriction maps for cells σ P τ are therefore connections of the (notationally awkward) form F σ P τ = ( F q σ P τ , F q σ P τ ) .The 0-cochains C ( X; F ) are choices of data on vertex stalks. There is a sensible definition of a Laplacianwhich mimics the Hodge Laplacian for sheaves of vector spaces. Given the role that the Tarski Fixed PointTheorem plays in what follows, it seems fitting to call this novel Laplacian by his name.D EFINITION
2. The
Tarski Laplacian for F : Fc ( X ) → Ltc is the lattice map L : C ( X; F ) → C ( X; F ) given by ( L x ) v = ^ e ∈ δ v F q v P e ^ w ∈ ∂ e F q w P e x w ! (8)This Lapacian, like the Hodge Laplacian of a cellular sheaf, defines a diffusion process in which infor-mation propagates via sheaf restriction maps.L EMMA The Tarski Laplacian decomposes into two parts, ( L x ) v = ^ e : v ↔ w F q v P e F q v P e x v !| {z } expanding ∧ ^ e : v ↔ w F q v P e F q w P e x w !| {z } mixing (9)P ROOF . The map F q v P e F q v P e (cid:23) id F v is expanding while F q v P e preserves meets. (cid:3) Information is propagated across the 1-skeleton of X as a combination of mixing with neighboring statesand expanding the local state, taking the meet of all operations. Since our lattices do not have weights,mixing and expansion are given equal priority.L
EMMA The Tarski Laplacian L is order-preserving on the product poset C ( X; F ) . P ROOF . Suppose x (cid:22) y in C ( X; F ) . Then, F q w P e x w (cid:22) F q w P e y w since F q is join-preserving, hence, order preserving. This implies ^ w ∈ ∂ e F q w P e x w (cid:22) ^ w ∈ ∂ e F q w P e y w ROBERT GHRIST AND HANS RIESS which, since F q is meet-preserving and thus order-preserving, implies F q v P e ^ w ∈ ∂ e F q w P e x w ! (cid:22) F q v P e ^ w ∈ ∂ e F q w P e y w ! .This is turn implies ^ e ∈ δ v F q v P e ^ w ∈ ∂ e F q w P e x w ! (cid:22) ^ e ∈ δ v F q v P e ^ w ∈ ∂ e F q w P e y w ! .Hence, ( L x ) v (cid:22) ( L y ) v for every v . (cid:3) Although the Tarski Laplacian as defined has the feel of a diffusion-typeoperator (Lemma 7), confirmation of its fitness as a Laplacian would be welcome. We provide such in theform of a Hodge-type theorem in grading zero.Recall the setting of a cellular sheaf F of inner-product spaces on X with Hodge Laplacian L . The factthat in degree zero Ker L = H ( X; F ) = Γ ( X; F ) means that the heat equation on C , d x dt = − α L x ; α > t → + ∞ . Discretizing this in time yields adiscrete-time system x t + = ( id − η L ) x ; η >
0. (10)This system likewise converges asymptotically to Fix ( id − L ) : harmonic 0-cocycles, global sections.To derive a similar result on the dynamics of cochains on Ltc -valued cellular sheaves, it will be neces-sary to use a discrete-time diffusion, given the nature of lattices. In addition, it will be necessary to forgetsome of the structure of the full
Ltc -sheaf and reduce from F to F = U ◦ F using the forgetful functor of(4). This makes possible a fixed point description of Γ ( X; F ) using an analogue of (10).L EMMA For L the Tarski Laplacian, ( id ∧ L ) x = x is equivalent to L x (cid:23) x . P ROOF . By definition. (cid:3) T HEOREM
Let F be a Ltc -valued cellular sheaf on X and L its Tarski Laplacian. Then, Fix ( id ∧ L ) = Γ ( X; F ) (11)P ROOF . Via Lemma 9, suppose x ∈ Γ ( X; F ) . Then, every meet-end in V w ∈ ∂ e F q w P e x w is equal by thedescription of global sections in (5). By Proposition 5, F q v P e F q v P e x v (cid:23) x v for every cobounding edge e .Hence, ( L x ) v = ^ e ∈ δ v F q v P e F q v P e x v (cid:23) x v and we conclude that global sections x satisfy L x (cid:23) x .For the converse, suppose ( L x ) v (cid:23) x v for every v ∈ X . Then, for every vertex v and cobounding edge e ∈ δ v , F q v P e ^ w ∈ ∂ e F q w P e x w ! (cid:23) ^ e ∈ δ v F q v P e ^ w ∈ ∂ e F q w P e x w ! (cid:23) x v Again, by Proposition 5, ^ w ∈ ∂ e F q w P e x w (cid:23) F q v P e x v ,which in turn implies for each w ∈ ∂ e , F q w P e x w (cid:23) F q v P e x v (12)Reversing the roles of v and w gives, via the same argument, a reversed inequality, so that F q v P e x v = F q w P e x w ,proving that x ∈ Γ ( X; F ) . (cid:3) C OROLLARY
For F as in Theorem 10, Γ ( X; F ) = Post ( L ) . P ROOF . Lemma 9 combined with Theorem 10. (cid:3)
ELLULAR SHEAVES OF LATTICES AND THE TARSKI LAPLACIAN 9 C OROLLARY
For F as in Theorem 10, with every vertex stalk complete, the limit lim (cid:18) Fc ( X ) F −−→ cSup (cid:19) = Γ ( X; F ) (13) exists and is a (nonempty) complete quasi-sublattice of C ( X; F ) . P ROOF . By Lemma 8, L is order-preserving which implies id ∧ L is order-preserving. The Tarski FixedPoint Theorem (Theorem 4) and Theorem 10 complete the proof. (cid:3) Note that Corollary 12 is strictly stronger than the existence of lim ( F : Fc ( X ) → cSup ) . Since Γ ( X; F ) is a complete quasi-sublattice, arbitrary joins and meets of global sections exist, and, in particular, there areunique maximum and minimum global sections, even when C ( X; F ) is not finite.E XAMPLE
1. For X a 1-dimensional cell complex (a graph), the combinatorial graph Laplacian can beseen as the Hodge Laplacian of the constant sheaf of a fixed vector space. As an endmorphism on C ( X ) ,the graph Laplacian has kernel equal to the (locally) constant 0-cochains, and its dimension is the numberof connected components of the graph.What is the Tarski analogue? Consider the constant sheaf on X whose stalks are all a fixed lattice withall restriction maps the identity. The Tarski Laplacian performs a local meet with neighbors. In this case, too,the harmonic 0-cochains are precisely those which are (locally) constant. The Tarski Laplacian generalizesthe graph Laplacian. Theorem 10 gives an argument for the Tarski Laplacian as the “right” defini-tion for
Ltc -valued sheaves; however, it only applies in grading zero. This is due to the difficulty of defininga natural nonabelian sheaf cohomology for
Ltc -valued sheaves (see § F a Ltc -valued cellular sheaf on X.D
EFINITION
3. The
Tarski Laplacian in degree k is the order-preserving map, L k : C k ( X; F ) −→ C k ( X; F ) acting on a k -cochain x and k -cell σ via ( L k x ) σ = ^ τ ∈ δσ F q σ P τ ^ σ ′ ∈ ∂τ F q σ ′ P τ x σ ′ ! . (14)D EFINITION
4. The
Tarski cohomology , TH • ( X; F ) , of a cellular sheaf F valued in Ltc is TH k ( X; F ) = Fix ( id ∧ L k ) = Post ( L k ) . (15)L EMMA If F is valued in complete lattices and connections, cLtc , then TH k ( X; F ) is a (non-empty) completequasi-sublattice of C k ( X; F ) . P ROOF . This follows immediately from the Tarski Fixed Point Theorem. (cid:3)
Fixed point theorems come in implicit [Brouwer, Lefschetz] and explicit [Banach]forms. Theorem 10 gives a fixed-point description of global sections which, along with higher Tarski co-homology, can be made constructive via diffusion dynamics – using the Laplacian to define a discrete-timeheat equation on cochains.D
EFINITION
5. Define the harmonic flow Φ : N × C • ( X; F ) → C • ( X; F ) as Φ t = Φ ( t , · ) = ( id ∧ L ) t .We say that cochain x converges with respect to harmonic flow, writing x → x ∗ , if there exist T ≥ Φ T ( x ) = Φ T + t ( x ) for all t ∈ N . Then, x → x ∗ if and only if x ∗ ∈ Fix ( id ∧ L ) = Post ( L ) = TH • ( X; F ) .T HEOREM
Let F be a Ltc -valued sheaf on a cell complex X such that (1) the number of k-cells is finite,and (2) the stalks over the k-cells satisfy the descending chain condition (DCC). Then, for some finite t > , Φ t is aprojection map from C • to TH • . P ROOF . Since each time-step of Φ involves a meet with id, an orbit of Φ is either descending or even-tually fixed. The hypotheses ensure finite termination of all initial conditions. (cid:3) This implies that finite or ranked stalks suffice to guarantee global finite-time convergence of the har-monic flow. Optimal bounds on the number of iterations is an interesting question; naive bounds given byheights of stalks are an exercise left to the reader.
4. Towards Hodge Cohomology
Given a cellular sheaf of lattices F over X, it would be satisfying to have a cochain complex ( C • ( X ; F ) , δ ) with sheaf cohomology from which a classical Laplacian L = δ ∗ δ + δδ ∗ could be defined and showed iso-morphic to the Tarski cohomology of § ]. Sheaves taking values in
Ltc have enough structure to do coho-mology. In
Ltc , the zero morphism , ( q , 0 q ) : X → Y , is the connection where 0 q : X q : Y
1. Amorphism f : X → Y has kernel and cokernel given by equalizer and coequalizer respectively:Ker f X YX Y
Cok f ker f f f f In Ltc , one checks that these satisfyKer f = ↓ f q = { x ∈ X : f q ( x ) = } ,Cok f = ↑ f q = { y ∈ Y : f q ( y ) = } .For a morphism f , define its normal image by the connection, ker ( cok f ) , and the quasi-sublattice,Nim f = ↓ f q Ltc is the cone
X X ∏ Y Y , p q (16)where X ∏ Y is the cartesian-product lattice, and where p q ( x , y ) = x , p q ( x ) = ( x , 1 ) , and q q ( x , y ) = y , q q ( y ) = ( y ) . The coproduct is the cocone X X ∐ Y Y , i j (17)where X ∐ Y is again the cartesian-product lattice, and where i q ( x ) = ( x , 0 ) , i q ( x , y ) = x , and j q ( y ) = ( y ) , j q ( x , y ) = y .Recall that a semi-additive category is a pointed category with a biproduct , denoted × : a special productthat is compatible with the coproduct and coincides with the product on objects. The category Ltc is semi-additive; compatibility is satisfied, as p ◦ i = id and q ◦ j = id, as well as p ◦ j = q ◦ i =
0. The objects X ∏ Y and X ∐ Y coincide by definition.L EMMA
Semi-additive categories are enriched in abelian monoids. P ROOF . The diagonal ( ∆ ) and codiagonal ( ∇ ) morphisms come out of the product and coproduct re-spectively. For f , g ∈ Hom ( X , Y ) , f + g is the composition in the diagram X X × X Y × Y Y ∆ f + gf × g ∇ (18)We leave the details of showing f + g = g + f and f + = f as an exercise to the reader. (cid:3) C OROLLARY In Ltc , ∆ is the connection, ∆ q ( x ) = ( x , x ) , ∆ q ( x , x ′ ) = x ∧ x ′ , and ∇ is the connection, ∇ q ( y , y ′ ) = y ∨ y ′ , ∇ q ( y ) = ( y , y ) . Furthermore, f + g is the following connection: ( f + g ) q ( x ) = f q ( x ) ∨ g q ( x ) , ( f + g ) q ( y ) = f q ( y ) ∧ g q ( y ) . ELLULAR SHEAVES OF LATTICES AND THE TARSKI LAPLACIAN 11
We denote sums of connections by f + g and ∑ α f α as appropriate. With this in place, one can examine cochain complexes ( C • , δ ) valued in Ltc and define a cohomology. This is implicit in the work of Grandis [ ]; we fill in certain details for clarity.As with ordinary cohomology, define the cocycles and coboundaries of a cochain complex valued in Ltc by Z • = Ker δ and B • = Nim δ respectively. (For simplicity, we omit the grading superscript on thecoboundary operator; in the remainder of this section, the reader should be careful when specifying to aparticular grading).L EMMA
There is an order inclusion of boundaries into cycles. P ROOF . Suppose x is a boundary, so that x (cid:22) δ q δ q x (cid:22) δ q δ q = δ ◦ δ = (cid:3) D EFINITION
6. The
Grandis cohomology of ( C • , δ ) ∈ Ch + ( Ltc ) is GH • ( C • ) = Cok ( B • ֒ → Z • ) . (19)Grandis cohomology is best computed as an interval.P ROPOSITION GH • is isomorphic to [ δ q δ q ] .P ROOF . Let x ∈ Cok ( B • ֒ → Z • ) so that x (cid:23) δ q
1. Simultaneously, x (cid:22) δ q x ∈ Ker δ k . (cid:3) The problem with defining a sheaf cohomologyfor sheaves valued in
Ltc lies in the definition of the coboundary map: for ordinary sheaf cohomology ofsheaves valued in
Vect , the abelian structure is vital to defining δ . However, if one begins with a sheafvalued in Vect , then it is possible to pass to
Ltc via a Grassmannian — converting all vector space data tothe lattice of subspaces. This is an interesting class of sheaves and, though not universal, does provide anarena in which to compare different sheaf cohomologies.Denote by Gr the transfer functor , Gr ( · ) : Vec → Mlc , which converts vector spaces and linear trans-formations to (modular) lattices of subspaces and exact connections. By abuse of notation, this extends tochain complexes in the appropriate categories as well. This transfer functor respects cohomology.T
HEOREM
For C • a cochain complex valued in Vect ,GH • ( Gr ( C • )) ∼ = Gr ( H • ( C • )) . (20)P ROOF . That Gr ( · ) is a functor is left as an exercise. It is clear that a connection induced by the cobound-ary maps in C • is (1) exact and (2) a coboundary map in Gr ( C • ) . By Proposition 18, GH k ( Gr ( C • )) ∼ = [ Gr ( δ q ) , Gr ( δ q )]= [ Im δ , Ker δ ] .Then, by the Fourth Isomorphism Theorem [ ][p. 394], [ Im δ , Ker δ ] ∼ = Gr ( Ker δ /Im δ ) . (cid:3) In the case of a sheaf F of vector spaces over X, we can pass to F = Gr ( F ) , the induced sheaf takingvalues in Mlc via subspaces.C
OROLLARY
For F = Gr ( F ) a sheaf of lattices induced by a sheaf of vector spaces, GH • ( X; F ) ∼ = Gr ( H • ( X; F )) . For such sheaves, the relationship between the Tarski-based and Grandis-based cohomologies is asfollows:P
ROPOSITION
For sheaves of the form Gr ( F ) , the zeroth Grandis cohomology is a quasi-sublattice of thezeroth Tarski cohomology. P ROOF . If x is a subspace in GH ( X; F ) , then by Corollary 20, x is a subspace of Γ ( X; F ) = H ( X; F ) . Inturn, this means that the images of x under the restriction maps F v P e , F w P e will coincide for every v , w ∈ ∂ e so that x ∈ Γ ( X; F ) = TH ( X; F ) . (cid:3) The inclusion is strict. Consider a twisted sheaf F of 1-dimensional stalks as in Fig. 1. Then, the lattice-valued sheaf F = Gr ( F ) is constant: the stalks are the lattice L = {
0, 1 } , and the restriction maps are allthe identity. Hence, TH ( X; F ) ∼ = L . However (as the reader may calculate), H ( X; F ) ∼ =
0, as there are nononzero global sections. By Theorem 19, GH ( X; F ) ∼ = .F IGURE
1. A twisted sheaf (left) demonstrates strict inclusion TH * GH due to the ab-sence of a nonzero section (right). Sheaves of lattices do not generically factor through the transfer functor. Forsuch sheaves, can their cohomology be read off of a cochain complex via Grandis cohomology? This isunclear if not unlikely. We therefore pursue a philosophically disparate lattice sheaf cohomology mimickingthe Hodge theory for vector-valued sheaves. We define a pseudo-coboundary connection ˜ δ : C • ( X; F ) → C • + ( X; F ) given by a sum of projections composed with restriction maps: sums naturally arise from thesemi-additive structure on Ltc ; projections onto a stalk ( F σ ) over a k -cell ( σ ∈ X k ) are connections, π σ : C k ( X; F ) → F σ , π σ q ( x ) = x σ π q σ ( x σ ) = (
1, . . . , 1, x σ , 1, . . . , 1 ) D EFINITION
7. Let F be a sheaf valued in Ltc with cochains C • ( X; F ) per usual. The pseudo-cochaincomplex of F is a sequence of connections˜ δ : C • ( X; F ) → C • + ( X; F ) given by ( ˜ δ x ) τ = ∑ σ P τ F σ P τ ◦ π σ ! ( x ) . (21)The connection ˜ δ is not a true coboundary as ˜ δ ◦ ˜ δ = EFINITION
8. For ( C • , δ ) a (bounded) cochain complex in Ltc , the
Hodge Laplacians are the pair L + , L − : C • → C • of order-preserving degree-zero maps L + k = δ q δ q : L − k = δ q δ q . (22)As with the cellular Hodge Laplacian, one calls L + the up-Laplacian and L − the down-Laplacian .For example, the Hodge Laplacians of sheaves factoring through the transfer functor are readily calcu-lated:P ROPOSITION
For F = Gr ( F ) and Gr ( C • , δ ) its sheaf cochain complex, the Hodge Laplacians are computedvia L + x = x ∨ Ker δ : L − x = x ∧ Im δ . ELLULAR SHEAVES OF LATTICES AND THE TARSKI LAPLACIAN 13 P ROOF . As Gr ( δ ) is an exact connection, δ q δ q x = x ∨ δ q δ q δ q x = x ∧ δ q δ q δ q (cid:3) More generally, we may compute the Hodge Laplacians of the pseudo-cochain complex for an arbi-trary lattice-valued sheaf. By slight abuse of notation, substitute the pseudo-coboundary ˜ δ of (21) into thedefinition of the up- and down-Laplacians of (22).P ROPOSITION
Let F be a sheaf valued in Ltc and ( C • , ˜ δ ) its pseudo-cochain complex. Then, the HodgeLaplacians (L + = ˜ δ q ˜ δ q , L − = ˜ δ q ˜ δ q ) are computed as ( L + x ) σ = ^ τ ∈ δσ F q σ P τ _ σ ′ ∈ ∂τ F q σ ′ P τ x σ ′ ! , (23) ( L − x ) σ = _ ρ ∈ ∂σ F q ρ P σ ^ σ ′ ∈ δρ F q ρ P σ ′ x σ ′ . (24)P ROOF . The definitions, Corollary 16, and a few minor details suffice. (cid:3)
A return to the fixed point perspectives of § Ltc -valued sheaf coho-mology.D
EFINITION
9. For the pseudo-cochain complex of F as above, the upper- and lower-Hodge cohomol-ogy lattices of F are + HH • ( X; F ) = Post ( L + ) (25) − HH • ( X; F ) = Pre ( L − ) (26)As a compression of cochains, this Hodge cohomology is less efficient than Tarski cohomology.P ROPOSITION
Tarski cohomology is a quasi-sublattice of upper-Hodge cohomology. P ROOF . Suppose x ∈ TH k ( X; F ) and σ P τ Q σ ′ is a pair of k -cell faces of a common ( k + ) -cell τ .Then, following the proof of Theorem 10, one has F q σ P τ x σ (cid:22) ^ σ ′ ∈ ∂τ F q σ ′ P τ x σ ′ (cid:22) F q σ ′ P τ x σ ′ By symmetry of σ P τ Q σ ′ , this implies F q σ P τ x σ = F q σ ′ P τ x σ ′ . Then, ^ τ ∈ δσ F q σ P τ _ σ ′ ∈ ∂τ F q σ ′ P τ x σ ′ ! = ^ τ ∈ δσ F q σ P τ F q σ P τ x σ (cid:23) x σ so that x ∈ + HH k ( X; F ) . (cid:3) For an explicit example where Tarski cohomology is strictly contained in upper-Hodge cohomology,consider the constant sheaf on X, a connected graph with three vertices ( u , v , w ) and two edges ( u ∼ v , v ∼ w ). The Tarski Laplacian is the endomorphism L x = ( x u ∧ x v , x u ∧ x v ∧ x w , x v ∧ x w ) ;the upper-Hodge laplacian is the endomorphism L + x = ( x u ∨ x v , ( x u ∨ x v ) ∧ ( x v ∨ x w ) , x v ∨ x w ) . TH is the lattice of constant sections. To see that TH ( + HH , consider cochain x = ( x , y , x ) such that y ≺ x . Then, L + x = ( x , x , x ) (cid:23) x . Thus, x ∈ Post ( L + ) = + HH , but is not a constant section.
5. Summary
We close with Table 1, summarizing our constructions for cellular sheaves of lattices. The Grandiscohomology – which is defined only when there is a cochain complex, such as in the case of factoringthrough
Vec — is the “smallest” cohomology, followed by the Tarski and then the Hodge theories: GH • ⊂ TH • ⊂ + HH • . (27)symb. target coboundary, ( δ k x ) τ cohomology Laplacian, ( L ± k x ) σ Cellular H k Hilb ∑ σ P τ [ σ : τ ] F σ P τ x σ Ker δ k /Im δ k − (cid:0) δ ∗ k δ k + δ k − δ ∗ k − (cid:1) σ Tarski TH k Ltc
Post ( L k x ) V τ ∈ δσ F q σ P τ (cid:0) V σ ′ ∈ ∂τ F q σ ′ P τ x σ ′ (cid:1) Grandis GH k Mlc * Gr ( δ ) Gr (cid:16) H k ( X; F ) (cid:17) x σ ∨ ( Ker δ k ) σ x σ ∧ ( Im δ k − ) σ Hodge + HH k Ltc (cid:0) ∑ σ P τ F σ P τ ◦ π σ (cid:1) ( x ) Post ( L + k ) V τ ∈ δσ F q σ P τ (cid:0) W σ ′ ∈ ∂τ F q σ ′ P τ x σ ′ (cid:1) − HH k Pre ( L − k ) W ρ ∈ ∂σ F q ρ P σ (cid:16) V σ ′ ∈ δρ F q ρ P σ ′ x σ ′ (cid:17) T ABLE
1. Cohomologies and their Laplacians for cellular sheaves of lattices. *Grandis cohomology is defined for sheaves valued in
Mlc whenever
Mlc factors through
Vec . Acknowledgements
S. Krishnan initially suggested to the authors a fixed-point type theorem for sheaf cohomology. Con-versations with G. Henselman-Petrusek, J. Hansen, K. Spendlove, and J.-P. Vigneaux were helpful. Thiswork was funded by the Office of the Assistant Secretary of Defense Research & Engineering through aVannevar Bush Faculty Fellowship, ONR N00014-16-1-2010.
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