Cosheaf Representations of Relations and Dowker Complexes
CCOSHEAF REPRESENTATIONS OF RELATIONS AND DOWKERCOMPLEXES
MICHAEL ROBINSON
Abstract.
The Dowker complex is an abstract simplicial complex that is con-structed from a binary relation in a straightforward way. Although there aretwo ways to perform this construction – vertices for the complex are eitherthe rows or the columns of the matrix representing the relation – the twoconstructions are homotopy equivalent. This article shows that the construc-tion of a Dowker complex from a relation is a non-faithful covariant functor.Furthermore, we show that this functor can be made faithful by enrichingthe construction into a cosheaf on the Dowker complex. The cosheaf can besummarized by an integer weight function on the Dowker complex that is acomplete isomorphism invariant for the relation. The cosheaf representation ofa relation actually embodies both Dowker complexes, and we construct a du-ality functor that exchanges the two complexes. Finally, we explore a differentcosheaf that detects the failure of the Dowker complex itself to be a faithfulfunctor.
Contents
1. Introduction 12. Recovery of a relation from a weight function on the Dowker complex 33. Functoriality of the Dowker complex 74. Functoriality of (co)sheaves on Dowker complexes 115. Duality of cosheaf representations of relations 216. Redundancy of relations 30Acknowledgments 32References 321.
Introduction
This article studies the structure of an abstract simplicial complex that is builtaccording to a binary relation between two sets, as originally described by Dowker[10].
Dowker complexes , as these simplicial complexes are now known, are simpleto both construct and apply, finding use in many areas of mathematics and datascience [11]. Dowker’s classic result is that there are two ways to build such anabstract simplicial complex, and that both of these complexes have the same ho-mology. This fact is a kind of duality , because it arises from the transpose of theunderlying relation’s defining matrix. It was later shown by Bj¨orner [4] that thetwo dual Dowker complexes have homotopy equivalent geometric realizations.This article explains how the Dowker complex can be augmented with an integerweight function, and develops this idea into several functorial representations of a r X i v : . [ m a t h . A T ] M a y MICHAEL ROBINSON the underlying relation. Although the integer weight is not functorial, we showhow it is the decategorification of a functorial, faithful cosheaf representation, andexplore some of the implications of that fact. In particular, Dowker’s famous dualityresult arises as a functor that exchanges the base space and the space of globalcosections of the cosheaf. Considering only the Dowker complex without the weightfunction yields a non-faithful functor, since many different relations can have thesame Dowker complex. The article ends with the non-functorial construction of acosheaf that captures the amount of redundancy present in a relation – a measureof how un-faithful the Dowker complex is for that particular relation.Probably because of the topological nature of the duality result in [10], mostof the literature discussing Dowker complexes focuses on their topological prop-erties. For instance, [12] links the construction of the Dowker complex from arelation to the order complex of a partial order, and proves a number of homotopyequivalences. Because Dowker complexes respect filtrations [7], they seem ripe foruse in topological data analysis, which typically focuses on the persistent homol-ogy of a filtered topological space. This line of reasoning recently culminated in afunctoriality result [8, Thm. 3], which establishes that Dowker duality applies tothe geometric realizations of sub-relations. Specifically, consider a pair of nestedsubsets R ⊆ R ⊆ ( X × Y ) of the product of two sets X and Y . The Dowkercomplexes D ( X, Y, R ) and D ( X, Y, R ) for R and R and their duals D ( Y, X, R T )and D ( Y, X, R T ) are related through a commutative diagram | D ( X, Y, R ) | (cid:47) (cid:47) ∼ = (cid:15) (cid:15) | D ( X, Y, R ) | ∼ = (cid:15) (cid:15) | D ( Y, X, R T ) | (cid:47) (cid:47) | D ( Y, X, R T ) | of continuous maps on their respective geometric realizations, in which the verticalmaps are homotopy equivalences.The paper [8] appears to have set off a flurry of interest in Dowker complexes.For instance, [6] showed how to use Dowker complexes instead of ˇCech complexesfor studying finite metric spaces. Since topological data analysis often takes a finitemetric space as an input, constructing a Vietoris-Rips complex is frequently anintermediate step; [16] shows how Dowker complexes and Vietoris-Rips complexesare related. Finally, [15] extended Dowker duality to simplicial sets, pointing theway to much greater generality.The present paper is also inspired by the functoriality result [8, Thm. 3], but ina somewhat different way. Instead of focusing on sub-relations, we show that theDowker complex construction is a functor from a category whose objects are rela-tions and whose morphisms are relation-preserving transformations (Definition 6).Furthermore, we show that the isomorphism classes of this category are completelycharacterized by two different weight functions on the Dowker complex, and thatthese are derived from a faithful cosheaf representation of the category.Given a relation between two sets, the main results of this article are as follows:(1) The existence of two integer weighting functions, differential and total weights, on the Dowker complex for the relation that are complete iso-morphism invariants (Theorems 1 and 2),(2) The Dowker complex is a functor from an appropriately constructed cate-gory of relations (Theorem 3), OSHEAF REPRESENTATIONS OF RELATIONS AND DOWKER COMPLEXES 3 a bd c
Dowker complex total weight t di erential weight d
121 3 24 1 12311 61110 3 176 4 5
Figure 1.
The Dowker complex for Example 1 (left), its totalweight (center), and its differential weight (right).(3) The existence of faithful functors that render the relation into a cosheaf(Theorem 4 and Corollary 4) or sheaf (Theorem 5), whose (co)stalks deter-mine the total weight function,(4) The space of global cosections of the cosheaf is the dual Dowker complexfor the relation (Theorem 6), and(5) There is a duality functor that exchanges the cosheaf’s base space and spaceof global cosections (Theorem 7).2.
Recovery of a relation from a weight function on the Dowkercomplex
Definition 1. An abstract simplicial complex X on a set V X consists of a set X ofsubsets of V X such that if σ ∈ X and τ ⊆ σ , then τ ∈ X . Each σ ∈ X is called a simplex of X , and each element of V X is a vertex of X . Every subset τ of a simplex σ is called face of σ .It is usually tiresome to specify all of the simplices in a simplicial complex.Instead, it is much more convenient to supply a generating set S of subsets of thevertex set. The unique smallest simplicial complex containing the generating set iscalled the abstract simplicial complex generated by S .Let R ⊆ X × Y be a relation between finite sets X and Y , which can be repre-sented as a Boolean matrix ( r x,y ). Definition 2.
The
Dowker complex D ( X, Y, R ) is the abstract simplicial complexgiven by D ( X, Y, R ) = { [ x i , . . . , x i k ] : there exists a y ∈ Y such that ( x i j , y ) ∈ R for all j = 0 , . . . , k } . The total weight is a function t : D ( X, Y, R ) → N given by t ( σ ) = { y ∈ Y : ( x, y ) ∈ R for all x ∈ σ } . The differential weight [2] is a function d : D ( X, Y, R ) → N given by d ( σ ) = { y ∈ Y : (( x, y ) ∈ R if x ∈ σ ) and (( x, y ) / ∈ R if x / ∈ σ ) } . It is immediate by the definition that the Dowker complex is an abstract simpli-cial complex.
MICHAEL ROBINSON a bd c
Dowker complex total weight t di erential weight d
001 0 11 1 1013 242 1 111 1 2
Figure 2.
The Dowker complex for Example 2 (left), its totalweight (center), and its differential weight (right).
Example 1.
Consider the sets X = { a, b, c, d } , and Y = { , , . . . , } and therelation R given by the matrix r = whose rows correspond to elements of X and columns correspond to elements of Y .The Dowker complex for this relation is generated by the simplices [ a, c, d ], [ a, b ],and [ b, c ], a fact witnessed by the columns marked with bold type. The Dowkercomplex D ( X , Y , R ) and its weighting functions are shown in Figure 1. Noticein particular that the differential weighting function counts the number of columnsof r of each simplex. The total weighting accumulates all of the counts of columnsfor its faces as well. Example 2.
If we keep the same set X = X as in Example 1, but change the Y set, with a different relation R given by the matrix r = we obtain the same Dowker complex, D ( X , Y , R ) = D ( X , Y , R ). However, asFigure 2 shows, the weight functions are different. Proposition 1.
The sum of the differential weight d on the Dowker complex D ( X, Y, R ) is the number of elements of Y . (Do not forget to count the differential weight of the empty simplex!) Proof.
Observe that sets of the form { y ∈ Y : (( x, y ) ∈ R if x ∈ σ ) and (( x, y ) / ∈ R if x / ∈ σ ) } are disjoint for different σ in D ( X, Y, R ). The sum of the differential weight is there-fore the cardinality of (cid:88) σ ∈ D ( X,Y,R ) d ( σ ) = (cid:91) σ ∈ D ( X,Y,R ) { y ∈ Y : (( x, y ) ∈ R if x ∈ σ ) and (( x, y ) / ∈ R if x / ∈ σ ) } , which completes the argument. (cid:3) Proposition 2.
The total weight t is a filtration on the Dowker complex D ( X, Y, R ) . OSHEAF REPRESENTATIONS OF RELATIONS AND DOWKER COMPLEXES 5
Proof.
This follows from showing that t is order-reversing in the following way: if σ ⊆ τ , then t ( σ ) ≥ t ( τ ).Suppose σ ⊆ τ , and that y ∈ Y satisfies ( x, y ) ∈ R for all x ∈ τ . Since σ ⊆ τ ,then ( x, y ) ∈ R for all x ∈ σ also. Thus { y ∈ Y : ( x, y ) ∈ R for all x ∈ τ } ⊆ { y ∈ Y : ( x, y ) ∈ R for all x ∈ σ } { y ∈ Y : ( x, y ) ∈ R for all x ∈ τ } ≤ { y ∈ Y : ( x, y ) ∈ R for all x ∈ σ } t ( τ ) ≤ t ( σ ) . (cid:3) Theorem 1.
Given the Dowker complex D ( X, Y, R ) and differential weight d , onecan reconstruct R up to a bijection on Y .Proof. The differential weight d ( σ ) simply specifies the number of columns of thematrix r for R that can be realized as an indicator function for each σ ∈ D ( X, Y, R ).Thus, we can construct r up to a column permutation. (cid:3) Theorem 2.
Given the Dowker complex D ( X, Y, R ) and the total weight t , onecan reconstruct R up to a bijection on Y .Proof. We construct the relation matrix r of R iteratively. Let t = t .(1) Set r to the zero matrix with no columns and as many rows as vertices of D ( X, Y, R ). That is, each row of r corresponds to an element of X , so letus index rows of r by elements of X .(2) If t n ( σ ) = 0 for all simplices σ ∈ D ( X, Y, R ), declare r = r n and exit.(3) Select a simplex σ with t n ( σ ) (cid:54) = 0 such that either there is no simplex τ containing σ as a face, or if such a τ exists, then t n ( τ ) = 0.(4) Define r n +1 to be the horizontal concatenation of r n with t n ( σ ) columns,each an indicator function for σ . That is, each new column is a Booleanvector v given by v x = (cid:40) x / ∈ σ x ∈ σ. (5) Define a new function t n +1 : D ( X, Y, R ) → N by t n +1 ( γ ) = (cid:40) t n ( γ ) − t n ( σ ) if γ ⊆ σt n ( γ ) otherwise.(6) Increment n (7) Go to step (3).Since t n +1 < t n on at least one simplex, and the relation R is finite, the algorithmwill always terminate.Secondly, the update step for r n +1 by adding columns, establishes that r relatesthe elements of X contained in a given simplex by the appended σ columns.Thirdly, notice that the apparent ambiguity in step (3) about selecting a simplex σ merely results in a column permutation, since two maximal simplices do notinteract with the update to t n +1 in step (5), since another maximal simplex is nota face of σ . (cid:3) MICHAEL ROBINSON total weight t r = ( ) t r = ( ) t r = ( ) t r = ( ) t r = ( ) t r = ( ) t r = ( ) Figure 3.
Recovering the relation from the total weight functionas described in Example 3.
Example 3.
Starting with the relation from Example 2 and its total weight func-tion, the algorithm described in the proof of Theorem 2 produces the relationmatrix r = which differs from the original matrix r by a cyclic permutation of the last threecolumns. Figure 3 shows the progression of the steps of the algorithm. Example 4.
Not every nonnegative integer filtration of an abstract simplicial com-plex corresponds to the total weighting of a Dowker complex of a relation. Althoughthe algorithm in Theorem 2 may appear to run, it can produce negative values forthe intermediate t • weights, which cannot correspond to a number of columns in arelation! For instance, the constant function on the simplicial complex generated by[ a, b ] and [ b, c ], as shown in Figure 4 is a filtration. However, running the algorithmon this filtration produces a negative value at [ b ], so we conclude that no relationcan have this as a total weight function. OSHEAF REPRESENTATIONS OF RELATIONS AND DOWKER COMPLEXES 7 ( ) abc r = ( ) abc r = 110 ( ) abc -1 r = 101001 Figure 4.
Attempting to recover the relation from a filtrationthat is not a total weight function can result in negative values, asdescribed in Example 4.3.
Functoriality of the Dowker complex
The Dowker complex D ( X, Y, R ) is a functor between an appropriately con-structed category of relations and the category of abstract simplicial complexes.We prove this fact in Theorem 3 along with a few other observations.
Definition 3.
Consider an arbitrary set P and a partial order ≤ on P . A partialorder is a relation between elements in P such that the following hold:(1) (Reflexivity) x ≤ x for all x ∈ P ,(2) (Transitivity) if x ≤ y and y ≤ z , then x ≤ z , and(3) (Antisymmetry) if x ≤ y and y ≤ x , then x = y .The category of partial orders Pos has every partially ordered set ( P, ≤ ) as anobject. Each morphism g : ( P, ≤ P ) → ( Q, ≤ Q ) consists of an order preservingfunction g : P → Q such that if x and y are two elements of P satisfying x ≤ P y ,then g ( x ) ≤ Q g ( y ) in Q . Morphisms compose as functions on their respective sets.To see that Pos is a category, notice that the identity function is always orderpreserving and that the composition of two order preserving functions is anotherorder preserving function. Associativity follows from the associativity of functioncomposition.
Definition 4.
The face partial order for an abstract simplicial complex X has thesimplices of X as its elements, and σ ≤ τ whenever σ ⊆ τ . Example 5.
The face partial order for the Dowker complex shown in Figure 1 isgiven by its Hasse diagram [ a, c, d ][ a, b ] [ b, c ] [ a, c ] (cid:59) (cid:59) [ a, d ] (cid:79) (cid:79) [ c, d ] (cid:99) (cid:99) [ b ] (cid:79) (cid:79) (cid:60) (cid:60) [ a ] (cid:98) (cid:98) (cid:60) (cid:60) (cid:53) (cid:53) [ c ] (cid:98) (cid:98) (cid:79) (cid:79) (cid:53) (cid:53) [ d ] (cid:79) (cid:79) (cid:59) (cid:59) Definition 5.
Suppose X and Y are abstract simplicial complexes with vertex sets V X and V Y , respectively. A function f : V X → V Y on vertices is called a simplicial MICHAEL ROBINSON map f : X → Y if it transforms each simplex [ v , · · · , v k ] of X into a simplex[ f ( v ) , · · · , f ( v k )] of Y , after removing duplicate vertices. The category Asc hasabstract simplicial complexes as its objects and simplicial maps as its morphisms.
Lemma 1.
Let f : X → Y be a simplicial map. For every pair of simplices σ , τ of X satisfying σ ⊆ τ , their images in Y satisfy f ( σ ) ⊆ f ( τ ) .Proof. Since f is a simplicial map, then f ( σ ) is a simplex of Y and so is f ( τ ). If σ ⊆ τ , then every vertex v of σ is also a vertex of τ . By the definition of simplicialmaps, f ( v ) is a vertex of both f ( σ ) and f ( τ ). Conversely, every vertex of f ( σ ) isthe image of some vertex w of σ . (cid:3) Proposition 3.
The face partial order is a covariant functor
F ace : Asc → Pos .Proof.
The construction of the face partial order from a simplicial complex estab-lishes how the functor transforms objects. Let us denote the face partial orderfor X by F ace ( X ). Lemma 1 establishes that every simplicial map f : X → Y induces an order preserving function F ace ( f ) : F ace ( X ) → F ace ( Y ) on the facepartial orders for X and Y . To verify that the functor is covariant, suppose thatwe have another simplicial map g : Y → Z . The composition of these is a sim-plicial map g ◦ f : X → Z that induces an order preserving map F ace ( g ◦ f ) : F ace ( X ) → F ace ( Z ) on the face partial orders for X and Z . On the other hand, F ace ( g ) ◦ F ace ( f ) : F ace ( X ) → F ace ( Z ) is also an order preserving map. Anysimplex σ in X can be reinterpreted as an element of F ace ( X ), which means thatthe simplex ( g ◦ f )( σ ) of Z corresponds to the same element of F ace ( Z ) as does g ( f ( σ )), thought of as the image of an element of F ace ( Y ). (cid:3) Definition 6. (which has [1, Sec 3.3] or [14, pg. 54] as a special case, and ismanifestly the same as what appears in [6]) The category of relations
Rel hastriples (
X, Y, R ) for objects, in which X , Y are sets and R ⊆ X × Y is a relation.A morphism ( X, Y, R ) → ( X (cid:48) , Y (cid:48) , R (cid:48) ) in Rel is defined by a pair of functions f : X → X (cid:48) , g : Y → Y (cid:48) such that ( f ( x ) , g ( y )) ∈ R (cid:48) whenever ( x, y ) ∈ R .Composition of morphisms is simply the composition of the corresponding pairsof functions, which means that Rel satisfies the axioms for a category. It will beuseful to consider the full subcategory
Rel + of Rel in which each object (
X, Y, R )has the property that for each x ∈ X , there is a y ∈ Y such that ( x, y ) ∈ R , andconversely for each y ∈ Y , there is an x ∈ X such that ( x, y ) ∈ R . Example 6.
Consider the relation R between the sets X = { a, b, c, d, e } and Y = { , , , , } , given by the matrix r = . Suppose that X = { A, B, C } and Y = { , , , , } , that f : X → X is givenby f ( a ) = A, f ( b ) = B, f ( c ) = C, f ( d ) = C, f ( e ) = C, and that g : Y → Y is given by the identity function, namely g (1) = 1 , g (2) = 2 , g (3) = 3 , g (4) = 4 , g (5) = 5 . OSHEAF REPRESENTATIONS OF RELATIONS AND DOWKER COMPLEXES 9
Then ( f, g ) is a
Rel morphism ( X , Y , R ) → ( X , Y , R ) if R is given by thematrix r = . Additionally if g (cid:48) : Y → Y is given by g (cid:48) (1) = 1 , g (cid:48) (2) = 2 , g (cid:48) (3) = 3 , g (cid:48) (4) = 3 , g (cid:48) (5) = 3 , then ( f, g (cid:48) ) is a Rel morphism ( X , Y , R ) → ( X , Y , R ) if R is given by thematrix r = . However, ( f, g ) is not a
Rel morphism ( X , Y , R ) → ( X , Y , R ), since ( f ( e ) , g (5)) =( C,
5) is not in the relation R even though ( e,
5) is in R . Theorem 3.
The Dowker complex defined in Definition 2 is a covariant functor D : Rel → Asc .Proof.
Given the construction of the Dowker complex D ( X, Y, R ) from R ⊆ X × Y ,we must show that(1) Each morphism in Rel translates into a simplicial map, and(2) Composition of morphisms in
Rel translates into composition of simplicialmaps.To that end, suppose that ( X , Y , R ), ( X , Y , R ), and ( X , Y , R ) are three ob-jects in Rel with f : X → X , g : Y → Y defining a morphism ( X , Y , R ) → ( X , Y , R ), and with f : X → X , g : Y → Y defining a morphism ( X , Y , R ) → ( X , Y , R ). The first claim to be proven is that f is the vertex function for asimplicial map D ( X , Y , R ) → D ( X , Y , R ). Suppose that σ is a simplex of D ( X , Y , R ). Under the vertex map f , the set of vertices of σ get transformedinto the set f ( σ ) = { f ( x ) : x ∈ σ } . But the defining characteristic of σ is that there is a y ∈ Y such that ( x, y ) ∈ R for each x ∈ σ . Using the function g and the fact that the pair ( f , g ) is a Rel morphism, we have that ( f ( x ) , g ( y )) ∈ R for every x ∈ σ . This means thatthe set f ( σ ) is actually a simplex of D ( X , Y , R ). Since σ was arbitrary, thisestablishes that f is a simplicial map.Since the composition of the Rel morphisms ( f , g ) ◦ ( f , g ) is defined to be( f ◦ f , g ◦ g ), this means that D is a covariant functor, since this composition of Rel morphisms becomes a composition of simplicial maps. (cid:3)
Example 7.
Continuing Example 6, the Dowker complex D ( X , Y , R ) is shownat left in Figure 5. The Dowker complexes D ( X , Y , R ) and D ( X , Y , R ) areidentical, and are shown at right in Figure 5. The Rel morphism ( f, g ) fromExample 6 induces a simplicial map according Theorem 3. The vertex function forthis simplicial map is shown in Figure 5 as well. The simplicial map collapses thesimplex [ c, d, e ] to the vertex [ C ], while it collapses the simplex [ b, c, d ] to the edge[ B, C ]. ab cd e AB CfD ( X , Y , R ) D ( X , Y , R ) Figure 5.
The simplicial map induced on the Dowker complexesby the
Rel morphism ( f, g ), which is used in Examples 6, 7, 10,and 15.Observe that
Pos can be realized as a (non-full) subcategory of
Rel : each ob-ject in this subcategory is a partially ordered set ( X, ≤ X ) realized as ( X, X, ≤ X ),and each order preserving function f : ( X, ≤ X ) → ( Y, ≤ Y ) corresponds to a Rel morphism ( f, f ) : (
X, X, ≤ X ) → ( Y, Y, ≤ Y ) since the axioms coincide. Beyond thisrelationship between Pos and
Rel , there is a different, functorial relationship.
Proposition 4.
There is a covariant functor
P osRep : Rel → Pos , called the poset representation of a relation, that takes each ( X, Y, R ) to a collection P osRep ( X, Y, R ) of subsets of X , for which A ∈ P osRep ( X, Y, R ) if there is a y ∈ Y such that ( x, y ) ∈ R for every x ∈ A . The elements of P osRep ( X, Y, R ) are ordered by subsetinclusion.Proof. P osRep translates morphisms in
Rel into order preserving maps amongpartially ordered sets. Specifically, the morphism (
X, Y, R ) → ( X (cid:48) , Y (cid:48) , R (cid:48) ) imple-mented by f : X → X (cid:48) and g : Y → Y (cid:48) is transformed into the function that takes A ∈ P osRep ( X, Y, R ) to f ( A ). By definition there is a y ∈ Y such that ( x, y ) ∈ R for every x ∈ A . Therefore, ( f ( x ) , g ( y )) ∈ R (cid:48) by construction. Since each x (cid:48) ∈ f ( A )is given by x (cid:48) = f ( x ) for some x , this means that ( x (cid:48) , g ( y )) ∈ R (cid:48) for all x ∈ f ( A ).Thus, f ( A ) ∈ P osRep ( X (cid:48) , Y (cid:48) , R (cid:48) ). Furthermore, the order relation among subsetsis evidently preserved.The same argument from the proof of Theorem 3 can be used mutatis mutandis to show that P osRep is a covariant functor, namely that composition of morphismsis preserved in order. (cid:3)
Proposition 5.
The composition of the Dowker functor D : Rel → Asc with theface partial order functor
F ace : Asc → Pos yields the
P osRep functor.
In brief, the diagram
Rel D (cid:47) (cid:47) P osRep (cid:35) (cid:35)
Asc
F ace (cid:15) (cid:15)
Pos of functors commutes.
OSHEAF REPRESENTATIONS OF RELATIONS AND DOWKER COMPLEXES 11 [ b , c , d ] [ c , d , e ][ c , d ][ b , c ] [ b , d ] [ c , e ] [ d , e ][ a , b ] [ a , c ][ a ] [ b ] [ c ] [ d ] [ e ] [ A , B ] [ A , C ] [ B , C ][ A ] [ B ] [ C ] Figure 6.
The order preserving map induced by the
Rel mor-phism ( f, g ) in Example 6 via the
P osRep functor. See Example8
Proof.
To establish this result, we merely need to observe that the set of elementsof (
F ace ◦ D )( X, Y, R ) is the same set as
P osRep ( X, Y, R ), with the same orderrelation (subset inclusion). Under that identification, the morphisms are the same,too. (cid:3)
Example 8.
Continuing Examples 6 and 7, the order preserving map induced bythe
Rel morphism ( f, g ) is a bit tedious to construct from the data in Example6. It is much more convenient to work from the simplicial map shown in Figure 5.Simply note that the only two nontrivial actions to be captured are related to thecollapse of the simplices [ b, c, d ] and [ c, d, e ]. All of the faces of [ c, d, e ] are mappedto [ C ], while the remaining two edges of [ b, c, d ] (those that are not also faces of[ c, d, e ]) are mapped to [ B, C ].4.
Functoriality of (co)sheaves on Dowker complexes
The Dowker functor D : Rel → Asc is not faithful: non-isomorphic relations canhave the same Dowker complex. The weighting functions distinguish between
Rel isomorphism classes.
Rel morphisms sometimes induce transformations betweenweighting functions, for instance if the morphism transforms only the rows, or if themorphism permutes columns. However, this does not always occur. For instance,if the columns of one relation are included into another while simultaneously therows are combined, the resulting transformation on the weighting functions is notdescribed in a convenient way. We really need a richer category for weighted Dowkercomplexes; this is found in the categories of cosheaves or of sheaves. Specifically,each relation can be rendered as a cosheaf of sets, whose costalk cardinality isthe total weight function on the Dowker complex. Alternatively, a relation can betransformed into a sheaf of vector spaces, whose stalk dimensions specify a totalweight function on the Dowker complex.
For convenience, let us begin by defining Y σ = { y ∈ Y : ( x, y ) ∈ R for all x ∈ σ } for a simplex σ of D ( X, Y, R ). The total weight function is simply the cardinalityof this set: t ( σ ) = Y σ . Lemma 2. If σ ⊆ τ are two simplices of D ( X, Y, R ) , then Y τ ⊆ Y σ .Proof. Suppose y ∈ Y τ , so that ( x, y ) ∈ R for all x ∈ τ . Since σ ⊆ τ , it follows that( x, y ) ∈ R for all x ∈ σ . Thus y ∈ Y σ . (cid:3) Lemma 3.
For each simplex σ of D ( X , Y , R ) , and each Rel morphism ( f, g ) :( X , Y , R ) → ( X , Y , R ) , g (( Y ) σ ) ⊆ ( Y ) f ( σ ) . Proof.
Suppose z ∈ g (( Y ) σ ), which means that z = g ( y ) for some y ∈ Y thatsatisfies ( x, y ) ∈ R for all x ∈ σ . Since ( f, g ) is a Rel morphism, this means that( f ( x ) , g ( y )) = ( f ( x ) , z ) ∈ R for all x ∈ σ as well. Therefore, z ∈ ( Y ) f ( σ ) . (cid:3) Corollary 1.
As a result of Lemmas 2 and 3, the diagram ( Y ) τ g (cid:47) (cid:47) ⊆ (cid:15) (cid:15) ( Y ) f ( τ ) ⊆ (cid:15) (cid:15) ( Y ) σ g (cid:47) (cid:47) ( Y ) f ( σ ) commutes when ( f, g ) is a Rel morphism.
Although the total and differential weight functions are complete isomorphisminvariants for
Rel , they are not functorial. To remedy this deficiency, these weightscan be thought of as summaries of a somewhat more sophisticated object: a cosheaf or a sheaf . Definition 7. [3] A cosheaf of sets C on a partial order ( X, ≤ ) consists of thefollowing specification: • For each x ∈ X , a set C ( x ), called the costalk at x , and • For each x ≤ y ∈ X , a function ( C ( x ≤ y )) : C ( y ) → C ( x ), called the extension along x ≤ y , such that • Whenever x ≤ y ≤ z ∈ X , C ( x ≤ z ) = ( C ( x ≤ y )) ◦ ( C ( y ≤ z )).Briefly, a cosheaf is a contravariant functor to the category Set from the categorygenerated by ( X, ≤ ), whose objects are elements of X and whose morphisms x → y correspond to ordered pairs x ≤ y .Dually, a sheaf of sets S on a partial order ( X, ≤ ) consists of the followingspecification: • For each x ∈ X , a set S ( x ), called the stalk at x , and • For each x ≤ y ∈ X , a function ( S ( x ≤ y )) : S ( x ) → S ( y ), called the restriction along x ≤ y , such that • Whenever x ≤ y ≤ z ∈ X , S ( x ≤ z ) = ( S ( y ≤ z )) ◦ ( S ( x ≤ y )).In this way, a sheaf is covariant functor to Set from the category generated by thepartially ordered set ( X, ≤ ). OSHEAF REPRESENTATIONS OF RELATIONS AND DOWKER COMPLEXES 13
Given that every abstract simplicial complex X corresponds to a partially or-dered set ( X, ⊆ ) via the face partial order F ace : Asc → Pos , we will often use (co)sheaves on an abstract simplicial complex as a shorthand for (co)sheaves on theface partial order of an abstract simplicial complex.This definition of a (co)sheaf is traditionally that of a pre (co)sheaf on a topo-logical space. The connection is that a (co)sheaf of sets of a partially ordered setis a minimal specification for a (co)sheaf on the partial order with the Alexandrovtopology, via the process of (co)sheafification [9]. Definition 7 is unambiguous inthe context of this article, since we only consider (co)sheaves on partially orderedsets.
Definition 8.
We can use the information in a relation (
X, Y, R ) to define a cosheaf R on the face partial order of D ( X, Y, R ) by
Costalks:
Each costalk is given by R ( σ ) = Y σ , and Extensions: If σ ⊆ τ in D ( X, Y, R ), then the extension R ( σ ⊆ τ ) : R ( τ ) →R ( σ ) is the inclusion Y τ ⊆ Y σ .In a dual way, we can also define a sheaf R by: Stalks:
Each stalk is given by R ( σ ) = span Y σ , and Restrictions: If σ ⊆ τ in D ( X, Y, R ), then the restriction R ( σ ⊆ τ ) : R ( σ ) → R ( τ ) is defined to be the projection induced by the inclusion Y τ ⊆ Y σ .Notice that the basis for each stalk of R is the corresponding costalk of R . Corollary 2.
For the cosheaf R or sheaf R constructed from a relation R asabove, t ( σ ) = R ( σ ) = dim R ( σ ) , and d ( σ ) = R ( σ ) − (cid:91) σ (cid:36) τ R ( τ ) = dim R ( σ ) − dim span σ (cid:36) τ R ( τ ) . The interpretation is that the total weight t computes how many columns of r start at σ , while the differential weight d counts columns of r that are related tothe elements of σ and no others.The main use of (co)sheaf theory is to formalize the notion of local and globalconsistency over some space, by way of identifying the data that are consistent withrespect to the (co)sheaf. These data are captured within global (co)sections . Definition 9.
For a cosheaf C on a partially ordered set ( X, ≤ ), consider thedisjoint union of all costalks (cid:71) x ∈ X C ( x ) . Let ∼ be the equivalence relation on this disjoint union generated by c x ∼ c y whenever there exists an x ≤ y in X that satisfies(1) c x ∈ C ( x ) and(2) c y ∈ C ( y ), such that To keep the notation from becoming burdensome, we will abuse notation by regarding theabstract simplicial complex D ( X, Y, R ) as a partially ordered set ( D ( X, Y, R ) , ⊆ ) rather thancarrying around the F ace functor. (3) c x = ( C ( x ≤ y )) ( c y ).The set of global cosections C ( X ) of the cosheaf C is given by the set of equiva-lence classes C ( X ) = (cid:32) (cid:71) x ∈ X C ( x ) (cid:33) / ∼ . Each element of C ( X ) is called a (global) cosection of C .Dually, the set of global sections of a sheaf S a on partially ordered set ( X, ≤ )is denoted by S ( X ) and is given by the subset S ( X ) = (cid:40) s ∈ (cid:89) x ∈ X S ( x ) : s y = ( S ( x ≤ y )) ( s x ) (cid:41) . Each element of S ( X ) is called a (global) section of S . Example 9.
Recall the relation R between X = { a, b, c, d } and Y = { , , , , , } from Example 2, which was given by the matrix r = . Using the partial order constructed for this relation in Example 5, the cosheaf R for the relation has the diagram R ([ a,c,d ]])= { } (cid:38) (cid:38) (cid:15) (cid:15) (cid:120) (cid:120) R ([ a,b ])= { } (cid:15) (cid:15) (cid:37) (cid:37) R ([ b,c ])= { } (cid:121) (cid:121) (cid:37) (cid:37) R ([ a,c ])= { , } (cid:121) (cid:121) (cid:15) (cid:15) R ([ a,d ])= { } (cid:116) (cid:116) (cid:15) (cid:15) R ([ c,d ])= { } (cid:120) (cid:120) (cid:116) (cid:116) R ([ b ])= { , } R ([ a ])= { , , } R ([ c ])= { , , , } R ([ d ])= { , } where the numbers specify elements of Y (also column indices of r ). The set ofglobal cosections of this cosheaf is precisely R ( X ) = { , , , , , } , since the extension maps are all inclusions. The equivalence classes involved merelyidentify equal elements of Y (column indices) in the the disjoint union. Each globalcosection of R therefore corresponds to an element of Y (equivalently, a columnof r ). OSHEAF REPRESENTATIONS OF RELATIONS AND DOWKER COMPLEXES 15
The costalk cardinalities in the above diagram agree exactly with the total weight t shown in Figure 2. Furthermore, the nonzero differential weights are d ([ a, c, d ]) = R ([ a, c, d ]) = 1 ,d ([ a, b ]) = R ([ a, b ]) = 1 ,d ([ a, c ]) = R ([ a, c ]) − R ([ a, c, d ]) = 2 − ,d ([ c ]) = R ([ c ]) − (cid:0) R ([ a, c, d ]) ∪ R ([ a, c ]) ∪ R ([ b, c ]) ∪ R ([ c, d ]) (cid:1) = 4 − { , , } = 4 − , and d ([ d ]) = R ([ c ]) − (cid:0) R ([ a, c, d ]) ∪ R ([ a, d ]) ∪ R ([ c, d ]) (cid:1) = 2 − { } = 2 − . By contrast, the sheaf is given by the diagram ℝ ℝ ℝ ℝ ℝ ℝ ℝ ℝ ℝℝ (1)(1) (1 0)(1 0)(0 1 0 0) (0 1 0) (1 0 0 0)(1 0 0)(1 0) (0 1) (1 0) ( ) ( )
We can interpret each global section of this sheaf as a formal linear combination ofelements of Y , or a formal linear combination of columns of r .To render cosheaves and sheaves into their own categories CoShv and
Shv ,respectively, we need to define morphisms . Typical definitions (for instance, [9,Def. 4.1.10]) require the construction of morphisms between cosheaves or sheaveson the same partial order, but it is important to be a bit more general in oursituation.
Definition 10. ([13] or [5, Sec. I.4]) Suppose that R is a cosheaf on a partiallyordered set ( X, ≤ X ) and that S is a cosheaf on a partially ordered set ( Y, ≤ Y ). A cosheaf morphism m : R → S along an order preserving base map f : ( X, ≤ X ) → ( Y, ≤ Y )consists of a set of component functions m x : R ( x ) → S ( f ( x )) for each x ∈ X suchthat the following diagram commutes R ( y ) R ( x ≤ X y ) (cid:15) (cid:15) m y (cid:47) (cid:47) S ( f ( y )) S ( f ( x ) ≤ Y f ( y )) (cid:15) (cid:15) R ( x ) m x (cid:47) (cid:47) S ( f ( x ))Dually, suppose that R is a sheaf on a partially ordered set ( X, ≤ X ) and that S is a sheaf on a partially ordered set ( Y, ≤ Y ). A sheaf morphism m : S → R along an order preserving base map f : ( X, ≤ X ) → ( Y, ≤ Y ) (careful: m and f go in opposite directions! ) consists of a set of component functions m x : S ( f ( x )) → R ( x )for each x ∈ X such that the following diagram commutes S ( f ( x )) m x (cid:47) (cid:47) S ( f ( x ) ≤ Y f ( y )) (cid:15) (cid:15) R ( x ) R ( x ≤ X y ) (cid:15) (cid:15) S ( f ( y )) m y (cid:47) (cid:47) R ( y )for each x ≤ X y .The category of cosheaves CoShv (or category of sheaves
Shv ) consists of allcosheaves (or sheaves) on partially ordered sets as the class of objects, with cosheafmorphisms (or sheaf morphisms) as the class of morphisms. Composition of mor-phisms in both cases is accomplished by simply composing the base map and com-ponent functions.
Lemma 4.
The transformation of a cosheaf to its underlying partial order is acovariant functor
Base : CoShv → Pos . Likewise, the transformation of a sheafto its underlying partial order is a contravariant functor
Base (cid:48) .Proof.
Both of these statements follow immediately from the definition. (cid:3)
Lemma 5.
The transformation of cosheaves to global cosections is a covariantfunctor
Γ :
CoShv → Set . Specifically, every cosheaf morphism p : R → S induces a function on global cosections.Proof. Suppose that p : R → S and q : S → T are cosheaf morphisms alongorder preserving base maps f : ( X, ≤ X ) → ( Y, ≤ Y ) and g : ( Y, ≤ Y ) → ( Z, ≤ Z ).Let us use these data to define a function p ∗ : R ( X ) → S ( Y ) (and a function q ∗ : S ( Y ) → T ( Z ) by the same construction) such that ( q ∗ ◦ p ∗ ) = ( q ◦ p ) ∗ . To thatend, consider a cosection c of R . This is an element of (cid:32) (cid:71) x ∈ X R ( x ) (cid:33) / ∼ . Because of this, c x ∈ R ( x ) for some x ∈ X . Define p ∗ ( c ) = p x ( c x ) . To verify that this is well-defined, suppose that c x (cid:48) ∈ R ( x (cid:48) ) for some other x (cid:48) ∈ X .Under the equivalence relation ∼ , the only way c x (cid:48) ∼ c x can happen is if x ≤ x (cid:48) or x (cid:48) ≤ x . But since c is a cosection, it happens that if x (cid:48) ≤ x ,( R ( x (cid:48) ≤ x )) ( c x ) = c x (cid:48) . Since p is a cosheaf morphism, this means that p x (cid:48) ( c x (cid:48) ) = ( p x (cid:48) ◦ ( R ( x (cid:48) ≤ x ))) ( c x ))= ( S ( f ( x (cid:48) ) ≤ f ( x )) ◦ p x ) ( c x ) , = ( S ( f ( x (cid:48) ) ≤ f ( x ))) ( p x ( c x )) , which implies that p x (cid:48) ( c x (cid:48) ) ∼ p x ( c x ) in S ( X ). On the other hand, if x ≤ x (cid:48) ( R ( x ≤ x (cid:48) )) ( c x (cid:48) ) = c x . OSHEAF REPRESENTATIONS OF RELATIONS AND DOWKER COMPLEXES 17
Since p is a cosheaf morphism, this means that p x ( c x ) = ( p x ◦ ( R ( x ≤ x (cid:48) ))) ( c x (cid:48) )= (( S ( f ( x ) ≤ f ( x (cid:48) ))) ◦ p x (cid:48) ) ( c x (cid:48) ) , which also implies that p x (cid:48) ( c x (cid:48) ) ∼ p x ( c x ) in S ( X ). Thus, p ∗ ( c ) is a well-definedglobal cosection of S .Repeating this construction with q , notice that( q ◦ p ) ∗ ( c ) = ( q ◦ p ) x ( c )= ( q x ◦ p x )( c )= ( q ∗ ◦ p ∗ )( c ) , which establishes covariance. (cid:3) Theorem 4.
The transformation ( X, Y, R ) (cid:55)→ R given in Definition 8 is a co-variant functor CoShvRep : Rel → CoShv . In particular, each
Rel morphisminduces a cosheaf morphism. Furthermore, if the domain is restricted to
Rel + , thefunctor becomes faithful.Proof. Suppose that ( f , g ) : ( X , Y , R ) → ( X , Y , R ) and ( f , g ) : ( X , Y , R ) → ( X , Y , R ) are two Rel morphisms. Suppose that R is the cosheaf associated to( X , Y , R ) according to the recipe given in Definition 8, and likewise R and R are the cosheaves associated to ( X , Y , R ) and ( X , Y , R ), respectively. We firstshow how to construct a cosheaf morphism m : R → R .Recognizing that the cosheaves R and R are written on the simplices of D ( X , Y , R ) and D ( X , Y , R ), recall that Theorem 3 implies that D ( f ) is asimplicial map D ( X , Y , R ) → D ( X , Y , R ), and that Propositions 4 and 5 im-ply that this can be interpreted as an order preserving function. This is the orderpreserving base map along which m is defined.Suppose that σ ⊆ τ in D ( X , Y , R ). As far as vertices are concerned, thediagram σ ⊆ (cid:15) (cid:15) f (cid:47) (cid:47) f ( σ ) ⊆ (cid:15) (cid:15) τ f (cid:47) (cid:47) f ( τ )commutes. Corollary 1 therefore states that( Y ) τ g (cid:47) (cid:47) ⊆ (cid:15) (cid:15) ( Y ) f ( τ ) ⊆ (cid:15) (cid:15) ( Y ) σ g (cid:47) (cid:47) ( Y ) f ( σ ) commutes. We therefore merely need to realize that according to Definition 8, thisis equal to the diagram R ( τ ) g (cid:47) (cid:47) R ( σ ⊆ τ ) (cid:15) (cid:15) R ( f ( τ )) R ( f ( σ ) ⊆ f ( τ )) (cid:15) (cid:15) R ( σ ) g (cid:47) (cid:47) R ( f ( σ )) (cid:1) ([ b , c , d ])={3} (cid:1) ([ c , d , e ])={4} (cid:1) ([ c , d ])={3,4} (cid:1) ([ b , c ])={3} (cid:1) ([ b , d ])={3} (cid:1) ([ c , e ])={4,5} (cid:1) ([ d , e ])={4} (cid:1) ([ a , b ])={1} (cid:1) ([ a , c ])={2} (cid:1) ([ a ])={1,2} (cid:1) ([ b ])={1,3} (cid:1) ([ c ])={2,3,4,5} (cid:1) ([ d ])={3,4} (cid:1) ([ e ])={4,5} (cid:2) ([ A , B ])={1} (cid:2) ([ A , C ])={2} (cid:2) ([ B , C ])={3} (cid:2) ([ A ])={1,2} (cid:2) ([ B ])={1,3} (cid:2) ([ C ])={2,3} Figure 7.
The cosheaf morphism induced by the
Rel morphism( f, g ) in Example 10 via the
CoShvRep functor.which establishes that m is a cosheaf morphism, with m σ = g | ( Y ) σ as components.Given that m : R → R can be constructed in the same way, the composition( f , g ) ◦ ( f , g ) of Rel morphisms induces the composition of component functions,which is precisely the composition m ◦ m of cosheaf morphisms.To show that this functor is faithful when restricted to objects in Rel + , a ratherdirect argument suffices. Suppose that ( f, g ) and ( f (cid:48) , g (cid:48) ) are two Rel morphisms( X , Y , R ) → ( X , Y , R ) in which ( X , Y , R ) is an object of Rel + . Recallthat the latter constraint means that for every x ∈ X , there is a y ∈ Y suchthat ( x, y ) ∈ R , and conversely for every y ∈ Y , there is an x ∈ X such that( x, y ) ∈ R . To establish faithfulness, let us suppose additionally that ( f, g ) and( f (cid:48) , g (cid:48) ) induce the same cosheaf morphism m : R → R .Let y ∈ Y be given. By assumption, there is an x ∈ X such that ( x, y ) ∈ R ,so there is also a simplex σ (usually several simplices, actually) for which y ∈ Y σ .But, since both ( f, g ) and ( f (cid:48) , g (cid:48) ) both induce the same cosheaf morphism m , thismeans that g ( y ) = g | ( Y ) σ ( y ) = m σ ( y ) = g (cid:48) | ( Y ) σ ( y ) = g (cid:48) ( y ) . Hence g = g (cid:48) .Now let x ∈ X be given. By assumption, there is a y ∈ Y such that ( x, y ) ∈ R ,so this means that [ x ] is a simplex of D ( X , Y , R ). Since both ( f, g ) and ( f (cid:48) , g (cid:48) )induce the same cosheaf morphism m , this means that both ( f, g ) and ( f (cid:48) , g (cid:48) ) inducethe same order preserving map on simplices of D ( X , Y , R ) → D ( X , Y , R ).When restricted to vertices, this map is simply f or f (cid:48) , respectively, so they mustalso be equal. (cid:3) OSHEAF REPRESENTATIONS OF RELATIONS AND DOWKER COMPLEXES 19
Example 10.
Consider again the relation R between the sets X = { a, b, c, d, e } and Y = { , , , , } , given by the matrix r = , from Example 6. However, this time let us consider a different morphism. Definethe relation R between X = { A, B, C } and Y = { , , } given by r = . The function f : X → X given by f ( a ) = A, f ( b ) = B, f ( c ) = C, f ( d ) = C, f ( e ) = C, and the function g : Y → Y given by g (1) = 1 , g (2) = 2 , g (3) = 3 , g (4) = 3 , g (5) = 3together define a Rel morphism ( X , Y , R ) → ( X , Y , R ).This relation morphism clearly maps each column of r to a column of r ,so it also acts on the costalks of the cosheaf representations. If we define A = CoShvRep ( X , Y , R ) and B = CoShvRep ( X , Y , R ), the resulting cosheafmorphism A → B is given by the diagram shown in Figure 7. Notice that eachdashed arrow represents a component map of the cosheaf morphism, and is givenby restricting the domain of g to each costalk, since this is how the columns aretransformed. Theorem 5.
The transformation R (cid:55)→ R given in Definition 8 is a contravariantfunctor ShvRep : Rel → Shv . When restricted to
Rel + → Shv , the functorbecomes faithful.
The proof of this Theorem starts out exactly dual to that of the proof of Theorem4, but then diverges due to differences in the algebraic structure of the stalks.The argument from that point looks different, but is actually the same (modulo atranspose, which is the duality) when restricted to basis elements of the stalk.
Proof.
Suppose that ( f , g ) : ( X , Y , R ) → ( X , Y , R ) and ( f , g ) : ( X , Y , R ) → ( X , Y , R ) are two Rel morphisms. Suppose that R is the sheaf associated to( X , Y , R ) according to the recipe from Definition 8, and likewise R and R arethe sheaves associated to ( X , Y , R ) and ( X , Y , R ), respectively. We first showhow to construct a sheaf morphism m : R → R . Given that m : R → R canbe constructed in the same way, we show that the composition ( f , g ) ◦ ( f , g ) of Rel morphisms induces the composition m ◦ m of sheaf morphisms.Recognizing that the sheaves R and R are written on the simplices of D ( X , Y , R )and D ( X , Y , R ), recall that Theorem 3 implies that D ( f ) is a simplicial map D ( X , Y , R ) → D ( X , Y , R ), and that Propositions 4 and 5 imply that this canbe interpreted as an order preserving function. This is the order preserving mapalong which m is defined. The component maps go the other way, and are expansions of the preimage of g . For a simplex σ of D ( X , Y , R ), the m ,σ : R ( f ( σ )) → R ( σ ) is given by theformula m ,σ (cid:88) z ∈ ( Y ) f ( σ ) a z z = (cid:88) z ∈ ( Y ) f ( σ ) , (cid:88) y ∈ g − ( z ) a z y. To show that this is indeed a sheaf morphism requires showing that it commuteswith the restriction maps. This follows from the diagram of Corollary 1, since thatdiagram explains how the basis vectors transform; the sheaf uses the dual of eachmap. To show this explicitly, it suffices to show this for a pair of simplices σ ⊆ τ in D ( X , Y , R ) and for a basis element z ∈ ( Y ) f ( σ ) , because we can extend bylinearity, (cid:0) R ( σ ⊆ τ ) ◦ m ,σ (cid:1) ( z ) = (cid:0) R ( σ ⊆ τ ) (cid:1) (cid:88) y ∈ g − ( z ) y = (cid:88) y ∈ g − ( z ) (cid:0) R ( σ ⊆ τ ) (cid:1) ( y )= (cid:88) y ∈ g − ( z ) and y ∈ ( Y ) τ y. According to Lemma 3, y ∈ ( Y ) τ implies that z ∈ ( Y ) f ( τ ) . Thus we may continuethe calculation along the other path (cid:0) m ,τ ◦ R ( f ( σ ) ⊆ f ( τ )) (cid:1) ( z ) = m ,τ ( z )= (cid:88) y ∈ g − ( z ) and y ∈ ( Y ) τ y, establishing commutativity of the diagramAs for composition ( f , g ) ◦ ( f , g ) of Rel morphisms, suppose that σ is asimplex of D ( X , Y , R ). We compute for z ∈ ( Y ) f ( f ( σ )) : (cid:0) m ,σ ◦ m ,f ( σ ) (cid:1) ( z ) = m ,σ (cid:88) y ∈ g − ( z ) y = (cid:88) w ∈ g − ( y ) , (cid:88) y ∈ g − ( z ) w = (cid:88) w ∈ ( g ◦ g ) − ( z ) w, which is precisely what is induced by ( f ◦ f , g ◦ g ).To show that this a faithful functor when restricted to Rel + , it suffices to recountthe same argument for the cosheaf given in the proof of Theorem 4, making theobservation that the components of the cosheaf morphism are simply the functionson the basis elements of the stalks of the sheaf, after a transpose. (cid:3) Actually, the cosheaf R seems a bit more natural than the sheaf R ! At least, R doesn’t entrain any linear algebraic machinery, which may be ancillary to themain point. On the other hand, the sheaf has cohomology, which may be worthexploring. OSHEAF REPRESENTATIONS OF RELATIONS AND DOWKER COMPLEXES 21
Example 11.
Notice that if we tried to define R as a sheaf of sets instead of acosheaf – using only the basis Y σ rather than its span – then the proof of Theorem5 fails to work correctly, even if we reverse the partial order on D ( X, Y, R ). Thisis not an accident, since any functor
Rel → Shv should compose with the globalsections functor Γ :
Shv → Set to ensure that
Rel morphisms induce functionson the space of global sections. This fails outright for a small example in which X = { A, B, C } , X = { A } , Y = Y = { a, b, c } , where R and R are given by thematrices r = (cid:18) (cid:19) and r = (cid:0) (cid:1) . Noting that there is only one option to define f : X → X , we define g = id Y .This is clearly a relation morphism as every pair ( x, y ) ∈ R ⊆ X × Y maps to apair that are related by R .Using the reverse partial order, the sheaf diagram of basis elements for ( X , Y , R )is { a } { b, c } { c }{ c } (cid:79) (cid:79) (cid:60) (cid:60) while the sheaf diagram for basis elements of ( X , Y , R ) has only one element { a, b, c } . (Both of these are identical to the cosheaf diagrams, since the unions inthe Alexandrov topology are not shown.) There is only one global section of thefirst sheaf, which consists of choosing a for the leftmost simplex, and b for the threeelements on the right. However, this cannot obviously be mapped to a global sectionof the second sheaf, since that needs to be a single element of { a, b, c } . Conversely,if we consider a global section of the second sheaf as being any one of its elements,this cannot correspond to a global section of the first sheaf. Corollary 3.
The composition of
CoShvRep : Rel → CoShv with the functor
Base : CoShv → Pos that forgets the structure of the costalks is
P osRep : Rel → Pos . This also works for the composition ( Base (cid:48) ◦ ShvRep ) : Rel → Shv → Pos .Briefly,
P osRep = Base ◦ CoShvRep = Base (cid:48) ◦ ShvRep . Notice that in the case of sheaves on partial orders, both functors are contravari-ant! 5.
Duality of cosheaf representations of relations
The most striking fact proven in Dowker’s original paper [10] is that the ho-mology of the Dowker complex is the same whether it is produced by the relationor by its transpose. Dowker provides a direct, if elaborate, construction of a mapinducing isomorphisms on homology. This construction was later enhanced to ahomotopy equivalence by [4]. More recently, [8] showed that the homotopy equiv-alence between these two complexes is functorial in a particular way. This sectionshows that the duality is also visible in a somewhat different way: one Dowkercomplex is the base space of a particular cosheaf, while the other is its space ofglobal cosections.Let us begin by connecting the relation to its transpose. a bd c
135 6 24 D ( X , Y , R ) D ( Y , X , R T ) Figure 8.
The Dowker complex for a relation R and its transposegiven in Example 12. Definition 11.
If (
X, Y, R ) is a relation, then its transpose is a relation (
Y, X, R T )given by ( y, x ) ∈ R T if and only if ( x, y ) ∈ R .Evidently, the matrix for the transpose of a relation is simply the transpose ofthe original matrix. Lemma 6.
The transformation ( X, Y, R ) (cid:55)→ ( Y, X, R T ) defines a fully faithfulcovariant functor T ransp : Rel → Rel .Proof.
Every
Rel morphism ( f, g ) : (
X, Y, R ) → ( X (cid:48) , Y (cid:48) , R (cid:48) ) is transformed to( g, f ) : ( Y, X, R T ) → ( Y (cid:48) , X (cid:48) , ( R (cid:48) ) T ). Composition is still composition of functionsand is preserved in order. (cid:3) Example 12.
Recall the relation R between X = { a, b, c, d } and Y = { , , , , , } from Example 2, which was given by the matrix r = . The transpose of this relation has the matrix r T = . Their Dowker complexes are shown in Figure 8. Clearly these complexes have thesame homotopy type!
Definition 12.
The category
CoShvAsc consists of the full subcategory of
CoShv whose objects are cosheaves on abstract simplicial complexes of abstract simplicialcomplexes, and whose extensions are simplicial inclusions. Briefly, an object of
CoShvAsc is a contravariant functor C from the face partial order of an abstractsimplicial complex to Asc , with the additional condition that each extension C ( σ ⊆ τ ) : C ( τ ) → C ( σ ) is a simplicial map whose vertex function is an inclusion. OSHEAF REPRESENTATIONS OF RELATIONS AND DOWKER COMPLEXES 23 (cid:1) ([ a , c , d ]) = [3] (cid:1) ([ a , d ]) = [3] (cid:1) ([ c , d ]) = [3] (cid:1) ([ d ]) = [3,5] (cid:1) ([ c ]) = [2,3,4,6] (cid:1) ([ a ]) = [1,3,6] (cid:1) ([ b ]) = [1,2] (cid:1) ([ a , b ]) = [1] (cid:1) ([ b , c ]) = [2] (cid:1) ([ a , c ]) = [3,6] Figure 9.
The diagram of the cosheaf R defined in Example 13for the relation ( X , Y , R ) defined in Example 2.
13 6 24 13 6 24Cosections on U Cosections on U Figure 10.
Some spaces of cosections of the cosheaf R defined inExample 13: (left) cosections on the set U , (right) cosections onthe set U . Definition 13.
The cosheaf representation of a relation ( X, Y, R ) is a cosheaf R = CoShvRep ( X, Y, R ) of abstract simplicial complexes, defined by the followingrecipe:
Costalks: If σ is a simplex of D ( X, Y, R ), then R ( σ ) = D (cid:0) Y σ , σ, ( R | σ,Y σ ) T (cid:1) , Extensions: If σ ⊆ τ are two simplices of D ( X, Y, R ), then the extension R ( σ ⊆ τ ) : R ( τ ) → R ( σ ) is the simplicial map along the inclusion Y τ (cid:44) → Y σ .The cosheaf R = CoShvRep ( X, Y, R ) for a relation (
X, Y, R ) defined in Sec-tion 4 is a sub-cosheaf of R = CoShvRep ( X, Y, R ). Evidently, R is an object of CoShvAsc . Example 13.
Recall the relation R between X = { a, b, c, d } and Y = { , , , , , } from Example 2, which was given by the matrix r = . The cosheaf R = CoShvRep ( X , Y , R ) was described in Example 9. The dia-gram for R = CoShvRep ( X , Y , R ) is shown in Figure 9, which incorporates allof the data from the previous examples into a single figure. Notice that each costalk shown in the diagram is a complete simplex. The space of cosections over the set U = { [ b ] , [ c ] , [ a, b ] , [ a, c ] , [ b, c ] , [ c, d ] , [ a, c, d ] } is an abstract simplicial complex on the vertex set that is the union R ([ b ]) ∪ R ([ c ]) = { , } ∪ { , , , } = { , , , , } , but is not the complete simplex on those vertices. Instead, it is the simplicialcomplex shown at left in Figure 10. Likewise the space of cosections over the set U = { [ a ] , [ c ] , [ a, b ] , [ b, c ] , [ a, c ] , [ a, d ] , [ c, d ] , [ a, c, d ] } is shown at right in Figure 10. From these two examples, it is clear that the spaceof global cosections is indeed D ( Y , X , R T ), as shown in Figure 8. Lemma 7.
For any simplex τ of D ( X, Y, R ) , the costalk R ( τ ) = D (cid:0) Y τ , τ, ( R | τ,Y τ ) T (cid:1) is always a complete simplex on the vertex set Y τ .Proof. Every subset { y , y , . . . , y n } consisting of elements y i of Y τ is a simplex of D (cid:0) Y τ , τ, ( R | τ,Y τ ) T (cid:1) , since that merely requires there to be at least one x ∈ τ toexist such that ( x, y i ) ∈ R for all i . (cid:3) Lemma 8.
The extensions defined for the cosheaf R for a relation ( X, Y, R ) inDefinition 13 are well-defined simplicial maps.Proof. Suppose that σ ⊆ τ are two simplices of D ( X, Y, R ). Lemma 7 establishesthat both R ( σ ) and R ( τ ) are complete simplices. Accordingly, consider the subset { y , y , . . . , y n } consisting of elements y i of Y τ . Notice that by the definition of Y τ ,for every x ∈ τ and every i , it follows that ( x, y i ) ∈ R . Therefore, since σ ⊆ τ , thiscondition also holds for every x ∈ σ . Thus, every simplex of D (cid:0) Y τ , τ, ( R | τ,Y τ ) T (cid:1) isalso a simplex of D (cid:0) Y σ , σ, ( R | σ,Y σ ) T (cid:1) whenever σ ⊆ τ . (cid:3) These two Lemmas together imply that Theorem 4 extends immediately to afunctoriality result for R . Corollary 4.
The transformation ( X, Y, R ) (cid:55)→ R is a covariant functor CoShvRep : Rel → CoShvAsc . If we restrict to
Rel + → CoShvAsc , this becomes a faithfulcovariant functor.
Theorem 6.
The space of global cosections of R is simplicially isomorphic to D ( Y, X, R T ) , the Dowker complex for the transpose.Proof. Before we begin, notice that the vertices of D ( Y, X, R T ) are elements of Y that are related to at least one element of X . These are also elements of thecostalks of R , and since the extensions of R are inclusions, we need not worry aboutconflicting names for elements of Y . Therefore, to establish this result, we simplyneed to show that every simplex σ ∈ D ( Y, X, R T ) appears in at least one costalkof R , and conversely that every simplex in every costalk of R is also a simplex of D ( Y, X, R T ).Suppose that σ = [ y , y , . . . , y n ] is a simplex of D ( Y, X, R T ). This means thatthere is an x ∈ X such that ( x, y i ) ∈ R for all i = 0 , . . . , n . Put another way, every y i ∈ σ is also an element of Y [ x ] . Therefore, the costalk R ([ x ]) contains σ .Suppose that σ = [ y , y , . . . , y n ] is a simplex of R ( τ ) for some simplex τ of D ( X, Y, R ). This means that σ is a simplex of D (cid:0) Y τ , τ, ( R | τ,Y τ ) T (cid:1) , by definition.That means that if we select any x ∈ τ , it follows that ( y i , x ) ∈ ( R | τ,Y τ ) T ⊆ R T .Therefore, σ is a simplex of D ( Y, X, R T ). (cid:3) OSHEAF REPRESENTATIONS OF RELATIONS AND DOWKER COMPLEXES 25 a b cd (cid:1) ([ a ]) (cid:1) ([ b ]) (cid:1) ([ c ]) (cid:1) ([ d ]) (cid:1) ([ a , b ]) (cid:1) ([ a , c ]) (cid:1) ([ b , c ]) (cid:1) ([ a , d ]) (cid:1) ([ c , d ]) (cid:1) ([ a , b , c ]) 1 23 1 34 1 3115 2 11 2 11 1 Base space Space of global cosectionsCosheaf diagram
Figure 11.
A cosheaf B of abstract simplicial complexes de-scribed in Example 14: (left) the base space of B , (center) thediagram of B , (right) the space of global cosections of B .What this means is that we have the following functorial diagram Asc G (cid:38) (cid:38) Rel D (cid:53) (cid:53) T ransp (cid:15) (cid:15)
CoShvRep (cid:47) (cid:47)
CoShvAsc
Base (cid:79) (cid:79) Γ (cid:15) (cid:15) CWRel D (cid:47) (cid:47) Asc G (cid:56) (cid:56) where CW is the category of CW complexes and homotopy classes of continuousmaps, G is the geometric realization of an abstract simplicial complex, Base is thefunctor that forgets the costalks of a cosheaf (Corollary 4), and Γ is the functorthat constructs the space of global cosections from a cosheaf (Lemma 5). Dowkerduality asserts that the top and bottom paths in this diagram are equivalent up tohomotopy.For a cosheaf C on an abstract simplicial complex X of abstract simplicial com-plexes whose extensions are inclusions, let us define a new cosheaf Dual ( C ) on thespace of global cosections of C . Noting that the space of global cosections C ( X )is also an abstract simplicial complex, suppose σ is a simplex of C ( X ). Define thecostalk ( Dual ( C )) ( σ ) to be the simplicial complex formed by the union of everysimplex α in X whose costalk C ( α ) contains σ . Abstractly, this is equivalent to aunion ( Dual ( C )) ( σ ) = (cid:91) { α ∈ X : σ ∈ C ( α ) } , which implies that Dual ( C ) is a well-defined cosheaf when the extensions are allchosen to be inclusions. ( Dual ( (cid:1) ))([1])( Dual ( (cid:1) ))([4]) ( Dual ( (cid:1) ))([3]) ( Dual ( (cid:1) ))([2])( Dual ( (cid:1) ))([1,4]) ( Dual ( (cid:1) ))([1,3])( Dual ( (cid:1) ))([3,4]) ( Dual ( (cid:1) ))([1,2]) ( Dual ( (cid:1) ))([2,3])( Dual ( (cid:1) ))([1,3,4])( Dual ( (cid:1) ))([5])( Dual ( (cid:1) ))([1,5]) db d d dc c cb d a b cd dc cb Figure 12.
The cosheaf
Dual ( B ) that is dual to the cosheaf B shown in Figure 11 and described in Example 14. Example 14.
Figure 11 shows a cosheaf B of abstract simplicial complexes onan abstract simplicial complex. Since each extension map is an inclusion, thiscosheaf is an object in CoShvAsc . The space of global cosections of this cosheaf isan abstract simplicial complex, which is shown at right in Figure 11. The cosheaf
Dual ( B ) can therefore be constructed on this new abstract simplicial complex usingthe definition above. The resulting cosheaf is shown in Figure 12, where it is clearthat each extension of this new cosheaf is an inclusion. It is also easily seen thatthe space of global cosections of Dual ( B ) is the base space of B . Lemma 9.
Dual is a covariant functor
CoShvAsc → CoShvAsc .Proof.
The
Dual functor exchanges the base space with the space of global cosec-tions. For a cosheaf R on X that is an object of CoShvAsc , Base ( Dual ( R )) = R ( X ) , OSHEAF REPRESENTATIONS OF RELATIONS AND DOWKER COMPLEXES 27 by definition andΓ(
Dual ( R )) = (cid:71) σ ∈ Base ( Dual ( R )) ( Dual ( R ))( σ ) / ∼ = (cid:71) σ ∈ R ( X ) ( Dual ( R ))( σ ) / ∼ = (cid:91) σ ∈ R ( X ) ( Dual ( R ))( σ )= (cid:91) σ ∈ R ( X ) (cid:91) { α ∈ X : σ ∈ R ( α ) } = { α ∈ X : there is a σ ∈ R ( X ) such that σ ∈ R ( α ) } = (cid:40) α ∈ X : there is a σ ∈ (cid:91) τ ∈ X R ( τ ) such that σ ∈ R ( α ) (cid:41) = X. A cosheaf morphism m : R → S along a simplicial map f : X → Y induces amap m ∗ : R ( X ) → S ( Y ) on each space of cosections (Lemma 5). We use thesedata to define a morphism w : Dual ( R ) → Dual ( S ). As such, the induced map m ∗ on the space of global cosections becomes the new base space map, along whichthe new cosheaf morphism w is written. The individual simplices map by way ofrestricting to the components of the old morphism, since m ∗ is a simplicial map.Conversely, the old base space map f defines the new component maps w σ byrestriction.Explicitly, if σ is a simplex of Base ( Dual ( R )) = R ( X ), we have that( Dual ( R ))( σ ) = (cid:91) { α ∈ X : σ ∈ R ( α ) } . The component w σ must be a function ( Dual ( R ))( σ ) → Dual ( S )( m ∗ ( σ )). Sinceevery element of ( Dual ( R ))( σ ) is an α ∈ X , and the domain of f is X , we maydefine w σ = f | ( Dual ( R ))( σ ) . This is well-defined because if σ ∈ R ( α ) then m ∗ ( σ ) ∈ S ( f ( α )), and( Dual ( S ))( m ∗ ( σ )) = (cid:91) { β ∈ Y : m ∗ ( σ ) ∈ S ( β ) } . To establish that these component maps form a cosheaf morphism, we need toestablish that the diagram below commutes for all simplices α ⊆ β in R ( X ):( Dual ( R ))( β ) ( Dual ( R ))( α ⊆ β ) (cid:15) (cid:15) w β (cid:47) (cid:47) ( Dual ( S ))( m ∗ ( β )) ( Dual ( S ))( m ∗ ( α ) ⊆ m ∗ ( β )) (cid:15) (cid:15) ( Dual ( R ))( α ) w α (cid:47) (cid:47) ( Dual ( S ))( m ∗ ( α ))This follows because the vertical maps are inclusions and the horizontal maps areboth restrictions of f to nested subsets.Finally, composition of morphisms is preserved because that is simply composi-tion of the base and global cosection functions. (cid:3) Theorem 7. (Cosheaf version of Dowker duality)
Dual is a functor that makesthe diagram of functors commute:
Rel
T ransp (cid:15) (cid:15)
CoShvRep (cid:47) (cid:47)
CoShvAsc
Dual (cid:15) (cid:15)
Rel
CoShvRep (cid:47) (cid:47)
CoShvAsc
Proof.
The way that
Dual ( R ) has been defined, we might end up with a simplicialcomplex as a costalk that is not a complete simplex – which is a problem accordingto Lemma 7 – but this does not happen in the image of CoShvRep . Supposethat R = CoShvRep ( X, Y, R ). We claim that for every simplex σ in R ( X ) = D ( Y, X, R T ), the set of simplices { α ∈ X : σ ∈ R ( α ) } has a unique maximal simplex in the inclusion order, so that the union in thedefinition of ( Dual ( R )) ( σ ) really is just that one simplex. To see that, supposethat α and β are simplices of X for which σ ∈ R ( α ) and σ ∈ R ( β ). Suppose thatany other simplex γ that contains α has σ / ∈ R ( γ ), so α is maximal in the senseof inclusion. We want to show that β ⊆ α . Going back to the definition of R , wehave that R ( α ) = D (cid:0) Y α , α, ( R | α,Y α ) T (cid:1) and R ( β ) = D (cid:0) Y β , β, ( R | β,Y β ) T (cid:1) . Both contain σ . What about the simplex δ whose vertices are the union of thevertices of α and β ? Suppose that y ∈ Y is a vertex of σ . This means that y ∈ Y α ∩ Y β , which means that ( x, y ) ∈ R for every x ∈ α ∪ β = δ . Thus σ ⊆ Y δ , orin other words σ ∈ R ( δ ) according to Lemma 7. On the other hand, if α (cid:36) δ , weassumed that σ / ∈ R ( δ ). So the only way this can happen is if α = δ , which implies β ⊆ α .With this fact in hand, we can observe that(( Dual ◦ CoShvRep )( X, Y, R )) ( σ ) = ( Dual ( R )) ( σ )= (cid:91) { α ∈ X : σ ∈ R ( α ) } = (cid:91) { α ∈ X : σ ⊆ Y α } = (cid:91) { α ∈ X : for all y ∈ σ and all x ∈ α, ( x, y ) ∈ R } = { x ∈ X : ( x, y ) for all y ∈ σ } = D ( X σ , σ, R | X σ ,σ )= (cid:0) CoShvRep ( Y, X, R T ) (cid:1) ( σ )= (( CoShvRep ◦ T ransp )( X, Y, R )) ( σ ) . (cid:3) Example 15.
Consider the relation morphism
Rel morphism ( f, g ) : ( X , Y , R ) → ( X , Y , R ) defined in Example 10. Recall that the relation R between the sets OSHEAF REPRESENTATIONS OF RELATIONS AND DOWKER COMPLEXES 29 (cid:1) ([ b , c , d ]) (cid:1) ([ c , d , e ]) (cid:1) ([ c , d ]) (cid:1) ([ b , c ]) (cid:1) ([ b , d ]) (cid:1) ([ c , e ]) (cid:1) ([ d , e ]) (cid:1) ([ a , b ]) (cid:1) ([ a , c ]) (cid:1) ([ a ]) (cid:1) ([ b ]) (cid:1) ([ c ]) (cid:1) ([ d ]) (cid:1) ([ e ]) (cid:2) ([ A , B ]) (cid:2) ([ A , C ]) (cid:2) ([ B , C ]) (cid:2) ([ A ]) (cid:2) ([ B ]) (cid:2) ([ C ])
21 31 21 31 32541 2 3 3 4543 4 1 2 3432 354 43
Figure 13.
The cosheaf morphism induced by
CoShvRep de-scribed in Example 15. (Dual( (cid:1) ))([1]) (Dual( (cid:1) ))([2]) (Dual( (cid:1) ))([3]) (Dual( (cid:1) ))([4]) (Dual( (cid:1) ))([5])(Dual( (cid:1) ))([1,2])(Dual( (cid:1) ))([1,3]) (Dual( (cid:1) ))([2,3,4,5])(Dual( (cid:1) ))([3,4]) (Dual( (cid:1) ))([4,5])(Dual( (cid:1) ))([2,3,4])(Dual( (cid:1) ))([2,3,5]) (Dual( (cid:1) ))([3,4,5])(Dual( (cid:1) ))([2,4,5])(Dual( (cid:1) ))([3,5])(Dual( (cid:1) ))([2,5])(Dual( (cid:1) ))([2,4])(Dual( (cid:1) ))([2,3]) ba ca b dc e dc eca b c c dc c ecc c cc (Dual( (cid:2) ))([1])(Dual( (cid:2) ))([2])(Dual( (cid:2) ))([3])(Dual( (cid:2) ))([1,2]) (Dual( (cid:2) ))([2,3])(Dual( (cid:2) ))([1,3])
BA CA CBA B Ccc
Figure 14.
The cosheaf morphism induced by
Dual ◦ CoShvRep described in Example 15. X = { a, b, c, d, e } and Y = { , , , , } , was given by the matrix r = , and the relation R between X = { A, B, C } and Y = { , , } was given by r = . The function f : X → X was given by f ( a ) = A, f ( b ) = B, f ( c ) = C, f ( d ) = C, f ( e ) = C, and the function g : Y → Y was given by g (1) = 1 , g (2) = 2 , g (3) = 3 , g (4) = 3 , g (5) = 3 . Let is define A = CoShvRep ( X , Y , R ) and B = CoShvRep ( X , Y , R ). Thecosheaf morphism induced by CoShvRep was shown in Figure 7, but what interestsus now is the cosheaf morphism A → B induced by CoShvRep and
Dual ( A ) → Dual ( B ) induced by Dual ◦ CoShvRep (or equally well, induced by
CoShvRep ◦ T ransp ). These two morphisms are shown in Figures 13 and 14, respectively.Notice in particular that each component map (in both morphisms) is a simplicialmap, so that whenever two vertices are collapsed (for instance g (3) = g (4) = g (5))the associated simplices are collapsed as well.6. Redundancy of relations
As has been explained earlier, the functor D : Rel → Asc is not faithful; manydistinct relations have the same Dowker complex. One way this can happen is ifcolumns (or rows) in the matrix for the relation are redundant , which means thata column (or row) has 1s in all the same places as another column (or row), sincethis simply adds additional copies of the same simplex to the Dowker complex orits dual. We can construct a (non-functorial) cosheaf to detect this redundancydirectly using a similar construction to our earlier ones.Since Lemma 7 establishes that D (cid:0) Y σ , σ, ( R | σ,Y σ ) T (cid:1) is always a complete simplexfor a relation ( X, Y, R ) – equivalently, the matrix for R | σ,Y σ is a block of all 1s –it does not have any useful information beyond the vertex set Y σ . In a sense, R = CoShvRep ( X, Y, R ) and R = CoShvRep ( X, Y, R ) are basically very similar;the only difference being the topology on their costalks.Taking a different perspective, the matrix for R | X \ σ,Y σ contains all the informa-tion about the elements of Y σ that is not a result of their relation to elements in σ . This observation means that we can also define a rather different cosheaf by itscostalks S ( σ ) = D (cid:0) Y σ , X \ σ, ( R | X \ σ,Y σ ) T (cid:1) , on each simplex σ of D ( X, Y, R ). As in the previous constructions, we may takethe extensions S ( σ ⊆ τ ) to be simplicial maps induced by inclusions. The ex-tensions are well defined because of Lemma 2. If [ y , y , . . . , y n ] is a simplexof D (cid:0) Y τ , X \ τ, ( R | X \ τ,Y τ ) T (cid:1) , then this means that there is an x / ∈ τ such that( x, y i ) ∈ R for all i = 0 , , . . . , n . Evidently, x / ∈ σ as well, so [ y , y , . . . , y n ] is asimplex of D (cid:0) Y σ , X \ σ, ( R | X \ σ,Y σ ) T (cid:1) as well.Elements of Y σ may not be related to any elements of X besides those alreadyin σ . This means that R = CoShvRep ( X, Y, R ) may not be a sub-cosheaf of S ,because a costalk of S may have fewer vertices than in the corresponding costalkof R . Moreover, the transformation ( X, Y, R ) (cid:55)→ S is not functorial. OSHEAF REPRESENTATIONS OF RELATIONS AND DOWKER COMPLEXES 31 (cid:1) b , c , d (cid:1) c , d , e (cid:1) c , d (cid:1) b , c (cid:1) b , d (cid:1) c , e (cid:1) d , e (cid:1) a , b (cid:1) a , c (cid:1) a (cid:1) b (cid:1) c (cid:1) d (cid:1) e (cid:2) A , B (cid:2) A , C (cid:2) B , C (cid:2) A (cid:2) B (cid:2) C
21 31 21 31 32543 3 44432 354 43∅ ∅ ∅ ∅ ∅ ∅ ∅
Figure 15.
The cosheaf morphism described in Example 17.(Compare with Figures 7 and 13, which are induced by the same
Rel morphism.)
Example 16.
Let (
X, Y, R ) be any relation, and let ( X (cid:48) , Y (cid:48) , R (cid:48) ) be the relationgiven by ( { a, b } , { , } , { ( a, , ( b, } ). If f : X → X (cid:48) is the constant function thattakes the value f ( x ) = a on all x ∈ X , then the constant function g : Y → Y (cid:48) with g ( y ) = 1 for all y ∈ Y will define a Rel morphism ( f, g ) : (
X, Y, R ) → ( X (cid:48) , Y (cid:48) , R (cid:48) ).Notice that the cosheaf T constructed by the recipe above on ( X (cid:48) , Y (cid:48) , R (cid:48) ) has T ([ a ]) = D (cid:16) Y [ a ] , X \ [ a ] , ( R | X \ [ a ] ,Y [ a ] ) T (cid:17) = D ( { } , { b } , ∅ ) = ∅ , but S ( σ ) = D (cid:0) Y σ , X \ σ, ( R | X \ σ,Y σ ) T (cid:1) may well be nonempty. Since f ( σ ) = [ a ] by construction, this means that thereis no way to construct a cosheaf morphism S → T along D ( f ) : D ( X, Y, R ) → D ( X (cid:48) , Y (cid:48) , R (cid:48) ), since the component m σ would be a function m σ : D (cid:0) Y σ , X \ σ, ( R | X \ σ,Y σ ) T (cid:1) → ∅ , which cannot exist unless the domain is empty.Finally, notice that restricting the domain to Rel + does not improve matters,since ( X (cid:48) , Y (cid:48) , R (cid:48) ) is already an object of Rel + .Even in the face of situations like Example 16, sometimes a cosheaf morphismcan be induced by a Rel morphism.
Example 17.
Consider the
Rel morphism ( f, g ) : ( X , Y , R ) → ( X , Y , R )given in Example 10. If we construct S from ( X , Y , R ) and T from ( X , Y , R )using the recipe above, this happens to induce a cosheaf morphism S → T , whichis shown in Figure 15. It is immediately apparent that the costalks on all of themaximal simplices are empty. This is a consequence of the definition: if σ is amaximal simplex of D ( X, Y, R ), then this means that for any y ∈ Y σ , y / ∈ Y τ for any strictly larger τ that contains σ . The presence of ∅ in various costalks is not aproblem for the morphism, since there always exists a unique function ∅ → A forany set A . Furthermore, even though in Example 16 the presence of empty sets inthe codomain caused a problem, they are benign in this case because the domaincostalks are also empty.A little inspection reveals that the costalks identify redundant simplices in D ( Y, X, R T ).Such a redundant simplex is generated by a row of the matrix for R that is a propersubset of some other row. This means that we can interpret the space of globalcosections of S (or T ) as being the collection of all redundant simplices – thosewhose corresponding elements of X can be removed without changing the Dowkercomplex. Acknowledgments
This material is based upon work supported by the Defense Advanced ResearchProjects Agency (DARPA) SafeDocs program under contract HR001119C0072.Any opinions, findings and conclusions or recommendations expressed in this ma-terial are those of the author and do not necessarily reflect the views of DARPA.
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Mathematics and Statistics, American University, Washington, DC, USA
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