(Co)homology theories for structured spaces arising from their corresponding poset
aa r X i v : . [ m a t h . A T ] A p r (Co)homology theories for structured spacesarising from their corresponding poset Manuel Norman
Abstract
In [1] we introduced the notion of ’structured space’, i.e. a space whichlocally resembles various algebraic structures. In [2] and [3] we studiedsome cohomology theories related to these space. In this paper we con-tinue in this direction: while in [2] we mainly focused on cohomologiesarising from f s and h , and in [3] we considered cohomologies for gener-alisations of objects which involved structured spaces, here we deal with(co)homologies coming from the poset associated to a structured spacevia an equivalence relation defined at the end of Section 4 in [1]. Moreprecisely, we will show that various (co)homologies for posets can also beapplied (under some assumptions) to structured spaces. The main idea of this paper is to assign to a structured space some cohomolo-gies which arise from the poset given by the relation ∼ in Section 4 of [1]. Wehave already found various cohomology theories related to these spaces, moreprecisely: • In [2] we have developed two cohomology theories starting from f s and h (these were suggested by the ”similarity” of sheaves, vector bundles and thetwo maps above, as explained in the cited article) • In [3] we have generalised various notions and we have developed the ’struc-tured versions’ of their cohomology theoriesWe now want to analyse structured spaces from another point of view: as wehave seen in Section 4 of [1], the map h is defined as follows (here x ∈ X , with( X, f s ) structured space): x h U p ∈ U : x ∈ U p } ⊆ U (1.1)where U is, as usual (see [1]), the domain of the structure map f s . The collectionof all the families of subcollection of U , excluding the empty collection, will be Author:
Manuel Norman ; email: [email protected]
AMS Subject Classification (2010) : 06A11, 55U10, 55N35
Key Words : structured space, poset, (co)homology L . In Section 4 of [1] we proved an important result onstructured spaces (see Theorem 4.1), which asserts that, if h : X → L satisfiessome properties, then there is a semilattice X/ ∼ (or a lattice) corresponding to X (we also proved the converse implication). Here, we will actually only needthe previous part: X/ ∼ is a poset (even if h is not surjective or meet) under ≤ , where we define: x ≤ y ⇔ h ( x ) ⊆ h ( y ) (1.2)and x ∼ y ⇔ h ( x ) = h ( y ) (1.3)(so ≤ is actually < ). Indeed, it is not difficult to see that ≤ is a partial orderon X/ ∼ . We will say that X/ ∼ is the ’poset corresponding/associated tothe structured space X ’. We will associate to X some cohomology theoriesrelated to its corresponding poset, and the obtained cohomologies will be called’ structured poset cohomologies ’. In each of the following sections, we will brieflyrecall some fundamental facts about the cohomologies for posets that we willneed to consider; then, we will relate them with the structured space X . We refer to Chapter 4 in [14] for simplicial homology. For simplicial (co)homologyof posets, we mainly refer to Section 1.5 and 1.6 of [9], and also to [17]. Briefly,given a poset P , we assign it its order complex ∆( P ), which is the abstractsimplicial complex defined as follows: • The vertices of ∆( P ) are the points belonging to P • The faces of ∆( P ) are the totally ordered subsets (usually called ’chains’) of P For instance, { x, y } is a face of ∆( P ), whichever are x, y ∈ P such that x ≤ y .The simplicial (co)homology of a poset P is the simplicial (co)homology of itsorder complex. This means that the chain complex we consider consists of the K -modules freely generated by n -chains of P , that is, C n ( P, K ) ( K is the ringof coefficients; we may take, for instance, K = Z ). We recall that an n -chain isthe following finite formal sum: s X i =1 a i σ i where a i ∈ K and σ i is an oriented n -simplex. The boundary ∂ n ( σ ) of anoriented n -simplex σ = ( t , t , ..., t n ) is defined as follows: ∂ n ( σ ) := n X i =0 ( − i ( t , ..., b t i , ..., t n )where as usual the cap b t i means that we delete the vertex t i (so, ( t , ..., b t i , ..., t n )is the i -th face of σ ). The map ∂ n : C n ( P, K ) → C n − ( P, K )2anuel Norman (Co)homology theories for ... from their corresponding posetis a homomorphism and ∂ n ◦ ∂ n +1 = 0. So we have a chain complex for the ordercomplex ∆( P ), and thus we have associated a homology theory to the poset P .As usual, the homology groups are: H n ( P, K ) = ker ∂ n / Im ∂ n +1 Then, a simple and usual way to obtain the cohomology groups is the applicationof the functor Hom( · , G ), for some K -module with identity G . See also [9] and[15]. Now we only have to apply this to our particular case P = X/ ∼ . Theorem 2.1.
Let X be a structured space, and define the equivalence relation ∼ as in (1.3) . Then, X/ ∼ is a partially ordered set (i.e. a poset) under ≤ (defined in (1.2) ). We can thus consider the order complex of this poset, ∆( X/ ∼ ) , and evaluate its simplicial (co)homology. The resulting (co)homologytheory is then called ’structured simplicial (co)homology’ of X . Remark 2.1.
We note that, in general, if there exists a maximal element inthe poset P , it is also useful to consider the Folkman complex of the poset,where such element is deleted (i.e. we consider P \
1, where 1 indicates themaximal element). Something similar can be done with the minimal element(usually denoted by 0). This ”cutting process” often simplifies the poset, andcould allow simpler evaluations of the (co)homologies. We refer to Chapter 4 in[18] and to Chapter 4.5 in [20] for more details.Moreover, we also notice that it is possible to obtain another (co)homologytheory for structured spaces which is again related to posets and simplicial(co)homology. More precisely, apply Theorem 2.1 in [22] to a structured space X (which is, by definition, a topological space): this (co)homology does notinvolve anymore ≤ and ∼ defined in (1.1) and (1.2); however, as explained in[22], the two (different) equivalence relations are, in some sense, similar in theirconstruction. Our main reference for this section is [11]. A poset P can be seen as a categorywhose objects are the elements of P and with the morphism x → y iff x ≤ y .Consider a presheaf F : P op → Ab (we may sometimes drop the notation P op ,writing more briefly P : this should not cause confusion, since the domain of thepresheaf certainly is the category P op , not the topological space itself). Thehigher limits of F are defined as the right derived functors of the limit: i lim ←− P := R i lim ←− P (3.1)We define the cohomology groups of P with values in F as follows: H p ( P, F ) := p lim ←− P F (3.2)3anuel Norman (Co)homology theories for ... from their corresponding posetThis clearly specialises to the case P = X/ ∼ .We now consider cellular cohomology. Again, our main reference is [11]. Letting x ≤ z in the poset P , if for any x ≤ y ≤ z we have either y = x or y = z , thenwe write x ≺ z . A poset P is graded if there is a function rk : P → Z such that( x < y means that x ≤ y and x = y ):(i) x < y ⇒ rk( x ) < rk( y )(ii) x ≺ y ⇒ rk( y ) = rk( x ) + 1Such a function is called a ’rank function’. If we suppose that some fixed rkis bounded above, say r := max P rk( x ), we can define the corank function | · | : P → Z by | x | := r − rk( x ). We now prove that, under some assumptions, P = X/ ∼ satisfies all the previous conditions. Proposition 3.1.
Let ( X, f s ) be a structured space and suppose that h is sur-jective onto L . Moreover, assume that U is finite (i.e. it consists of a finitenumber of sets). Then, the function rk( x ) := | h ( x ) | (the cardinality of thecollection, defined as for sets; see also Section 4 of [2]) is a rank function, r := max X/ ∼ rk( x ) = 2 |U| − , and thus X/ ∼ is a graded poset with a corankfunction.Proof. The fundamental aspect to keep in mind here is that h is surjective. Weimmediately notice that, because of the equivalence relation ∼ defined in (1.3), x ≤ y is actually always a strict inequality, i.e. we always have x < y . Thisimplies that, if x < y , then h ( x ) ( h ( y ), and thus rk( x ) < rk( y ). Furthermore,since h is surjective, we can say that:1) If x ≤ y ≤ z implies either y = x or y = z , then either h ( x ) = h ( y ) ⊆ h ( z ) or h ( x ) ⊆ h ( y ) = h ( z ).2) By surjectivity, we know that the cardinality of h does not ”jump” any value,i.e. if there are x , x such that | h ( x ) | = n and | h ( x ) | = t (say, with n ≤ t ; theother case can be treated analogously) for some natural numbers n, k (whichare certainly = 0), then there must exist some points y , y , ..., y k such that | h ( y ) | = n + 1, | h ( y ) | = n + 2, ..., | h ( y k ) | = t − x ≺ y ⇒ rk( x ) =rk( y ) + 1. Indeed, suppose that this does not hold. It is clear that we then haverk( y ) > rk( x ) + 1 (by definition of ≤ , and by definition of cardinality). But thisimplies that there exists at least one point a such that rk( x ) < rk( a ) < rk( y ).Consequently, it is not anymore true that whenever x ≤ z ≤ y implies either x = z or z = y , because z = a is a counterexample. This is absurd, and thusthe statement above holds.This rank function is bounded above, because U is finite and we know that(as noted in Section 4 of [1]) L is the ”power collection” without the emptycollection, which clearly implies that the maximum cardinality is |U| −
1. Wetherefore conclude that X/ ∼ is graded and it has a corank function.Thanks to this result, we know that, under its hypothesis, we can applythe theory in Section 2 of [11]. We first need to recall some other facts and Since h is surjective, there is at least one x such that h ( x ) = U . P bythe corank defined on it are: P k := { x ∈ P : | x | ≤ k } (3.3)with k ∈ N . Consequently, we have P ⊂ P ⊂ P ⊂ ... , and a presheaf F on P can be defined on each P k thanks to the inclusions P k ֒ → P . Now, it ispossible to compute the higher limits defined by (3.1) via some groups, whichwill be indicated by HS p . A description of this computational method can befound, for instance, in Section 1.1 of [11]. It can be proved that ( ∼ = means’isomorphic’): p lim ←− P F ∼ = HS p ( P, F ) (3.4)for all p . We can also define (see Section 1.4 in [11]) a relative cohomology HS p ( P , P , F ). This allows us to finally define the cellular cochain complexassociated to a poset P . The groups in the cochain are: C n ( P, F ) := HS n ( P n , P n − , F ) (3.5)while the coboundary maps δ n are defined via Lemma 4 in [11]; see also equation(9) in the cited paper. We can thus conclude with the follwing important: Theorem 3.1.
Let ( X, f s ) be a structured space and consider a presheaf F onthe poset X/ ∼ . Then, the higher limits defined in (3.1) allows us to definethe cohomology of X/ ∼ with values in the presheaf F via (3.2) . Moreover, if h : X → L is surjective and the domain U of f s is finite (i.e. it contains a finitenumber of sets), X/ ∼ is a graded poset with corank, with the rank defined by rk( x ) := | h ( x ) | (the cardinality of the collection). It is then possible to definealso the cellular cohomology of X/ ∼ , see (3.5) and the comments below it. Allthese cohomologies are called ’structured cohomologies with values in a presheaf ’of X . Remark 3.1.
The surjectivity of h implies the existence of at least one element x ∈ P such that h ( x ) = U . This implies that there exists at least one maximalelement, which is unique when T U p ∈U U p has one and only one element (see alsothe next Section). Thus, it can be useful to consider also the Folkman complex(see Remark 2.1). See also Section 4.2 in [11] and [21]. In this section we consider a (co)homology for coloured posets. Our main ref-erence here is [10]. By Definition 1 in the cited paper, we know that a colouredposet is a couple ( P, F ), where P is a poset with a unique maximal element(i.e. there exists an element e x which is not smaller than any other element in P ;moreover, this element is required to be unique: the possibility of having morethan one maximal element is due to the partial order), and F : P → Mod R (the category of R -modules, for some unital ring R ) is a covariant functor. F Proposition 4.1.
Let ( X, f s ) be a structured space and suppose that \ U p ∈U U p = { e x } (precisely one element). Then, e x is the unique maximal element of the poset X/ ∼ (under ≤ ).Proof. Since \ U p ∈U U p = { e x } we know that h ( e x ) = U (because it intersects all the fixed neighborhoods).Moreover, it is clear that there is no other element in X/ ∼ such that thishappens. The fact that h ( y ) ( h ( e x ) ∀ y ∈ X/ ∼ then implies that e x is amaximal element, and the uniqueness follows from the observation above.Of course, there are also other cases for which a unique maximal elementexists. However, here we will mainly consider the above one. We can thusdefine: Definition 4.1.
Given a structured space ( X, f s ) , the couple ( X/ ∼ , F ) , where F : X/ ∼ → Mod R is a covariant functor, and X/ ∼ has a unique maximalelement, is a structured coloured poset. A ’maximal structured coloured poset’is a couple ( X/ ∼ , F ) where \ U p ∈U U p = { e x } which is a particular case of structured coloured poset by Proposition 4.1. The name ’maximal’ comes from the fact that | h ( e x ) | = |U| has the maximumcardinality among all the collections h ( y ) for y ∈ X . The unique maximalelement e x of a structured coloured poset will be indicated, as usual in thiscontext, by 1.Following [10], if ( P, F ) is a coloured poset, we can define the chain complex( n > S n ( P, F ) := M x x ··· x n , x i ∈ P \ F ( x ) (4.1)and S ( P, F ) := F (1), where x = x x · · · x n means x ≤ x ≤ ... ≤ x n . For n >
0, an element of the chain complex can be written as: X x λ · x x of lenght n , and λ ∈ F ( x ). The differ-entials d n : S n ( P, F ) → S n − ( P, F ) are defined as follows ( n > d n ( λx x · · · x n ) := F x x ( λ ) x · · · x n − n X i =2 ( − i λx · · · b x i · · · x n (4.2)and d ( λx ) := F x ( λ ). Something similar can be done using strict inequalities < instead of ≤ . The resulting complex is denoted by C n ( P, F ). In [10] variousresults involving also this complex are obtained. Here, our main interest was toshow that this construction also applies to some structured spaces, so we willnot go deeper. We can define a cohomology via the usual application of thefunctor Hom( · , R ) (where R is the same as before). We then have: Theorem 4.1.
Let ( X, f s ) be a structured space. Suppose that X/ ∼ has aunique maximal element, indicated by (see, for instance, maximal structuredcoloured posets) and let F : X/ ∼ → Mod R be a covariant functor ( R is someunital ring). Then, ( X, F ) is a structured coloured poset to which we can asso-ciate a coloured homology theory (see (4.1) and (4.2) with P = X/ ∼ ) and acohomology theory obtained by applying to the homology, as usual, the functor Hom( · , R ) . We conclude this paper with two (co)homologies for a particular kind of poset-stratified spaces, called ’structured stratified spaces’. Recall that (see, for in-stance, [25-30]) a poset-stratified space is a structure consisting of a topologicalspace X , a poset P with the Alexandroff topology and a continuous surjection s : X → P . Our previous discussions suggest an interesting case from the pointof view of the theory of structured spaces: to consider a structured space X andthe corresponding poset X/ ∼ endowed with the Alexandroff topology. Definition 5.1.
A structured stratified space consists of a structured space X endowed with the topology in Proposition 1.1 in [1] (more precisely, the one con-structed in Example 1.1 of such paper), its corresponding poset X/ ∼ endowedwith the Alexandroff topology, and a continuous surjection s : X → X/ ∼ . We can apply the theory of stratified spaces to structured spaces thanks tothe above definition; this gives another tool to study these spaces. We brieflysketch some possible directions to do this, even though we will soon return tothe main topic of this paper. Following [25], it is clear (by Definition 4.2.1) thatstructured stratified spaces are indeed a particular case of stratified spaces. ByConstruction 4.2.3, we can endow the structured space X with a preorder givenby the considered map s as follows: x ≤ s y ⇔ sx ≤ sy in X/ ∼ ≤ s | • on the preorderedspace X via restrictions. This implies that ( X, ≤ s | • ) is a prestream (see Defini-tion 5.1.1). We will call it the ’prestream associated to the structured stratifiedspace’. It is also possible to consider other precirculations for structured strat-ified spaces, which could turn out to make the prestream into a stream. Thereare various possible definitions of this concept: Definition 5.1.14 involves Hau-court streams (see also [27]); another definition, which can be found in Remark5.1.19, is given by Krishnan in [28]. We can also define the ’d-space associatedto a structured stratified space’, which is the d-space ( X, d ≤ s | • X ) defined as in5.1.11 (see 5.1.10 for the definition of d-space).We now return to our main topic: we first give a ”standard” example of struc-tured stratified space. In fact, the following example will be called ’standardstructured stratified space associated to a structured space’, and we will alwaysrefer to it if not specified otherwise. Proposition 5.1.
Let X be a structured space endowed with the topology inProposition 1.1 (more precisely, Example 1.1) of [1], and let X/ ∼ be its corre-sponding poset endowed with the Alexandroff topology. Moreover, suppose that sup x ∈ X | h ( x ) | < + ∞ . Then, X , X/ ∼ and the map s : X → X/ ∼ given by s ( x ) := [ x ] form a structured stratified space.Proof. We only need to verify that s is continuous and surjective. Surjectivity isclear, since all the elements [ x ] in X/ ∼ are reached at least by x ∈ X . In orderto prove continuity, recall that (see, for instance, Definition 4.1.13 in [25]), bydefinition of Alexandroff topology, if U ⊆ X/ ∼ is open, then whenever p ∈ U and t ∈ X/ ∼ , with p ≤ t , we have t ∈ U . Now consider some open U in X/ ∼ ;we have s − ( U ) = { x ∈ X : s ( x ) ∈ U } . By definition of the equivalence classesin X/ ∼ , we know that all the elements x ∈ X such that s ( x ) = [ x ] satisfy thefollowing property: h ( x ) = { U p } for some U p ∈U for some fixed U p ’s in U (they are precisely the same for all these x , and no other y ∈ X is such that h ( y ) is equal to the above collection). This means that all thepoints x with the same equivalence class form a set A = T p U p \ ( S t U t ), wherethe intersection is over the above U p ’s, while the U t ’s are sets not belongingto h ( x ) but which intersect with the set T p U p . However, since an open set U ⊆ X/ ∼ satisfies the property previously noticed (thanks to Alexandrofftopology), we know that s − ( U ) will be precisely the union of some intersectionsas the above one, because the gaps due to the differences of sets will be filledby the points y such that x ≤ y (which belong to U ). Consequently, s − ( U )is a union of open sets, since (under the assumption that the number of opensets in all the considered interesections is finite) the intersections are open. TheProposition follows.Now, following [26], to each structured stratified space X we can associate asimplicial set, called ’stratified singular simplicial set of X ’. This set is denoted The cardinality of a collection of sets is defined similarly to sets; see, for instance, Section4 in [2]. SS ( X ); see Definition 7.1.0.3. We can evaluate the homology of this simpli-cial set and associate it to the structured stratified space (also recall that thishomology is isomorphic to the singular homology of the geometric realisationof the simplicial set). More precisely, we can associate to a simplicial set Y thechain: C n = Z [ Y n ](the free abelian group on Y n ) and the boundary maps: X i ( − i d i : C n → C n − (the alternating sum of the face maps). Another possible homology can beobtained choosing some simplicial set A and considering the homology of thesimplicial set sset ( A, SS ( X )) ( X structured stratified space); see also Lemma7.3.0.1. We can obtain cohomologies in the usual way. We thus conclude,summing up what we have said up to now: Theorem 5.1 ((Co)homology for structured stratified spaces) . Let ( X, X s −→ X/ ∼ ) be a structured stratified space. We can define at least two (co)homologiesfor such a space: one is given by the (co)homology of the simplicial set SS ( X ) ,while another one is given by the (co)homology of the simplicial set sset ( A, SS ( X )) (after choosing some simplicial set A ). Everything clearly specialises to the particular case of the standard struc-tured stratified space: