Extra-fine sheaves and interaction decompositions
Daniel Bennequin, Olivier Peltre, Grégoire Sergeant-Perthuis, Juan Pablo Vigneaux
aa r X i v : . [ m a t h . A T ] S e p EXTRA-FINE SHEAVES AND INTERACTIONDECOMPOSITIONS
DANIEL BENNEQUIN, OLIVIER PELTRE, GR´EGOIRE SERGEANT-PERTHUIS,AND JUAN PABLO VIGNEAUX
Abstract.
We introduce an original notion of extra-fine sheaf on a topologicalspace, for which ˇCech cohomology in strictly positive dimension vanishes. Weprovide a characterization of such sheaves when the topological space is apartially ordered set (poset) equipped with the Alexandrov topology. Thenwe further specialize our results to some sheaves of vector spaces and injectivemaps, where extra-fineness is (essentially) equivalent to the decomposition ofthe sheaf into a direct sum of subfunctors, known as interaction decomposition,and can be expressed by a sum-intersection condition. We use these results tocompute the dimension of the space of global sections when the presheaves arefreely generated over a functor of sets, generalizing classical counting formulaefor the number of solutions of the linearized marginal problem (Kellerer andMat´uˇs). We finish with a comparison theorem between the ˇCech cohomologyassociated to a covering and the topos cohomology of the poset with coefficientsin the presheaf, which is also the cohomology of a cosimplicial local system overthe nerve of the poset. For that, we give a detailed treatment of cosimpliciallocal systems on simplicial sets. The appendixes present presheaves, sheavesand ˇCech cohomology, and their application to the marginal problem.
Contents
1. Introduction 12. Fine, extra-fine, super-local and acyclic 53. Alexandrov topologies and sheaves 94. Interaction decomposition 125. Factorization of free sheaves 186. Nerves of categories and nerves of coverings 23Appendix A. Topology and sheaves 37Appendix B. ˇCech cohomology 39Appendix C. Finite probability functors 40References 441.
Introduction
This article develops cohomological tools to study collections of data associatedto hypergraphs, or to more general partially ordered sets (posets). The kind of datawe will consider is organized in families of sets indexed by the elements of the poset,forming covariant and contravariant functors with respect to the partial ordering,
Mathematics Subject Classification.
Primary 14F05, 55N30, 55U10 Secondary 06A11,60B99 . which are called respectively copresheaves and presheaves over the poset. Suchfunctors have been applied to several problems at the crossroad of data analysis,information theory, coding theory, logic, computation, and bayesian learning. Wewill mention below some of these problems and develop several applications of thecohomological approach.In this work, we see a partially ordered set (poset) A as a small category suchthat:(1) there is at most one morphism between two objects;(2) if a → b and b → a , then a = b .An hypergraph is a particular case of poset, whose objects are some finite subsets ofan index set I , and there exist a morphism S → S ′ whenever S ′ ⊆ S . An abstractsimplicial complex is an hypergraph K that satisfies an additional property: if S belongs to K , then every subset of S belongs to K too.A presheaf on a category A is a contravariant functor F from A to the cate-gory of sets S , in other terms it is a covariant functor on the opposite category F : A op → S . A copresheaf is just a covariant functor F : A → S . The presheavesof classical sheaf theory on topological spaces [15] are obtained when A is the cat-egory of open sets of some topological space, which is an example of poset.Abstract simplicial complexes play a prominent role in persistent homology [8,14], a technique to extract topological features that is a cornerstone of appliedalgebraic topology. The basic idea is to replace a sequence of data points in ametric space by an abstract simplicial complex induced by a proximity parameter(e.g. the ˇCech complex or the Vietoris-Rips complex). Then homological tools(spectral sequences) are applied to an increasing family of complexes for defininginvariant quantities of the data.Curry’s dissertation [10] showed that persistent homology can be extended inseveral directions involving sheaves on posets of parameters.Curry [10] also gave a systematic treatment of sheaves defined on another kindof complexes, the cellular complexes (giving cellular sheaves and cosheaves), whichhe traces back to Zeeman’s Ph.D. thesis [47]. A spectral theory of such sheaveswas later developed by Hansen and Ghrist [18]. Those works list several situationsthat can be modeled by cellular (co)sheaves, which include network coding, sensornetworks, distributed consensus, flocking, synchronization and opinion dynamics,among other things.Along similar lines, a series of works by Robinson and collaborators [29, 31,30] argued that sheaves are a canonical model for the integration of informationprovided by interconnected sensors. In those works, the vertices of an abstractsimplicial complex represent heterogeneous data sources and the abstract simplexessome sort of interaction between these sources. It is claimed that sheaves constitutea canonical data structure if one requires sufficient generality to represent all sensorsof interest and the ability to summarize information faithfully. A similar approachis taken by Mansourbeigi in his doctoral dissertation [23].Independently, Abramsky and his collaborators (see e.g. [2, 1]) have used sheavesand cosheaves on simplicial complexes to study contextuality . In this situation, the XTRA-FINE SHEAVES AND INTERACTION DECOMPOSITIONS 3 vertices represent observables, the simplices represent joint measurements (mea-surement contexts) and the maximal faces of the complex are called maximal con-texts . The functor associates to each context a set of possible outcomes or a set ofprobabilities on those outcomes.
Contextuality refers to the fact that it can happenthat some sections of the probability functor (i.e. coherent collections of “local”probabilities) are not compatible with a globally defined probability law. In thisarticle, we refer to this problem as the probabilistic marginal problem . There arealso linearized versions of this problem, as well as ”possibilistic” versions.In all these examples, homology and cohomology is used to determine the ”shape”of the simplicial complex or the relevant geometrical invariants of the associatedsheaves.Simplicial complexes are particularly convenient because they have a geometricrealization as CW-complexes, so they can be studied using standard tools in al-gebraic topology e.g. standard homology and cohomology theories. Hypergraphswere introduced in combinatorics, not in geometry, hence their geometrical study isless straightforward. There have been several proposals to define (co)homologicalinvariants of hypergraphs. A recent paper by Bressan, Li, Ren and Wu [7] definesthe embedded homology of an hypergraph H , which equals the homology of thesmallest abstract simplicial complex that contains H . A specific cohomology of k -regular hypergraphs (i.e. containing only subsets of cardinality k ) was introducedby Chung and Graham [9] motivated by some problems in combinatorics.The present article develops an alternative approach, based on sheaf theory andsimplicial methods. We equip the poset A with the lower or upper Alexandrofftopology (see Section 3), obtaining the topological space X A or X A , respectively.There is a bijection between covariant (resp. contravariant) set-valued functors on A and sheaves on X A (resp. X A ) i.e.(1.1) [ A , S ] ∼ = Sh( X A ) , [ A op , S ] ∼ = Sh( X A ) . In other words, we can see a (co)presheaf on A as a usual sheaf on a topologicalspace, where ˇCech cohomology can be used. This cohomology is convenient froma computational viewpoint and well adapted to study the global sections of thesheaf.The article presents and studies in detail several equivalent definitions of thiscohomology, from simplicial methods involving nerves of categories, and from topostheory (i.e. derived algebra and geometry), all presenting a particular interest forsome specific problem.Here, we are particularly interested in the following setting, which is adapted toa wide variety of problems, as mentioned above. One introduces an hypergraph A with vertex set I . The elements of I represent elementary observables or sources,and the elements α of A represent interactions or joint measurements. To takeinto account the internal degrees of freedom of each object of A , one introduces acovariant set-valued functor E : A → S of possible outcomes, associating to eachobject α of A a set E α , and to each arrow α → β a surjective map E α → E β .The local probabilities on each E α or the functions over each E α give rise to otherimportant functors, that can be covariant or contravariant. For instance, the vectorspaces { V α } α ∈A of numerical functions on the sets E α , and the inclusion j αβ : V β → V α form a contravariant functor (an example of an injective presheaf , see Section4). D. BENNEQUIN, O. PELTRE, G. SERGEANT-PERTHUIS, AND J.P. VIGNEAUX
In particular, the study of the special case of real-valued functions of the prob-ability laws on finite sets E α over a simplicial complex A gives a natural interpre-tation in terms of topos theory and cohomology [3] of the information quantitiesdefined by Shannon and Kullback, or by Von Neumann in the quantum case, cf.[6, 41]. These results were later extended to presheaves of functions of statisticalfrequencies, and to gaussian laws in Euclidean space [42]. The cohomologies whichwere used here are not of the type of ˇCech, they are based on the action of variableson probabilities by conditioning, expressed as non-trivial modules in the topos ofpresheaves over A . A conjecture is that computing cohomology in degrees higherthan one will give entirely new information quantities.Furthermore, there exist several notions of morphisms from a module ( A ; V )to a module ( B ; W ). A natural hope is the existence of convenient categories ofsheaves on hypergraphs that would be suitable to construct a new kind of geomet-ric topology and homotopy theory in this setting. A similar approach was takenby Friedman in the case of graphs in order to prove Hanna Neumann’s conjecture[13]: the category of directed graphs over a given directed graph G can be faith-fully embedded in Sh( G ), but such embedding is not full and the kernels of somenew morphisms ( ρ -kernels) play a fundamental role in the proof (which essentiallyreduces the problem to the vanishing of the homology of those kernels).We also expect that sheaf-theoretic constructions on hypergraphs will give a bet-ter understanding of certain algorithms in Statistics or Machine learning. In thisdirection, Olivier Peltre (cf. [27] and his thesis [28]) has developed a cohomologicalunderstanding of the Belief Propagation Algorithm (in the generalized version of[46]); the algorithm appears as a non-linear dispersion flow. Higher dimensionalanalogs are promising tools. Gr´egoire Sergeant-Perthuis (cf. [34, 35, 36, 37] andhis forthcoming thesis) focussed on defining the thermodynamical limit in the cate-gory of Markov Kernels, extending several constructions of statistics and statisticalphysics such as the decomposition into interaction subspaces, first introduced forfactor spaces [20, 40], the space of Hamiltonians, infinite-volume Gibbs state, andthe renormalisation group.In both these works, a same result appears: the vanishing of sheaf cohomology(in the toposic form, or in ˇCech form respectively) in degree larger than one (i.e.acyclicity, without contractility) for the case of an injective presheaf V over A ,under a certain condition relating the intersections and the sums of the subspacesgiven by the faces: the condition G in Section 4. The main goal of this article is toenunciate and prove this result, and to place it in a topological context.The condition G is satisfied for hypergraphs and freely generated modules whenthe hypergaph is closed by intersection. As a corollary we get a proof by homologyof the Marginal Theorem of [19], showing the existence of perhaps non-positivemeasures on the joint measurement represent by the index set I , with prescribedcompatible marginals along the arrows of A , and computing their dimensions withthe M¨obius function of A , cf. also [24]. Remark that [43] showed that the neces-sary and sufficient condition for extension by positive measures of every collectionof compatible marginals is the regularity of the complex A , which is a convenientnotion of contractility in this context (cf. [28]).Section 2 defines an original notion of extra-fine presheaf over a topological space X , that is reminiscent of the classical notion of fine sheaf. Then we prove, as it was XTRA-FINE SHEAVES AND INTERACTION DECOMPOSITIONS 5 the case for fine presheaves on paracompact spaces, that extra-fine implies acyclic for the ˇCech cohomology (Theorem 1) on any topological space. In Section 3 wecharacterize extra-fine sheaves on the Alexandrov spaces X A or X A by the propertyof interaction decomposition .In Section 4 we consider presheaves V of K vector spaces (over any field K )and injective maps. For such presheaves, we give an alternative characterizationof extra-fineness through the sum-intersection condition G; this is the main resultof the article (Theorem 4.4). We do that without any finiteness condition on thevector spaces, and only weak finiteness conditions for the poset. Then we studyduality, proving the acyclicity theorem for the weak dual cosheaves.Section 5 contains the definition of free presheaves generated by a covariantset-valued functor E over a commutative field K (the usual case in data analysisover hypergraphs). We establish the condition G for the injective presheaf V offunctions from E to K , when the poset A has conditional coproducts (meaningstable by non-empty intersections in the case of hypergraphs), then its acyclicity(Theorem 5.7). The acyclicity is deduced for the sheaf generated by E over K on X A . Then we compute the cohomology on hypergraphs (Theorem 5.10): it isthe sum of the ordinary cohomology of A in all degrees, and of the cohomology ofdegree zero of a restricted sheaf of functions (where the sum of coordinates is zero).We also prove a version of the marginal theorem (surjection in ˇCech cohomology,Theorem 5.15), which seems to be new in this generality. We deduce an indextheorem for the Euler characteritic of the marginal sheaves (Theorem 5.16).Finally, Section 6 comes back to a general topological space X and preshaves ofabelian groups, to provide the homotopy equivalence of the ˇCech cochain complexof an open covering of a presheaf with the cochain complex of the nerve of thecategory generated by the covering; this is done for a general notion of cosimplicialcoefficients (Theorem 6.31). This answers a natural question in our framework, butthe proof is surprisingly cumbersome, which is reminiscent of the known fact thatthere exists an homotopy equivalence between a finite simplicial complex and itsbarycentric subdivision but non-canonically.In all the above sections we take care of morphisms between presheaves andnaturality behaviors, or functoriality.Three appendices are added at the end, where we summarize the main objectsand constructions involved in the article: sheaves, ˇCech cohomology, and M¨obiusfunctions, among other things.2. Fine, extra-fine, super-local and acyclic
In this section, we consider presheaves of abelian groups over a topological space X . See Appendix A for some basic topological definitions and notations. Weuse ˇCech cohomology as presented in any standard reference, e.g. [39], but allrelevant definitions can also be found in Sections 6.1 and 6.2 under the formalismof cosimplicial local systems. Definition 2.1 ( Fine presheaf , cf. [39, Sec. 6.8]) . A presheaf F of abelian groupsover a topological space X is said to be fine if for every open covering U of X ,there exists a family { e V } V ∈U of endomorphisms of F (i.e. natural transformations e V : F → F , whose components e V ( W ) we denote by e V | W ), such that: D. BENNEQUIN, O. PELTRE, G. SERGEANT-PERTHUIS, AND J.P. VIGNEAUX (i) For all V ∈ U and every open set W , one has e V | W | ( W \ ¯ V ) = 0. (ii) For every open W and every x ∈ F ( W ), there exists only a finite number ofelements V ∈ U such that e V | W ( x ) = 0, and we have x = P V ∈U e V | W ( x ).Under the second condition, the family { e V } V ∈U is named a partition of unity (or partition of identity ) adapted to U . If the two conditions are satisfied we say thatthe partition of unity is supported by U .Fine presheaves are part of the classical literature on sheaf theory, see also [15,Sec. 3.7] and [16, p. 42], although the classical definitions require U to be locallyfinite. Positive dimensional cohomology of a paracompact topological space withcoefficients in a fine presheaf vanishes [39, Thm. 6.8.4], and this fact has importantimplications in the comparison of Alexander and ˇCech cohomology. We proposehere a specialization of this notion that plays a fundamental role in our investiga-tions. Definition 2.2 ( Extra-fine presheaf ) . A presheaf F of abelian groups over thetopological space X is said to be extra-fine if for every open covering V of X , thereexists a finer open covering U and a partition of unity { e V } V ∈U adapted to U (i.e.2.1-(ii) is satisfied), such that(i’) for all V ∈ U and W ∈ U , e V | W = 0 implies W ⊆ V ;(iii) for all V, W ∈ U such that V = W, e V ◦ e W = e W ◦ e V = 0. Lemma 2.3.
If a partition of unity satisfies condition (iii), then for all V ∈ U theequality e V ◦ e V = e V holds.Proof. For any s V ∈ F ( V ), we have a finite decomposition s V = P W e W ( s V ), then e V ( s V ) = P W e V ◦ e W ( s V ). Therefore e V ( s V ) = e V ◦ e V ( s V ), because all the otherelements in the sum are zero. (cid:3) Thus a partition of unity { e V } V ∈U that satisfies (iii) is a family of projections,decomposing the presheaf F is a direct sum; we refer to this as a local orthogonaldecomposition of the functor. If (i’) is also satisfied, we speak of a super-localorthogonal decomposition .The condition (i’) for a partition of unity is named super-locality ; it is certainlyexceptional for usual topologies, but useful for the particular topologies we areinterested in in this text.When a presheaf admits a partition of unity satisfying (i’) in addition to (ii),but perhaps not (iii), we say that the presheaf F is fine and super-local . Proposition 2.4. If F is a presheaf of abelian groups on X , then it is is fineif, for any open covering V of X , there exists a finer covering U , and a partitionof unity { e U } U ∈U of endomorphisms of F supported by U (i.e. satisfying (i) and(ii)). Moreover, every open covering V admits a partition of unity of F which isorthogonal as soon as it is true for the finer covering U .Proof. Suppose given a partition of unity { e U } U ∈U (resp. an orthogonal decom-position) and an arbitrary open cover V of X coarser than U ; one can build apartition of unity { e V } V ∈V subordinated to V (resp. an orthogonal decomposition)as follows. Here ·| ( W \ ¯ V ) denotes postcomposition e V | W ◦ F ( ι ) with the map F ( ι ) : F ( W \ ¯ V ) → F ( W )induced by the inclusion ι : W \ ¯ V ֒ → W . XTRA-FINE SHEAVES AND INTERACTION DECOMPOSITIONS 7
For each U ∈ U , we choose an element V ( U ) of V such that U ⊆ V ( U ). For each V ∈ V , let A V be the set of U in U such that V ( U ) = V . The subsets { A V } V ∈V are two by two disjoint and their union is U . Define e V = P U ∈ A V e U .Let us show that the resulting { e V } V ∈V form a partition of the unity (Defi-nition 2.1). Let W be an open set in X , and v ∈ F ( W ). Let A ( v ) be the setof U ∈ U such that e U | W ( v ) = 0; by hypothesis, this set is finite, and we have v = P U ∈ A ( v ) e U | W ( v ). Since the sets { A V } V ∈V partition U , the set(2.1) V ( v ) = { V ∈ V | A V ∩ A ( v ) = ∅} is also finite. Now, if an element V in V is such that e V | W ( v ) = 0, the correspondingset A V is non-empty, then V = V ( U ) for some U ∈ U , and A V contains at leastone U such that e U | W ( v ) = 0, thus V belongs to V ( v ). This shows that the axiom(ii) of 2.1 is satisfied. Moreover, if V ∈ V and W is open, consider the open set W ′ = W \ ( W ∩ V ). For any U ⊆ V , we have W ′ ⊆ W \ U . But for every U ∈ U , wehave e U ‘ | W | W \ U = 0, then for each U ∈ A V , by naturality of e U , we have(2.2) e U | W | W ′ = 0 . This proves the condition ( i ).If the decomposition e U ; U ∈ U is orthogonal, for two different elements V, W of V , the sets A V an A W are disjoint, then the above definition of e V and e W showsthat e V ◦ e W = 0. (cid:3) The proposition above does not extend to the super-locality; this property cannotbe transferred to coarser coverings. From this result, we see that a fine presheaf canbe defined analogously to an extra-fine presheaf , by the existence of a finer coveringwhich supports a partition of unity. But extra-fine presheaves cannot be definedon the model of fine presheaves i.e. by the existence of an adapted super-localpartition of identity for every open covering.
Remark . Let f : X → Y be a continuous map and F a fine presheaf of abelian groups over X , the presheaf G = f ∗ F on Y is fine [39,Thm. 6.8.3], but it can happen that F is extra-fine on X and that G = f ∗ F is notextra-fine. The problematic property is super-locality. For the inverse image of apresheaf G over Y , both fine and extra-fine fail to be transmitted from G to f − G .We shall see that positive dimensional ˇCech cohomology of a super-local presheafvanishes. To fix some notations, we summarize here the construction of ˇCech co-homology; more details can be found in Sections 6.1 and 6.2.Let U be an open covering of a topological space X , and for each n ∈ N , let K n ( U )denote the set of sequences of length n + 1, u = ( U , ..., U n ), of elements of U suchthat the intersection U u = U ∩ ... ∩ U n is non-empty. For n ∈ N , a ˇCech cochain of F of degree n with respect to U is a element { c ( u ) } u ∈ K n ( U ) of Q u ∈ K n ( U ) F ( U u ).The set of n -cochains is denoted C n ( U ; F ); it is an abelian group.A coboundary operator δ : C n ( U ; F ) → C n +1 ( U ; F ) is then introduced, as alinear map such that(2.3) ( δc )( U , ..., U n +1 ) = i = n +1 X i =0 ( − i c ( U , ..., c U i , ..., U n +1 ) | U ∩ ... ∩ U n +1 , where c U i means that U i is omitted. When we want to be more precise we write δ = δ n +1 n at degree n . D. BENNEQUIN, O. PELTRE, G. SERGEANT-PERTHUIS, AND J.P. VIGNEAUX
It is well known that δ ◦ δ = 0, which allows one to define the ˇCech co-homology of F over U in degree n as the quotient abelian group H n ( U ; F ) =ker( δ n +1 n ) / im( δ nn − ). As explained in Appendix B, the set of open coverings of X with the relation of refinement is a directed set . And the ˇCech cohomology of F over X is defined as(2.4) ∀ n ∈ N , H n ( X ; F ) = lim −→ H n ( U ; F ) . See [15, Ch. 5], [39, Sec. 6.7.11] or [16, Sec. 0.3].From the definition of δ , it is clear that the group H ( U ; F ) can be identifiedwith the group of global sections of F over X , for any open covering U . Hence H ( X ; F ) coincides with every H ( U ; F ) and also corresponds to global sections.A presheaf is called acyclic if its cohomology is zero for every degree n ≥ Theorem 2.6.
A presheaf F of abelian groups which is fine and super-local isacyclic. More precisely, for every open covering V , and every integer n ≥ , thereexists an open covering U finer than V such that the cohomology group H n ( U ; F ) is zero.Proof. We adapt a more elaborate argument given by Spanier in the case of para-compact spaces [39, Thm. 6.8.4].Given V , we take for U the covering which satisfies (i’), the condition of super-locality.Consider a cochain ψ for U and F of degree q ≥
1, which is a cocycle i.e. δψ = 0.Then for every collection U , U , ..., U q , U q +1 of elements of U , we have(2.5) ψ ( U , ..., U q +1 ) | U ∩ ... ∩ U q +1 = q +1 X k =1 ( − k +1 ψ ( U , ..., c U k , ..., U q +1 ) | U ∩ ... ∩ U q +1 . Set U = U . We deduce from (2.5) that when U contains U ∩ ... ∩ U q +1 ,(2.6) e U ψ ( U , ..., U q +1 ) | U ∩ ... ∩ U q +1 = q +1 X k =1 ( − k +1 e U ψ ( U, U , ..., c U k , ..., U q +1 ) | U ∩ ... ∩ U q +1 , and when U does not contain U ∩ ... ∩ U q +1 , the super-locality implies that(2.7) e U ψ ( U , ..., U q +1 ) = 0 . For any U ∈ U , we define a ( q − φ U for F and the covering U as follows:given V , ..., V q − ∈ U , if V ∩ ... ∩ V q − ⊆ U then(2.8) φ U ( V , ..., V q − ) = e V ( ψ ( U, V , ..., V q − ) | V ∩ ... ∩ V q − ) , and if V ∩ ... ∩ V q − " U then(2.9) φ U ( V , ..., V q − ) = 0 . By definition of the coboundary operator, in both cases we have(2.10) ( δφ U )( U , ..., U q +1 ) = q +1 X k =1 ( − k +1 φ U ( U , ..., c U k , ..., U q +1 ) | U ∩ ... ∩ U q +1 ; XTRA-FINE SHEAVES AND INTERACTION DECOMPOSITIONS 9 which gives, when U contains U ∩ ... ∩ U q +1 ,(2.11)( δφ V )( U , ..., U q +1 ) = q +1 X k =1 ( − k +1 e V ( ψ ( U, U , ..., c U k , ..., U q +1 ) | U ∩ ... ∩ U q +1 ) , and, when U does not contain U ∩ ... ∩ U q +1 , gives ( δφ V )( U , ..., U q +1 ) = 0.Consequently, in any case we get(2.12) δφ U ( U , ..., U q +1 ) = e V ( ψ ( U , ..., U q +1 )) . Then we define φ by summing over the open sets U in U , and using (ii), we obtain δφ = ψ , which proves the theorem. (cid:3) Alexandrov topologies and sheaves
Basic definitions. A partially ordered set (poset) is set with a binary relation ≤ that is reflexive, antisymmetric and transitive. Equivalently, it is a small category A such that:(1) for any pair of objects α , β , there is at most one morphism from α to β ,and(2) if there is a morphism from α to β and a morphism from β to α , then α = β .Starting with a partially ordered set Ob A , there exists an arrow α → β if and onlyif β ≤ α . This convention is chosen to agree with other studies on categories ofrandom variables, such that arrows are in the sense of fine to coarse, cf. [41]. Acovariant functor between two posets is simply a monotone map. We write α ∈ A instead of α ∈ Ob A if there is no risk of ambiguity.The categorical coproduct between two objects α and α ′ of A is an object β such that α → β and α ′ → β , that additionally satisfies the following property:for any ω ∈ A , if α → ω and α ′ → ω , then β → ω . Such β is denoted α ∨ α ′ and called coproduct (or sup ) of α and α ′ ; it is unique. We shall not suppose thatour categories have all finite coproducts, but sometimes we impose the following conditional existence of coproducts : for any α, α ′ ∈ A , if there exists ω ∈ A suchthat α → ω and α ′ → ω , then α ∨ α ′ exists.The dual notion is the product α ∧ α ′ of α and α ′ , called meet . In [41], Vigneauxintroduced posets subject to conditional existence of meets under the name of conditional meet semilattices ; they are the fundamental ingredient to introduce information cohomology . Example 3.1.
Let K be an abstract simplicial complex i.e. a family of subsets of agiven set I such that if α ∈ K , then every subset of α is also in K . In this structureall coproducts exist, α ∨ β = α ∩ β , but meets only exists conditionally.P. S. Alexandrov introduced a natural topology on the set of objects of a poset A , given by a basis of open sets U α = { β | α → β } , indexed α ∈ A . We willname this topology the lower Alexandrov topology (A-topology) of A , and denote X A the topological space obtained in this way. To justify the definition, one must verify that an intersection U α ∩ U α ′ is a union of sets U β , β ∈ B ; but if α → β and α ′ → β , we have U β ⊆ U α ∩ U α ′ , then U α ∩ U α ′ = S β ∈ U α ∩ U α ′ U β .The same argument shows that the intersection of every family of open sets is an open set. Dually, the upper sets U β = { α | α → β } , indexed by objects β ∈ A , formthe basis of a topology that we call upper A-topology of A . The correspondingtopological space is denoted X A . Clearly, it is the lower A-topology of the oppositecategory A op .Remark that if α → β then U α ⊇ U β and U α ⊆ U β . Also, whenever A possessesconditional coproducts and U α ∩ U α ′ is non-empty, one has U α ∩ U α ′ = U α ∨ α ′ ; theelement in U α ∩ U α ′ is a common upper bound of α and α ′ .A general reference for Alexandrov spaces, finite topological spaces, and theirrelations to simplical complexes is [5]. For instance, the reader can find there thefollowing result. Lemma 3.2 ([5, Prop. 1.2.1]) . Let A , B be posets. A map f : Ob A → Ob B isorder preserving (equivalently, defines a covariant functor from A to B ) if and onlyif f is continuous for the lower (or upper) A-topology. The functors on a poset A can be seen as classical sheaves on the associatedtopological space. Proposition 3.3 (cf. [10, Thm. 4.2.10]) . Every covariant functor F from A to thecategory of sets, can be extended to a sheaf on X A , and this extension is unique.Proof. Let F be a covariant functor on A . Suppose that F extends to a sheaf F on X A . For any open set U = S α ∈ U U α , we must have F ( U ) = lim −→ α ∈ U F ( α ), which isthe set of collections ( s α ) α ∈ U , with s α ∈ F ( α ), such that for any pair α, α ′ in U andany element β in U α ∩ U α ′ , the images of s α and s α ′ in F ( β ) coincide (“coherentcollection”). This proves the uniqueness of the extension. In any case, this formuladefines a presheaf F on X A i.e. for the lower A-topology.Let us verify that F is a sheaf. First, let U be a covering of an open U , and s, s ′ two elements of F ( U ) such that s | V = s ′ | V for all V ∈ U ; in this case, foreach α ∈ U , the components s α and s ′ α (in F ( α )) of s and s ′ are necessarily thesame, so s = s ′ . Concerning the second axiom of a sheaf, suppose that a collection s V is defined for V ∈ U , and that s V | V ∩ W = s W | V ∩ W whenever V, W ∈ U havenonempty intersection, then by restriction to the U α for α ∈ U we get a coherentsection over U . This proves the existence of the extension. (cid:3) Neither the (conditional) existence of coproducts or products nor any finitenesshypothesis are used in the previous proof. A similar proposition holds for the uppertopology, but in this case the sheaves are in correspondence with contravariantfunctors on A (i.e. presheaves on A ). Remark . In the case of posets and their associated Alexandrovtopologies, the direct images and inverse images of sheaves (or presheaves) are easyto handle.Let f : A → B be a morphism of posets, i.e. an increasing map; f is continuousfor the lower and the upper topologies.If G is a sheaf of sets on B for the lower A -topology, its inverse image is definedat the level of germs of sections by the formula ( f ∗ G )( α ) = G ( f ( α )), which givesthe stack in α .If F is a sheaf of sets on A for the lower A -topology; its direct image is defined by( f ∗ F )( β ) = ( f ∗ F )( U β ) = ( f − F )( U β ) = F ( f − ( U β )), where the open set f − ( U β )is the set of elements α ∈ A such that f ( α ) ⊆ β . XTRA-FINE SHEAVES AND INTERACTION DECOMPOSITIONS 11 If β does not meet the image of f , this is the empty set. For sheaves of abeliangroups, and β non-intersecting f ( A ), we have f ∗ F ( β ) = 0 β . For instance, if A is asub-poset of B , and J the injection: J ∗ F coincides with F on A and is zero in itscomplement.Analog results hold true for the upper topology and for the contravariant func-tors on A and B .3.2. Extra-fine presheaves on posets.
Consider a poset A , and the inducedtopological space X A whose underlying set is Ob A , equipped with the lower A-topology. Let us denote by U A the covering of X A by the open sets U α , α ∈ A .By definition of the lower A-topology, U A refines any other open covering. Soby taking the injective limit on the category of coverings pre-ordered by refinement(cf. Appendix B), Theorem 1 implies that if F is an extra-fine sheaf on X A , forany n ≥ H n ( X ; F ) = 0.Due to the maximality of the open covering U A , the existence of a super-localorthogonal decomposition for F subordinated to U A implies that F is extra-fine.Proposition 2.4 tells that also the notions of fine sheaf and of orthogonality can betested on U A .However, in general, U A is not the only finest open covering of A . For instance,one can take all the intersections (or finite intersections) of the elements of U A ;when A is not stable by arbitrary coproducts (resp. finite coproducts), the resultingcovering U + A is strictly larger than U A , but U A is also a refinement of U + A . In suchcase, the relation of refinement only defines a pre-order.Therefore it can happen that a sheaf F is extra-fine, but that F is not super-localfor U A .In the applications, the covering U A is super-local for the sheaf F ; in this casewe say that F over X A is canonically extra-fine . This property implies that F isextra-fine, because every open covering is less fine than U A , and extra-fine impliescanonically extra-fine when A is stable by any non-empty coproducts.If F is canonically extra-fine, we can describe completely the group H ( X ; F ) =0:if F is canonically extra-fine, there is a super-local orthogonal decomposition { e α } α ∈A associated to the covering U A . In this situation, the covering U ′ A of the axiom (i’)can be replaced by U A itself, which is finer. From the axioms, the images S α = im e α define sub-sheaves of F such that F = L α ∈A S α .Moreover, for any open set U of X A , e α ( U ) is the projection on S α ( U ) parallel to L β : β = γ S β ( U ).We will see the relation with the interaction decomposition in the next section.This can be summarized in the following result. Proposition 3.5.
Let F be a canonically extra-fine sheaf over X A , where X A denotes the topological space defined by a poset A equipped with its lower A-topology.Then, H ( X A ; F ) = M α ∈A H ( U A ; S α ) . Proof.
Recall that H ( X ; F ) = H ( U ; F ) for any open covering. The naturality ofthe e α implies that S α ( β ) is mapped to S α ( γ ) by the map F ι induced by ι : β → γ .Hence the set of global sections of F can be computed as the direct sum of sectionsof S α . (cid:3) Remark . In the above results, the groups F ( α ) = F ( U α ), for α ∈ A , or F ( U ),for U ∈ U , are not supposed finitely generated. This is a good point because,starting with a covariant functor F of finitely generated abelian groups over A ,it can happen that the sheaf extending F to X A in Proposition 1 is not made offinitely generated abelian groups.3.3. Finiteness conditions.
In what follows we will sometimes consider posets A that satisfy some finiteness condition.We say that A is of locally finite relative length if for any arrow α → γ , thereexist a natural number r ∈ N upper-bounding the length of every chain withoutrepetition α → ... → γ beginning at α and ending at γ . The smallest number r with this property is called the height of α with respect to γ or the depth of γ withrespect to α . This is the weakest condition that we will consider.A stronger hypothesis is locally finite : for every arrow α → β , the intersection U βα = U α ∩ U β is finite. In other terms, there exist only a finite number of chainswithout repetition beginning at α and ending at β .Even stronger is (lower) closure finite , meaning that every U α is finite. This isthe case for the poset associated to a CW complex (cf. [45]).A more convenient condition for us will be the hypothesis of locally finite dimen-sion . For any α in A , there exists a minimal object ω such that α → ω , and thereexists a natural number d , such that for every minimal object ω under α the heightof α over ω is smaller than d . The smallest such d is called the dimension of α .If A is of locally finite dimension, then it is also of locally finite relative dimension.This is easily verified: consider α → β and β → ω where ω is terminal, if the heightof α over β were infinite, it should be the same for the height of α over ω , then forthe dimension of α .Note that the conditions of local finiteness and locally finite relative length areself-dual, i.e. they hold for A if and only they hold for A op . This is not the case forclosure finiteness or locally finite dimension. The posets A and A op are both closurefinite if and only if they are finite. The posets A and A op are both of locally finitedimension if and only if there exists a number d such that any sequence α → ... → β of length bigger than d + 1 has a repetition; in this case we say that A has finitedimension, or finite depth.The most elegant finiteness condition is Lower Well Foundedness: there existsno infinite chain without repetition (cf. [36]).In the case of finite posets and sheaves of finitely generated abelian groups, wecan assert that the cohomology is finitely generated.4. Interaction decomposition
Condition G and the equivalence theorem.
Let A be an arbitrary poset,and let V be a contravariant functor on A , valued in the category of vector spacesover a commutative field K . We suppose that for each ρ : α → β in A , the map j αβ = V ( ρ ) : V ( β ) → V ( α ) is injective. We call V an injective presheaf . An object ω of A is minimal if ω → α implies that ω = α . XTRA-FINE SHEAVES AND INTERACTION DECOMPOSITIONS 13
To get a sheaf on a topological space from V , we must consider the upper A-topology and not the lower one, because U β ⊇ U α whenever α → β . In whatfollows we denote by X A the set Ob A equipped with the upper A-topology.We write V αβ instead of j αβ ( V β ). For a partition of unity associated to V , if itexists, e α | β is an endomorphism of V β . Definition 4.1. An interaction decomposition of an injective presheaf V is a familyof vector sub-spaces S γ of V γ , indexed by γ ∈ A , such that(4.1) ∀ α ∈ A , V α = M β ⊆ α j αβ S β . Let us introduce, for every γ ∈ A , the vector space(4.2) ∀ α ∈ A , S γ ( α ) = ( S αγ = j αγ S γ if α → γ α γ , this defines a presheaf S γ on A . The interaction decomposition corresponds to adecomposition of the presheaf V :(4.3) V = M γ ∈A S γ . Remark . The name interaction decomposition comes from Statistical Physics,where the spaces { V α } α ∈A are spaces of functions depending on local variablesover a lattice. An important old example corresponds to Wick’s theorem, usedin remormalization theory and Wiener analysis; a particular case is the decom-position of functions in sum of Bernoulli polynomials or Hermite polynomials, cf.Sinai’s Theory of Phase Transition: rigorous results [38]. The notion of interactiondecomposition also plays a fundamental role in other domains of Probability andStatistics, cf. [20].For injective presheaves, the concepts of canonical extra-fine and interactiondecomposition are equivalent, as shown by the following lemma in combinationwith the construction at the beginning of Section 3.2.
Lemma 4.3. If { S γ } γ ∈A defines an interaction decomposition of the injectivepresheaf V , the family of endomorphisms { e β } β ∈A such that e β | α : V ( α ) → V ( α ) is the projection onto S αβ parallel to L β ′ : β = β ′ S αβ ′ forms a super-local orthogonaldecomposition.Proof. By the maximality of the covering U A of X A by the { U α } α ∈A , it is sufficientto verify the axioms (i’), (ii) and (iii) for this covering U A . For the condition (i’), if U α * U β , then α β , which in turn implies that S αβ = 0 so e β | α = 0. For (ii), letus consider x ∈ V α ; to have e γ | α ( x ) = 0, we must have γ → α , but the definition ofinteraction decomposition tells that x belongs to the direct sum of the spaces S αβ ,thus only a finite number of the e γ | α ( x ) are different from zero. For (iii) consider β = β ′ lower than α ∈ A , then by definition of the projector e β | α the space S α | β ′ belongs to its kernel. (cid:3) Condition G :(G) ∀ α, β ∈ A such that α → β, V αβ ∩ X γ : α = −→ γγ β V αγ ⊂ X γ : α = −→ γβ = −→ γ V αγ , where β = −→ γ means that β → γ and β = γ . Theorem 4.4.
Let V be an injective presheaf on a poset A . (1) If the the condition G is satisfied and A is of locally finite dimension (inthe lower direction), the sheaf defined by V on X A is canonically extra-fine; (2) If the sheaf induced by V on X A is canonically extra-fine, the condition Gis satisfied.Proof. In view of the preceding results, we establish the first claim showing thatthere exists an interaction decomposition associated to the presheaf V .For each α ∈ A , we define the boundary sum V ′ α = P β : α → β,α = β V αβ , and wechoose any supplementary space S α of it. Hence it remains to prove that V ′ α is thedirect sum L β : α → β,β = α S αβ . We prove this by recurrence in the dimension of α .First, if α has dimension zero, then it is maximal, which means that α → β implies α = β . So V ′ α = 0 and the claim is then trivially true.Let us suppose now that the recurrence hypothesis hods true in dimension smalleror equal than r −
1, for some r ≥
1, and consider α of dimension r .Let B be the set of maximal cells β such that α → β and α = β . Andfor β ∈ B , consider x ∈ V αβ , and suppose it also belongs to the algebraic sum P γ : α → γ,β = γ,γ = α V αγ . As β is maximal, we have(4.4) x ∈ X γ : α = −→ γγ β V αγ . Then, applying the condition G to x , we deduce that x belongs to the sum of V αγ over the γ ∈ A such that α = −→ γ and β = −→ γ , which coincides with V ′ αβ = j αβ V ′ β ,consequently(4.5) V ′ α = S αβ ⊕ ( V ′ αβ + X β ′ = β,β ′ ∈ B V αβ ′ ) . For β ′ ∈ B , β ′ = β , consider x ′ ∈ V αβ ′ and suppose it also belongs to the algebraicsum P β ” ∈ B,β ” = β,β ” = β ′ V αγ . As β ′ is maximal, we have(4.6) x ∈ X γ : α = −→ γγ β ′ V αγ Then, applying the condition G to x ′ , we deduce that x ′ belongs to the sum of V αγ over the γ ∈ A such that α = −→ γ and β ′ 6 = −→ γ , which is the space V ′ αβ ′ = j αβ ′ V ′ β ,consequently(4.7) V ′ α = S αβ ⊕ S αβ ′ ⊕ ( V ′ αβ + V ′ αβ ′ + X β ” ∈ B \{ β,β ′ } V αβ ” ) . XTRA-FINE SHEAVES AND INTERACTION DECOMPOSITIONS 15
By (possibly transfinite) induction, we get(4.8) V ′ α = M β ∈ B S αβ ⊕ X β ∈ B V ′ αβ . Then we conclude by applying the recurrence hypothesis to the spaces V ′ β , and thetransitivity, j αγ = j αβ ◦ j βγ .We prove now the second claim. As we saw above, if the sheaf V on X A is extra-fine then there exists a super-local partition of unity subordinated to the covering U A , say { e α } α ∈A where e α = e U α . Setting S γ = im e γ , one has F = L γ ∈A S γ andthe formulae (4.1) hold with S γ = S γ ( U γ ), so we have an interaction decomposition.Let us fix α ∈ A , look at α → β , and consider a vector x in S αβ . Suppose that thisvector is equal to a finite sum y + ... + y m of elements of V αβ , ..., V αβ m respectively,with α → β i but β β i . Applying to each of these vectors y i the projector e β | α = e U β ( U α ), we find zero, in reason of super-locality; however e β | α ( x ) = x bydefinition of S αβ , thus x = 0. Now consider any vector z in V αβ ; by interactiondecomposition, z is (in a unique way) the sum of a vector x ∈ S αβ , and a vector y in the space V ′ αβ , equal to the sum of the S αγ for β = −→ γ , which is included in thesum of all the V αγ for β = −→ γ . Then the condition G is proved. (cid:3) In this generality, the theorem above appears for the first time in [36], by G.Sergeant-Perthuis. It holds true if we replace the property of local finite dimensionby the noetherian property of well-foundedness.We also refer to condition G as sum-intersection property . Before the work ofG. Sergeant-Perthuis, a particular case of this property appeared in the book ofLauritzen (see Proposition B.5 in the Appendix B of [20]), as a corollary of theinteraction decomposition. Lauritzen considers a finite poset A and a presheaf offinite dimensional vector spaces { V a } a ∈A that admits an interaction decomposition;the property is stated for two open subsets U, V of the lower space X A (namedgenerating classes, the topology was not mentioned) in the following form:(4.9) X c ∈ R ( U ∩ V ) V c = X a ∈ R ( U ) V a ∩ X b ∈ R ( V ) V b , where R ( U ) denotes the set of all elements a of A such that U a is included in U .In [36] it is assumed that all the V ( α ), α ∈ A , belong to a fixed vector space V ,this is not a restriction, because it is always possible to inject all of them in thecolimit of V , seen as a diagram over A op . A traditional name for this special colimitis direct limit , denoted lim −→ A op V or simply lim −→ V . It is the quotient of the directsum L α ∈A V α subject to the relations j αβ ( x β ) = x β , for each arrow α → β . Everyspace V α goes naturally into lim −→ V ; we denote by j α this map. The space lim −→ V is universal in the sense that if there exist a vector space W and homomorphisms f α : V α → W ; α ∈ A , such that f β = f α ◦ j αβ every time it has a meaning, thereexists a unique homomorphism f : lim −→ V → W such that ∀ α ∈ A , f ◦ j α = f α .From this description, it is clear that lim −→ V corresponds to set H ( A , V ) of globalsection of V over A , which is in turn isomorphic to H ( X A , V ). If V is an injective presheaf, then every j α is injective. Moreover, if V has aninteraction decomposition S , we have(4.10) lim −→ V = M α ∈ A j α S α . Duality.
We suppose that the sheaf V on X A is extra-fine. For each α ∈ A ,let V ∗ α be the (algebraic) dual vector space of the vector space V α = V ( α ); thetranspose maps t j αβ ; β ⊆ α define a covariant functor on A , then a presheaf for thelower topology on A . But the transposed maps t e α ; α ∈ A give a decomposition ofthis presheaf into the product of the presheaves S ∗ α ; α ∈ A , not into a direct sum.In general V ∗ is not extra-fine, the condition of super-locality fails for the lowertopology. Only the condition (iii) is satisfied. Therefore we need to follow anotherway to dualize Theorem 4.4.This can be done adding a further hypothesis. Let us suppose that there exists acovariant functor F on A (equivalently, a topological sheaf on X A ), with surjectivearrows π βα for α → β , such that V α = F ∗ α and j αβ = t π βα for all pairs α, β with α → β . Then for every α ∈ A , the space F α embeds naturally in V ∗ α , in such amanner that, for every pair α, β with α → β , j αβ induces the map π βα . Let usdenote by e ∗ α the restriction of t e α to F . Given the following lemma, this givesa family of orthogonal projectors from F to the dual copresheaf V ∗ (by the sameargument given in the proof of Lemma 2.3). We do not ask that t e α preserves F . Lemma 4.5. id F = P α ∈A e ∗ α , in the sense of finite sum when applied to a givenvector g ∈ F β for any β ∈ A .Proof. Consider γ ∈ A and a basis { f j | j ∈ J } of F γ as a vector space over K , thespace V γ = F ∗ γ is isomorphic to the product K J , in such a manner that the dualityis given by the natural evaluation. The space F γ itself is isomorphic to the spaceof scalar functions on J which are zero outside a finite subset.For j ∈ J , note x j = f ∗ j the element of V γ corresponding to f j (in the dualbasis). The set A j of elements β ∈ A such that e β ( x j ) = 0 is finite, and we have(4.11) x j = X β ∈ A j e β | γ ( x j ) . Choose g ∈ F γ . For any α ∈ A , we have(4.12) h x j , e ∗ α | γ ( g ) i = h e α | γ ( x j ) , g i , where the bracket denotes the form of incidence from V ∗ γ × V γ to K . Then(4.13) h x j , g i = X β ∈ A j h e β | γ ( x j ) , g i = h x j , X β ∈ A j e ∗ β | γ ( g ) i . If α does not belong to A j , we have(4.14) 0 = h e β | γ ( x j ) , g i = h x j , e ∗ α | γ ( g ) i , i.e. x j vanishes at e ∗ α | γ ( g ). Therefore, for every j ∈ J and g ∈ F ,(4.15) h x j , g i = h x j , X β ∈A e ∗ β | γ ( g ) i . XTRA-FINE SHEAVES AND INTERACTION DECOMPOSITIONS 17
Let us denote by B g a finite set of indexes k ∈ J such that(4.16) g = X k ∈ B g g k f k . Equivalently, if j does not belong to B g , we have(4.17) 0 = h x j , g i = h x j , X β ∈A e ∗ β | γ ( g ) i , and if j ∈ B g ,(4.18) g j = h x j , g i = h x j , X β ∈A e ∗ β | γ ( g ) i . Now, consider any element x ∈ V γ = F ∗ γ , it is identified with the numerical functionthat assigns x ( j ) ∈ K to j ∈ J . Then, using the above equations (4.17) and (4.18),we get h x, g i = h x, X k ∈ B g g k f k i = X k ∈ B g g k x ( k ) = X k ∈ B g h x ( k ) x k , X β ∈A e ∗ β | γ ( g ) i = h x, X β ∈A e ∗ β | γ ( g ) i , which implies the desired result. (cid:3) Note that the axiom (i’) is not verified for F and the family of projectors e ∗ α | β .Then, for this dual situation neither the extra-fine condition nor the interactiondecomposition hold, but something else holds true, which is sufficient in manyapplications.The images of e ∗ α | β for β describing A define a sub-sheaf of V ∗ , that we denote T α . And we denote by T α its stack at α ∈ A . Corollary 4.6.
The presheaf F (for the lower A-topology) defined above is acyclicand H ( X A , F ) ≈ L α ∈A H ( X A , T α ) = L α ∈A T α .Proof. It is sufficient to prove the acyclicity for the covering by the { U α } α ∈A .For any γ ∈ A , the presheaf T γ is zero outside U γ and is ( j αγ S γ ) ∗ for α ∈ U γ .The ˇCech cohomology of a direct sum of sheaves is the direct sum of their coho-mology; this follows by projection of the cochain and naturality of the coboundaryoperator. Therefore the corollary 1 results from the following lemma. (cid:3) Lemma 4.7.
Let T be a presheaf on A , equipped with the lower A-topology, thatis supported on a set U γ for γ ∈ A . If for every α, β ∈ U γ such that α → β themorphism π βα is an isomorphism, then T is acyclic and H ( X A , T ) = T γ .Proof. Again, it is sufficient to prove the acyclicity for the covering by the { U α } α ∈A .Every space T α is zero except if α → γ , then we can consider that every cochaintakes its value in T γ , whatever being its degree. Considering a cochain c of degree n , if it is a cocycle, for any family α , ..., α n +1 in A , we have, in T γ :(4.19) c ( α , ..., α n +1 ) = n +1 X k =1 ( − k +1 c ( γ, α , ..., c α k , ..., α n +1 ) , which tells that c is equal to δφ , where φ is the ( n − ∀ β , ..., β n ∈ A , φ ( β , ..., β n ) = c ( γ, β , ..., β n ) . This establishes the lemma. (cid:3)
In the following section we will need a variant of this lemma, concerning therelative cohomology. Suppose that A is a sub-poset of B , and that we have apresheaf T on X B (i.e. for the lower A -topology B ), which is supported on a set U γ for γ ∈ B , such that every morphism π βα with α → β → γ is an isomorphism.Then we consider the sheaf S over A , obtained by restriction. Lemma 4.8.
Under the above hypotheses, ∀ n ≥ , H n ( B , A ; T, S ) = 0 . See Appendix B for the definition of relative cohomology.
Proof.
It is sufficient to prove the result for the cohomology of the covering bythe open sets { U β } β ∈B , and their traces on A . By definition, a relative cochain c ∈ C n ( B , A ; T, S ) takes the value 0 on every family of n + 1 elements of A . If it isa cocycle, for any family α, α , ..., α n +1 of n + 2 elements in B , we have, in T γ :(4.21) c ( α , ..., α n +1 ) = n +1 X k =1 ( − k +1 c ( α, α , ..., c α k , ..., α n +1 ) , which tells that c is equal to δφ , where φ is the ( n − ∀ β , ..., β n ∈ B , φ ( β , ..., β n ) = c ( α, β , ..., β n ) . If U γ has empty intersection with A , taking α = γ , we have φ ∈ C n − ( B , A ; T, S ),and c = dφ . And if α belongs to A ∩ U γ , the cochain φ belongs to C n − ( B , A ; T, S ),and c = dφ . This establishes the lemma. (cid:3) Factorization of free sheaves
Free presheaves and intersection properties.
In many applications toStatistical Physics or Bayesian Learning, the presheaves that appear are free mod-ules, generated by subsets of a fixed set.A set I is given (non-necessarily finite) and the poset A is a sub-poset (i.e. asubcategory) of the poset ( P f ( I ) , → ) of finite subsets of I , ordered in such a waythat A → B iff B ⊆ A . The poset A is automatically of locally finite dimension.The pair ( A , I ) is named an hypergraph . We consider a covariant functor (a.k.a.copresheaf) of sets E on A , such that, for every α ∈ A , the set E α = E ( α ) can beidentified with the cartesian product Q i ∈ α E i by surjective maps π iα = E ( α → i ).By naturality, all the maps π βα : E α → E β are surjective. If the empty set ∅ belongs to A , the set E ∅ is a singleton ∗ = {∅} . In this case, for every element α ∈ A , there exists a unique map π ∅ α : E α → E ∅ .Note that E is a sheaf of sets for the lower A -topology on A , and for every arrow α → β , the map π βα is the restriction of sections from the open set U α to the openset U β .A commutative field K of any characteristic is given. For every α ∈ A , we define V α as the space of all functions from E α to K . We say that V is the free presheaf generated by E . If ∅ ∈ A , the space V ∅ is canonically isomorphic to K . If α → β , i.e. β ⊆ α , we get a natural application j αβ : V β → V α , which is linear and injective.As before, V αβ designates the image of j αβ in V α . Using the projection π βα we XTRA-FINE SHEAVES AND INTERACTION DECOMPOSITIONS 19 can identify V αβ with the space of numerical functions of x α that depend only onthe variables x β , these functions are named the cylindrical functions with respectto π βα . Definition 5.1 (Reduced functor) . The sub-functor of constants K A maps each α ∈ A to the one dimensional vector subspace K α of constant functions, embeddedin V α . The reduced functor (or reduced free presheaf) V α ; α ∈ A is made of thequotient vector spaces V α / K α .If ∅ ∈ A , for every α ∈ A , we have K α = V α ∅ . Definition 5.2 (Intersection property) . The hypergraph ( A , I ) satisfies the strong(resp. weak) intersection proprety , if, for every pair ( α, α ′ ) in A (resp. every pairhaving non-empty intersection in P ( I )), the intersection α ∩ α ′ belongs to A . Remark . If A satisfies the strong intersection property, all the coproducts α ∨ α ′ exist; if A satisfies the weak intersection property, the coproducts exist condition-ally, i.e. α ∨ α ′ exists as soon as α and α ′ have a common majorant (under therelation → ).If A has non-intersecting elements, the strong intersection property implies thatthe empty set ∅ belongs to A , then A possesses a unique final element, that is ∅ . If A satisfies the weak intersection property, it possesses conditional coproducts (hereintersections) in the categorical sense of Section 3. Proposition 5.4. If A has the strong intersection property, the condition G issatisfied by the free presheaf V .Proof. Consider α → β in A (i.e. β ⊆ α ), and a vector v in V αβ that satisfies(5.1) v = X γ : α → γ,γ = α,γ β v γ , for some v γ ∈ V αγ .The above decomposition tells that for every x β ∈ E β , and for any collection ofelements { y j } j ∈ α \ β , where y j ∈ E j , we have(5.2) v ( x β , y α \ β ) = X γ : α → γ,γ = α,γ β v γ ( x γ );where on the right, the components of x γ are x i with i ∈ β ∩ γ and y j with j ∈ ( α \ β ) ∩ γ .For each index k ∈ α \ β , we choose a fixed y k , and replace everywhere in theformula the variable x k by this value. The formula continues to hold true. Inthe expression v γ ( x γ ), the variables x i that do not belong to β ∩ γ , are constants y k ; k ∈ α \ β . Moreover the intersection of β and γ is a strict subset of β , because β is assumed to be not included in γ .This gives(5.3) v ( x β , y α \ β ) = X γ : α → γ,γ = α,β → β ∩ γ,β = β ∩ γ v β ∩ γ ( x β ∩ γ ) . And, for all possible ω ∈ A , ω ⊂ β , β = ω , if we bring together the γ such that α → γ, γ = α, β ∩ γ = ω , this gives(5.4) v ( x β , y α \ β ) = X ω : α → ω,ω = α,β → ω,β = ω w ω ( x ω ) . Which is the expected result. (cid:3)
Remark . Without the strong intersection property the results is false. Take forinstance, I = { i, j } , A = { i ; j ; α = ( i, j ) } , a non-zero constant function belongs to V αi , but cannot belong to the image of a strict subset of { j } . Proposition 5.6. If A has the weak intersection property, the condition G is sat-isfied by the reduced functor V .Proof. Repeat the proof of Proposition 5.4, but distinguish the cases where β ∩ γ is empty or not. When it is empty the respective function v β of ( x γ , y γ ′ ) belongsto the constants. (cid:3) Now remind that, by construction, the poset A is of locally finite dimension (itis even locally finite), then the following proposition results directly from the prop.5.4 (resp. 5.6) and the Theorem 4.4. Theorem 5.7. If A has the strong (resp. weak) intersection property, the sheaf V (resp. V ) is extra-fine for the upper A -topology. Remind that under the strong (resp. weak) condition of intersections, extra-fineis equivalent to canonically extra-fine.Theorem 5.7 generalizes a theorem of existence of an interaction decompositionfor factor spaces that, under different forms, has been known for long time in prob-ability theory, but only for finite posets and finite dimensional vector spaces, cf.[19, 24, 40, 20].As in the preceding section, denote by V ′ α the sum of the V αβ over β ( α , (resp. V ′ α the sum of the V αβ over β ( α ) and take a supplementary subspace S α of V ′ α in V α (resp. S α of V ′ α in V α ). The interaction decomposition gives(5.5) ∀ α ∈ A , V α = M β ⊆ α S β , resp.(5.6) ∀ α ∈ A , V α = M β ⊆ α S β . Duality: Free copresheaves.
Note F α = K ( E α ) the space of functions withfinite supports, which can be seen as the vector spaces freely generated by the set E α over the field K . Its dual space is V α = K E α and the transpose of the naturalmap π βα : F α → F β is j αβ . The vector spaces F α and the maps π βα define acovariant functor (i.e. a copresheaf) over A (resp. a sheaf on X A ) named the freecopresheaf (resp. the free sheaf ) generated by E .We can apply Corollary 6.32 in the preceding section to get the following result. XTRA-FINE SHEAVES AND INTERACTION DECOMPOSITIONS 21
Proposition 5.8.
When A satisfies the strong intersection property, F is acyclicand H ( X A , F ) ≈ L α ∈A S ∗ α . Respectively, denote by F α the subspace of F α = K ( E α ) defined by annihilatingthe sum of the coordinates in the canonical basis. Its dual space is V α = K E α / K α .The transpose of the natural map π βα : F α → F β is again j αβ . This forms a sheafover X A , named the restricted free sheaf generated by E . As before, we obtain thefollowing. Proposition 5.9.
When A satisfies the weak intersection property, F is acyclicand H ( X A , F ) ≈ L α ∈A S ∗ α . In the case of finite sets { E i } i ∈ I and I finite, this result was established in H.G.Kellerer [19]. See also [24] and Appendix C below. Theorem 5.10.
If the hypergraph ( A , I ) satisfies the weak intersection hypoth-esis, for any covariant functor of sets E on the category A , the ˇCech cohomol-ogy H ∗ ( X A ; F ) of the induced free sheaf F is naturally isomorphic to the sum of H • ( X A ; F ) which is concentrated in degree zero, and of the full ˇCech cohomology(with trivial coefficients K ) of the topological space X A (i.e. the poset A equippedwith the lower Alexandrov topology).Proof. The sheaf F over A is decomposed into the sum of the sheaf F and theconstant sheaf K A ; this induces a decomposition in direct sum of the cochain com-plexes. One of them gives gives H • ( X A ; F ), which is concentrated in degree zeroas just said by the preceding proposition, whereas the other one gives the standardˇCech cohomology of A . (cid:3) When A is the poset of a finite simplicial complex, it satisfies the weak inter-section property, and the standard ˇCech cohomolgy (with constant coefficients) on X A is isomorphic to the singular or simplicial cohomology with coefficients in K .5.3. Relative cohomology.
In addition to
A ⊆ P f ( I ), consider another poset B satisfying the same kind of hypotheses, with respect to a set J , i.e. B ⊆ P f ( J ). Definition 5.11. A strict morphism from ( A , I ) to ( B , J ) is the pair ( f, f I ) of afunctor (i.e. an increasing map) f : A → B , and a map f I : I → J , such that ∀ i ∈ I and all α ∈ A such that i ∈ α , one has f I ( i ) ∈ f ( α ) ⊆ J . For simplicity, we willdenote f I = f .As before, let E be the sheaf of sets over A given by products of the sets { E i } i ∈ I i.e. such that α E α ∼ = Q i ∈ α E i ; we call the E i basic sets . Consider a strictmorphism f : A → B . For every j ∈ J , let us define E ′ j as the product of the E i for i ∈ I such that f ( i ) = j . Proposition 5.12.
The direct image f ∗ E over B is given by the products of thebasic sets { E ′ j } j ∈ J .Proof. For β ∈ B , the set f ∗ E ( U β ) (in the lower A -topology) is the subset of theproduct of the E α over α ∈ f − ( β ) formed by the families s α ; f ( α ) ⊆ β , thatare compatible on the intersections U α ∩ U α ′ . Each set E α is the product of thesets E i ; i ∈ α , and the compatibility condition tells that for any pair α, α ′ with f ( α ) ⊆ β and f ( α ′ ) ⊆ β , the restriction of s α and s α ′ to their common terminalpoints coincide. This implies that E ( f − ( U β )) is the product of the E i ; i ∈ I suchthat f ( i ) ∈ β , then it is the product of the E ′ j for j ∈ β . (cid:3) In particular, E ′ j coincides with the set ( f ∗ E ) j which corresponds to the directimage of sheaves. Definition 5.13. A simplicial morphism from A to B is a strict morphism f : A → B , such that ∀ α ∈ A , the restriction of f I to the set α ∈ P f ( I ) is surjectiveonto the set f ( α ) ∈ P f ( J ). Proposition 5.14.
Let f : A → B be injective and simplicial, and let F ′ be a sheafon B , given by products of the basic sets E ′ j ; j ∈ J . The inverse image f ∗ F ′ over A is given by the products of the basic sets { E i = E ′ f ( i ) } i ∈ I .Proof. For α ∈ A , by definition of f − F ′ (which coincide with f ∗ F ′ in the case ofposets), ( f ∗ E ′ ) α = E ′ f ( α ) is the product of the sets E ′ j ; j ∈ f ( α ), and this productcoincide with the product of the sets E ′ f ( i ) for i ∈ α because f is simplicial andinjective. (cid:3) In the following result, we consider the restricted subsheaves F and F ′ , and weassume that both A and B verify the weak intersection property. Theorem 5.15.
Let J : A → B be an inclusion of posets, strict and simplicial.If F ′ is a restricted free copresheaf over B , then the inverse image F = J ∗ F ′ over A is restricted, and we have a natural surjection from F ′ to J ∗ F , and the inducednatural map in cohomology J ∗ : H ( B ; F ′ ) → H ( A ; F ) is surjective.Proof. Along A , the stalk of F ′ and J ∗ F coincide. From Theorem 4.4 and thelong exact sequence in ˇCech cohomology (Appendix B), we get the following exactsequence:(5.7) 0 → H ( B , A ; F ′ , F ) → H ( B ; F ′ ) → H ( A ; F ) → H ( B , A ; F ′ , F ) → . Then the theorem is equivalent to the vanishing of H ( B , A ; F ′ , F ) = 0. To provethe latter, we proceed as in the proof of Corollary 6.32: we decompose F ′ over B and then F accordingly over A in direct sums of sheaves T β ; β ∈ B and S α ; α ∈ A respectively, which satisfy the hypotheses of Lemma 4.8. Then we conclude byapplying Lemma 4.8 and the natural isomorphism in cohomology between A (resp. B ) and A (resp. B ) for the restricted sheaves. (cid:3) In the context of finite probabilities, reducing corresponds to the tangent equa-tion of the probability restriction of sum 1, and Theorem 5.15 is equivalent to aresult of H.G. Kellerer [19].5.4.
Marginal theorem.
Given a set I , let { E i } i ∈ I be a collection of sets. Forany subset β of I , define E β = Q i ∈ β E i . The (discrete) probabilistic marginalproblem is the following: given a subposet A of P f ( I ) and a family of probabilitylaws { P α } α ∈A over the respective sets { E α } α ∈A , which satisfy the compatibilityconditions over all the intersections U α ∩ U β , for α, β ∈ A , determine if there existsa probability Q : E = Q i ∈ I E i → [0 ,
1] such that for every α ∈ A ,(5.8) ∀ α ∈ A , ∀ x α ∈ E α , X x I \ α ∈ E I \ α Q ( x α , x I \ α ) = P α ( x α ) , The linearized marginal problem asks for a function Q : E → R that is not neces-sarily positive. XTRA-FINE SHEAVES AND INTERACTION DECOMPOSITIONS 23
In the last appendix below, based on the preceding sections, we prove the fol-lowing index formula, which generalizes the result of Kellerer [19] and Mat´uˇs [24].
Theorem 5.16.
If the poset A is finite and satisfies the weak intersection property,and if the { E i } i ∈ I are finite sets, then (5.9) χ ( A ; V ) = ∞ X k =0 ( − k dim K H k ( U A ; V ) = X α,β ∈A µ αβ N β ; where µ α,β is the M¨obius function of A , and, for each α ∈ A , N α denotes thecardinality of E α . We will also prove that(5.10) χ ( A ; V ) = dim K H ( U A ; V ) + χ ( A );where χ ( A ) denotes the Euler characteristic of A , in every possible sense: as a met-ric subspace of the simplex P ( I ), as the lower or upper Hausdorff topological spacein ˇCech cohomology, or as an abstract poset; this is also the Euler characteristic ofthe nerve of the category A .6. Nerves of categories and nerves of coverings
Any contravariant functor G of abelian groups on a poset A produces a sheaf,also denoted by G , on the topological space X A , whose underlying set is Ob A ,equipped with the upper A-topology. This is equivalent with the dual statementfor covariant functors on A op . But G is also an abelian object in the topos PSh( A ),cf. [4], [26].And in the context of topos theory, it is customary to study another cohomology,that is the graded derived functor H • ( A , − ) of(6.1) Γ A ( − ) = Hom Ab( A ) ( Z , − ) ∼ = Hom PSh( A ) ( ∗ , − );cf. [44]. In the following lines, we give a more explicit and topological definition ofthis functor, according to [4], [26].The nerve of a small category C is the simplicial set whose n simplices are se-quences c → · · · → c n of composable arrows in C , and whose face operators are(6.2) d i ( c f → · · · f n → c n ) = c → · · · c n if i = 0 c → · · · c i − f i +1 ◦ f i → c i +1 → · · · → c n if 0 < i < nc → · · · → c n − if i = n . For background and details, see Section 6.1 below. This permits to define a canoni-cal cochain complex ( C n ( A, G ) , d ) whose cohomology is precisely H • ( A , G ), cf. [26,Prop. 6.1]. This complex comes from a canonical projective resolution of the con-stant presheaf Z [4, Ex. V.2.3.6].The n -cochains are(6.3) C n ( A , G ) = Y a n →···→ a in A G ( a n ) = Y a →···→ a n in A op G ( a n ) and the coboundary δ : C n − ( A , G ) → C n ( A , G ) is given by(6.4) ( δg ) a →···→ a n = n − X i =0 ( − i g d i ( a →··· a n ) + ( − n G ( ϕ n ) g d n ( a →··· a n ) , where ϕ n is the A -morphism from a n to a n − in the sequence a n → · · · → a . Remark . This complex and its analog for a covariant homology were rediscov-ered by O. Peltre in the context of his doctoral work [28], which gives a homologicalinterpretation of the generalized Belief Propagation algorithm [46], which is appliedin statistical physics, bayesian learning and decoding processes. One of the initialmotivations behind the present article was to understand better the connections ofit with ˇCech cohomology and sheaf cohomology.In this section, we want to compare this cohomology with the topological ˇCechcohomology of the sheaf G on X A that we have studied in the previous sections.In fact we will prove that they are naturally isomorphic.When G is the constant sheaf Z , H • ( C ( N ( A ) , Z ) , d ) corresponds to the sim-plicial cohomology of | N ( A ) | , the geometric realization of the nerve N ( A ) (seeRemark 6.7); it is known to be naturally isomorphic to the singular cohomology of | N ( A ) | (cf. [11]). In turn, the ˇCech cohomology ˇ H • ( X A , Z ) is isomorphic to thesingular cohomology of X A . Hence the isomorphism between H • ( C ( N ( A ) , Z ) , d ) ∼ =ˇ H • ( X A , Z ) is implied by the homotopy equivalence between | N ( A ) | and X A , seeMay [25]. Thus we are looking here for an extension of this result in the context ofsheaves.For that purpose, we introduce a general framework of cosimplicial local systems on simplicial sets. We will remind below the definition of simplicial sets and simpli-cial objects in a category. The nerve K • ( U ) of a covering U introduced in Section 2and the nerve N ( C ) of a category C are examples of simplicial sets. Cosimplicial lo-cal systems are functorial assignments of local data to the simplexes and morphismsof a simplicial set. It appears that both ˇCech cohomology and the cohomology in-troduced by (6.3)-(6.4) become particular cases of this general construction andcan be compared in this framework. Remark . It is not excluded that spectral sequences, as defined in Segal [33],can be used for establishing the comparison, but we have not seen how this can bedone directly.6.1.
Simplicial sets and nerves of coverings.
Simplicial sets can be traced backto Eilenberg and Zilber [12]—under the name “complete semi-simplicial sets”. Theybecame ubiquitous in algebraic topology, due to the works of Segal, Grothendieck,Kan, Quillen, May and many others. The subject was treated in great detail byMay in [25]; also [44, Ch. 8] is a good introduction.Let ∆ be the category whose objects are the finite ordered sets [ n ] = { < < ... < n } , for each n ∈ N , and whose morphisms are nondecreasing monotonefunctions. Given any category C , a simplicial object S in C is a contravariant functorfrom ∆ to C i.e. S : ∆ op → C . When C is the category of sets, S is called a simplicialset . One can define analogously simplicial groups, modules, etc.Although ∆ has many morphisms, which seem complicated at first sight, theycan be conveniently expressed in terms of certain morphisms known as face and XTRA-FINE SHEAVES AND INTERACTION DECOMPOSITIONS 25 degeneracy maps . For each n ∈ N and i ∈ [ n ], the face map d ni : [ n ] → [ n + 1] isgiven by(6.5) d ni ( j ) = j if j < i, d ni ( j ) = j + 1 if j ≥ i. Similarly, for each i ∈ [ n + 1], the degeneracy map s n +1 i : [ n + 1] → [ n ] is(6.6) s n +1 i ( j ) = j if j ≤ i, s n +1 i ( j ) = j − j > i. Normally the super-index is dropped.Given a morphism ϕ : [ m ] → [ n ] of ∆, let ı , ..., i s be the elements of [ n ] not in ϕ ([ m ]), in reverse order, and let j , ..., j t , in order, be the elements of [ m ] such that ϕ ( j ) = ϕ ( j + 1). Then(6.7) ϕ = d i · · · d i s s j · · · s j t . Remark that m − t + s = n . This factorization is unique [25, Sec. I.2].The face and degeneracy maps satisfy some relations: ∀ n ∈ N , ≤ j < k ≤ n, s n +1 j ◦ d nk = d nk − ◦ s n +1 j , (6.8) ∀ n ∈ N , ≤ j ≤ n + 1 , s n +1 j ◦ d nj = s n +1 j +1 ◦ d nj = id n +1 , (6.9) ∀ n ∈ N , n + 1 ≥ j > k + 1 ≥ , s n +1 j ◦ d nk = d nk ◦ s n +1 j − . (6.10)A (simplicial) morphism from a simplicial set S to a simplicial set S ′ is a naturaltransformation of functors: a collection of maps { f n : S ([ n ]) → S ′ ([ n ]) } n ∈ N suchthat, for each morphism ϕ : [ m ] → [ n ] in ∆, the diagram(6.11) S ([ n ]) S ([ m ]) S ′ ([ n ]) S ′ ([ m ]) f n Sϕ f m Sϕ commutes. Example 6.3 (Simplex) . A basic example of simplicial set is the k -simplex ∆ k [25,Def. I.5.4], which is the presheaf represented by [ k ]. This means that ∆ kn = ∆ k ([ n ])equals Hom([ n ] , [ k ]), and the map ∆ k ϕ : Hom([ m ] , [ k ]) → Hom([ n ] , [ k ]) induced by ϕ : [ m ] → [ n ] is given by precomposition with ϕ . Example 6.4 (Nerve of a covering) . Let X be a topological space and U an opencovering of X . The nerve of the covering U is the set K ( U ) of finite sequences of ele-ments of U having a non-empty intersection. It has a natural structure of simplicialset: K n = K ([ n ]) is the set of sequences of length n + 1, denoted ( U , ..., U n ), andfor any nondecreasing function ϕ m,n from m to n , there is a map ϕ ∗ m,n : K n → K m given by(6.12) ϕ ∗ m,n ( V , ..., V n ) = ( V ϕ (0) , ..., V ϕ ( m ) ) . In other terms, K n ( U ) is the set of maps u : [ n ] → U such that the intersection ofthe images are non-empty, and if ϕ : [ m ] → [ n ] is a morphism and v ∈ K n ( U ), then ϕ ∗ m,n ( v ) = v ◦ ϕ .Hence the map s ∗ i = K ( s n +1 i ) is given by(6.13) s ∗ i ( U , ..., U n ) = ( U , ..., U i − , U i , U i , U i +1 , ..., U n );is also called degeneracy map , whereas d ∗ i = K ( d n +1 i ) is given by(6.14) d ∗ i ( U , ..., U n +1 ) = ( U , ..., U i − , c U i , U i +1 , ..., U n +1 ) , and called face map. For each u = ( U , ..., U n ), we denote by U u the intersection U ∩ ... ∩ U n . It iseasily verified that, for every morphism ϕ m,n : [ m ] → [ n ] and every v ∈ K n ( U ), onehas U v ⊆ U ϕ ∗ ( v ) . In particular, U ϕ ∗ ( v ) is non-empty if U v is non-empty. Example 6.5 (Nerve of a category) . To any small category C is naturally as-sociated a simplicial set N ( C ), named its nerve: the elements of N n ( C ) are thecovariant functors from the poset [ n ] to C , and the morphisms are obtained byright composition.Concretely an element of degree n is a sequence(6.15) a = α → α → ... → α n . The action s ∗ i of s i is the repetition of the object α i via the insertion of an identityid α i ; the action d ∗ i of d i is the deletion of α i via the composition of α i − → α i and α i → α i +1 .More generally if(6.16) b = β → β → ... → β m belongs to N m ( C ), and ϕ : n → m is non-decreasing, then(6.17) ϕ ∗ ( b ) = β ϕ (0) → β ϕ (1) → ... → β ϕ ( n ) . Example 6.6 (Barycentric subdivision of the nerve of a covering) . Consider thecategory C ( U ) which has for objects the non-empty intersections of the elementsof U , and for morphisms the inclusions, then the nerve N ( C ( U )) is the barycentricsubdivision of the simplicial set K ( U ). This was remarked by Segal [33], interpreting[12]. Remark . It is reassuring to know that any simplicialset gives rise to a CW-complex, even if this is not directly used in the present text.The geometric realization | K | of the simplicial set K is a topological space obtainedas the quotient of the disjoint union of the products K n × ∆( n ), where K n = K ([ n ])and ∆( n ) ⊂ R n +1 is the geometric standard simplex, by the equivalence relationthat identifies ( x, ϕ ∗ ( y )) and ( ϕ ∗ ( x ) , y ) for every nondecreasing map ϕ : [ m ] → [ n ],every x ∈ K n and every y ∈ ∆( m ); here f ∗ is K ( f ) and f ∗ is the unique linear mapfrom ∆( n ) to ∆( m ) that maps the canonical vector e i to e f ( i ) . For every n ∈ N , K n is equipped with the discrete topology and ∆( n ) with its usual compact topology,the topology on the union over n ∈ N is the weak topology, i.e. a subset is closedif and only if its intersection with each closed simplex is closed, and the realizationis equipped with the quotient topology. In particular, even it is not evident at firstsight, the realization of the simplicial set ∆ k is the standard simplex ∆( k ). See [25,Ch. III].The cartesian product of two simplicial sets K and L is taken as it must be forfunctors to E , that is term by term: ( K × L )([ n ]) = K ([ n ]) × L ([ n ]) at the level ofobjects, and similarly for the maps. Definition 6.8.
Let f : K → L and g : K → L be two simplicial maps, a simplicial homotopy from f to g is a simplicial map h : K × ∆ → L , such that f = h ◦ (id K × d ) and g = h ◦ (id K × d ).Simplicial homotopy is an equivalence relation, compatible with composition ofsimplicial maps. XTRA-FINE SHEAVES AND INTERACTION DECOMPOSITIONS 27
Example 6.9 (Homotopy induced by a projection of coverings) . A covering U iscalled a refinement of another covering U ′ when every set of U is contained in someset of U ′ . In that case, there exists a map λ : U → U ′ , called projection , such thatfor every U ∈ U one has U ⊆ λ ( U ). It is also said that U is finer than U ′ [11].A projection map λ : U → U ′ induces a simplicial morphism λ ∗ from the simpli-cial set K ( U ) to the simplcial set K ( U ′ ):(6.18) λ ∗ ( u ) = λ ◦ u ; Proposition 6.10. If U is a refinement of U ′ , two projections λ, µ from U to U ′ induce homotopic simplicial maps λ ∗ , µ ∗ from K ( U ) to K ( U ′ ) .Proof. Let u = ( U , ..., U n ) be an element of K n ( U ), and ϕ i = (0 , .., , , ..., ∈ ∆ n with the first 1 at place i between 0 and n + 1, we put(6.19) h ( u, ϕ i ) = ( λ ( U ) , ..., λ ( U i − ) , µ ( U i ) , ..., µ ( U n )) . (cid:3) Cosimplicial local systems and their cohomology.
We present here ageneral definition of cohomology for cosimplicial local systems on simplicial sets.
Definition 6.11. A cosimplicial local system of sets F over the simplicial set K isa family F u indexed by the elements u of K , and, for any morphism ϕ : [ m ] → [ n ]and any v ∈ K n , a given application F ( ϕ, v ) : F u → F v , where u = ϕ ∗ m,n ( v ), suchthat F ( ψ, w ) ◦ F ( ϕ, v ) = F ( ψ ◦ ϕ, w ), for ϕ : [ m ] → [ n ], ψ : [ n ] → [ p ], w ∈ K p , v = ψ ∗ n,p ( w ), u = ϕ ∗ m,n ( v ). Remark . A definition of simplicial local systems appeared in the work ofHalperin [17]. In his case the arrows are in the reverse direction, i.e. for ϕ :[ m ] → [ n ], v ∈ K n , a map ϕ ∗ v : F v → F ϕ ∗ ( v ) . Example 6.13 ( ˇCech system) . Take a presheaf F over the topological space X and consider an open covering U of X . Then for u ∈ K ( U ), define F u = F ( U u ), andfor ϕ : [ m ] → [ n ], v ∈ K n , take for F ( ϕ, v ) the restriction from F ( U ϕ ∗ v ) to F ( U v ).This defines a cosimplicial system over K ( U ). Example 6.14 (Upper and lower systems associated to a functor) . Let F be acontravariant functor from C to the category of sets E . We can define a cosimpliciallocal system F ∗ over the nerve N ( C ) (see Example 6.5 for the definition and thenotation), named the upper system, by taking F ∗ ( a ) = F ( α n ), and for a morphism ψ : n → p , and an element b = β → ... → β p in N p ( C ), denoting by α n the lastelement of a = ψ ∗ ( b ), we have α n = β ϕ ( n ) , and this comes with a canonical arrow f going to β p , then we take ψ ∗ b = f ∗ , going from F b = F ( β p ) to F a = F ( α n ).In the dual manner, if F is covariant, we can define the lower cosimplicial system F ∗ , by taking F ∗ ( a ) = F ( α ). Taking again the element b and the morphism ψ ,we use now the fact that the first element of a = ψ ∗ ( b ) is α = β ϕ (0) , which comeswith a canonical arrow g in C from b to it, and we can take ϕ ∗ b = g ∗ from F ∗ ( b ) to F ∗ ( a ).Replacing C by the opposite (or dual) category C op , we exchange contravariantfunctors with covariant ones, and lower systems with upper ones. Remark . Introduce the category S ( K ), having for objects the elements of K ,and for arrows between two elements v ∈ K n and u ∈ K m the elements ϕ of ∆( m, n ) such that ϕ ∗ ( v ) = u . Then a cosimplicial local system F over K is a contravariantfunctor (i.e. a presheaf) from S ( K ) to the category of sets. Definition 6.16.
Let F be a cosimplicial local system over the simplicial set K ;for each n ∈ N , a simplicial cochain of F of degree n is an element ( c u ) u ∈ K n of theproduct C n ( K ; F ) = Q u ∈ K n F u .When F is a local system of abelian groups, C n ( K ; F ) has a natural structureof abelian group. In what follows, we stay in this abelian context.The coboundary operator δ : C n ( K ; F ) → C n +1 ( K ; F ) is defined by(6.20) ( δc )( v ) = n +1 X i =0 ( − i F ( d i , v )( c ( d ∗ i ( v ))) , for any element v of C n +1 ( K ; F ). In the expression, d ∗ i is K ( d i ); remark that thesum takes place in F v . When we want to be more precise, we write δ = δ n +1 n atdegree n . The operator δ is also named the differential of the cochain complex C n ( K ; F ) , n ∈ N . Proposition 6.17.
For all n ∈ N , the equality δ n +2 n +1 ◦ δ n +1 n = 0 holds. In short, δ ◦ δ = 0 .Proof. The expression of δ ◦ δ ( c )( w ) is the sum of elementary terms of the form( − k F ( d n +1 j ◦ d ni , w )( c ( d ∗ i ◦ d ∗ j w )), with i = j and k = i + j if j < i , k = i + j + 1if j > i .It is easy to verify that the maps d satisfy the relation d j d i = d i d j − if i < j [44,Ex. 8.1.1]. It follows that the terms in the sum cancel two by two. (cid:3) A sequence { C n } n ∈ N of abelian groups with an operator δ of degree +1 andsquare zero, is named a differential complex, or a cochain complex. Definition 6.18 (Cohomology of a cosimplicial local system) . The cohomologygroup in degree n ∈ N of the local system F over the simplicial set K is thequotient abelian group H n ( K ; F ) = ker( δ n +1 n ) / im( δ nn − ) . By convention δ − = 0.Equivalently, the cohomology H • ( K ; F ) of F over K , seen as graded vectorspace, is the cohomology of the complex of simplicial cochains ( C • ( K, F ) , δ ).As usual, the elements of ker δ n are called n -cocycles, and those in the image of δ n − are the n -coboundaries.For example, a 0-cochain is a collection ( c u ) u ∈ K , and it a 0-cocycle if for any v ∈ K ,(6.21) 0 = F ( d , v )( c d ∗ v ) − F ( d , v )( c d ∗ v ) . In the particular case of an open covering U of a topological space X , and apresheaf of abelian groups F on X , the group H ( K U ; F ) (for the associated localsystem on the nerve of the covering U ) coincide with the set of global sections of F on X , i.e. the families ( c U ) ∈ Q U ∈U F ( U ) of local sections of F whose restrictioncoincide on the non-empty intersections U ∩ V , U, V ∈ U . This is a short argument with big consequences. When did it appear for the first time? Whocame up with it? Euler, Poincar´e, Noether, Lefschetz, Alexander?
XTRA-FINE SHEAVES AND INTERACTION DECOMPOSITIONS 29
Remark . Let G be a contravariant functor over a poset A . The simpli-cial cochain complex ( C n ( N ( A op ) , G ∗ ) , δ ) associated to the upper local system of G : A op → E , in the sense of the preceding definitions, is precisely the cochaincomplex introduced by (6.3)-(6.4). One could also compute this cohomology fromthe complex ( C n ( N ( A ) , G ∗ ) , δ ): the cochains are the same, and the differential onlydiffers by a sign when n is odd.Similarly, if G is covariant over A , then one can see it as a presheaf on A op ;its sheaf cohomology can be computed as the cohomology of ( C n ( N ( A ) , G ∗ ) , δ ) or( C n ( N ( A op ) , G ∗ ) , δ )As mentioned before, these complexes were rediscovered by O. Peltre [27] forunderstanding geometrically the generalized belief propagation algorithm of [46].Given two cochain complexes of abelian groups C • and D • , whose differentialoperators have degree +1 (i.e. ascending complexes) a cochain map (or cochainmorphism) is a collection { f n : C n → D n } n ∈ Z of morphism of groups that commutewith the differentials. In other words, it is a morphism of graded abelian groups ofdegree zero that commute with the differentials.A chain map between two cochain complexes sends coboundaries to coboundariesand cocycles to cocycles, thus it induces a map at the level of cohomology. Example 6.20.
Let U be a refinement of U ′ , and λ : U → U ′ an adapted projection.The simplicial morphism λ ∗ of the simplicial set K ( U ) to the simplicial set K ( U ′ )in the last section induces, for each integer n , a map λ ∗ from C n ( U ′ ; F ) to C n ( U ; F )defined by λ ∗ ( c ′ ) = c ′ ◦ λ ∗ . More concretely,(6.22) ( λ ∗ c ′ )( U , ..., U n ) = c ′ ( λ ( U ) , ..., λ ( U n )) . This map commutes with the ˇCech differential, then it induces a map in cohomology(6.23) λ ∗ : H n ( U ′ ; F ) → H n ( U ; F ) . Given two cochain maps f • , g • , a cochain homotopy from f • to g • is a morphismof graded groups h from C • to D • of degree − d ◦ h + h ◦ d = g − f. This defines an equivalence relation on cochain maps (of degree zero) which iscompatible with the composition of maps.
Proposition 6.21 ([11, Thm. 4.4]) . If two cochain morphisms are homotopic, theygive the same application in cohomology.Proof. If c is a cocycle of C , and if h is an homotopy from f to g , then dh ( c ) = g ( c ) − f ( c ), thus g ( c ) and f ( c ) have the same classes in cohomology. (cid:3) Definition 6.22 (Lift of simplicial map to a local system) . let F (resp. G ) be acosimplicial local system over the simplicial set K (resp. L ), and f : K → L asimplicial map, a lift e f of f from G to F is family of maps e f u : G f ( u ) → F u , foreach u ∈ K , such that, for any morphism ϕ : m → n , any v ∈ K n , u = ϕ ∗ v ∈ K m ,(6.25) F ( ϕ, v ) ◦ e f u = e f v ◦ G ( ϕ, f ( v )) . An example is given by a morphism ( f, ϕ ) between a presheaf F over X anda presheaf G over Y , when we consider two open coverings U and V of X and Y respectively, such that U is finer than the open covering f − ( V ). In this case,we choose a projection λ from U to V i.e. ∀ U ∈ U , U ⊆ f − λ ( U ). As we have seen, this defines a simplicial map from K ( U ) to K ( V ), that we write f λ ; then, for u = ( U , ..., U n ), the group G ( f λ ( u )) can be identified with G ( T ni =0 ( λ ( U i )), and forevery element g in this group, we pose(6.26) f f λu ( g ) = ϕ ( g ) ∈ F ( n \ i =0 ( U i ));because ϕ can be seen as a morphism of f − G to F . Definition 6.23.
Two pairs ( f, e f ), ( g, e g ) of morphisms and lifts are simpliciallyhomotopic if there exists a simplicial homotopy h : K × ∆ → L from f to g , anda family of maps e h u,s , u ∈ K, s ∈ ∆ from G h ( u,s ) to F u , such that e f = e j ◦ e h and e g = e j ◦ e h , where j = id K × s and j = id K × s .As a consequence of Proposition 6.10, two choices of projections in the construc-tion of the map of local systems associated to a morphism of presheaves gives twohomotopic morphisms in the simplicial sense.Suppose given two local systems F, G over
K, L respectively, and a lift e f of f : K → L . The following formula defines a natural morphism e f ∗ from C • ( L ; G )to C • ( K ; F ):(6.27) e f ∗ ( c L )( u ) = e f u ( c L ( f ( u )) . Lemma 6.24. e f ∗ commutes with the differentials. Definition 6.25.
Let G be a cosimplicial local system over a simplicial set L , and f a simplicial map from K to L , the family F u = G f ( u ) , for u ∈ K , with the maps F ( ϕ, v ) = G ( ϕ, f ( v )) is a cosimplicial local system over K , named the pull-back of G , and denoted by f ∗ ( G ). Example 6.26.
Start with a cosimplicial local system F over a simplicial set K ;then, over the product K × ∆ , we define a local system π ∗ F by taking, for u ∈ K n and s ∈ ∆ , π ∗ F ( u, s ) = F ( u ). Then consider the two injections j = Id K × s and j = Id K × s from K = K × ∆ to K × ∆ ; the two pull-back j ∗ π ∗ F and j ∗ π ∗ F coincide with F . Thus we have evident lifts e j and e j from F to π ∗ F . Theyare homotopic in the simplicial sense, the map h from K × ∆ to itself being theidentity, and the lift being the natural identification.From C • ( K × ∆ ; π ∗ F ) to C • ( K ; F ), the two chain maps e j ∗ and e j ∗ are givenby the following formulas, for c be a n -cochain of K × ∆ with value in π ∗ F ), and u an element of K n e j ∗ c ( u ) = c ( u, (0 , ..., , (6.28) e j ∗ c ( u ) = c ( u, (1 , ..., . (6.29) Lemma 6.27. e j ∗ and e j ∗ are homotopic as chain maps.Proof. Let C be a ( n + 1)-cochain of K × ∆ with value in π ∗ F , and u an elementof K n , we pose(6.30) H ( C )( u ) = n +1 X j =0 ( − j F ( s j , u )( C ( s ∗ j ( u ) , n +1 j )) , XTRA-FINE SHEAVES AND INTERACTION DECOMPOSITIONS 31 where, for n ∈ N and j ∈ [ n + 2], 1 n +1 j denotes the element (0 , ..., , , ...,
1) of∆ n +1 , where the first 1 is at the place j . That gives 1 n +1 n +2 = (0 , ...,
0) = s (0) and1 n +10 = (1 , ...,
1) = s (0).This defines an endomorphism H of degree − C • ( K ; F ). Now we compute, for c ∈ C n ( K × ∆ ; π ∗ F ):(6.31) H ( δc )( u ) = n +1 X j =0 n +1 X k =0 ( − j + k F ( s j , u ) ◦ F ( d k , s ∗ j u ) c ( d ∗ k s ∗ j u, d ∗ k n +1 j ) , and(6.32) δ ( H ( c ))( u ) = n X j =0 n X k =0 ( − j + k F ( d k , u ) ◦ F ( s j , d ∗ k u ) c ( s ∗ j d ∗ k u, n +1 j ) . Let us add H ( δc )( u ) and δ ( H ( c ))( u ), in virtue of the relations (6.8), most of theterms annihilate. The only ones that survive correspond to the terms ( s j d j ) ∗ and( s j d j − ) ∗ . Note that d ∗ j n +1 j = 1 nj and that d ∗ j − n +1 j = 1 nj − , then, due to the signs,they annihilate two by two, except the extreme terms, for j = 0 and j = n + 1,giving(6.33) δ ( H ( c ))( u ) + H ( δc )( u ) = c ( u, n ) − c ( u, nn +1 )= c ( u, (1 , ..., − c ( u, (0 , ..., . Then H is a chain homotopy operator from e j ∗ to e j ∗ , as we desired. (cid:3) Proposition 6.28.
Let f, g be two simplicial maps from the simplicial set K tothe simplicial set L , let F, G be cosimplicial systems over K and L respectively, and e f , e g two lifts over K ; suppose that the pais ( f, e f ) , ( g, e g ) are simplicially homotopic,then the induced maps of cochains complexes are homotopic.Proof. The map e h is a pullback of the simplicial map h to the local systems π ∗ F on K × ∆ and G on L . Thus wee get a chain-map(6.34) e h ∗ : C • ( L ; G ) → C • ( K × ∆ ; π ∗ F ) . On the other side, we have two natural cochain maps, for k = 0 , e j k ∗ : C • ( K × ∆ ; π ∗ F ) → C • ( K ; F ) . Applying the Lemma 6.27, there exists an homotopy from e j ∗ to e j ∗ :(6.36) H : C • ( K ; F ) → C •− ( K ; F );therefore, applying the Lemma 6.24:(6.37) e g ∗ − e f ∗ = e h ∗ ◦ ( e j ∗ − e j ∗ )= e h ∗ ◦ ( d ◦ H + H ◦ d ) = d ◦ ( e h ∗ ◦ H ) + ( e h ∗ ◦ H ) . (cid:3) Corollary 6.29.
The induced morphisms in cohomology are the same.
Theorem 6.30 (cf. [11, Ch. IX]) . If U is a refinement of U ′ , two projections λ, µ from U to U ′ give the same application in cohomology. Proof.
The simplicial maps λ ∗ and µ ∗ from K ( U ) to K ( U ′ ) are homotopic in thesimplicial sense, then the maps λ ∗ and µ ∗ are homotopic in the sense of maps ofdifferential complexes, cf. last section). (cid:3) Comparison theorems.
Given a covering U of a topological space X , let A ( U ) denote the poset whose objects are finite intersections of elements of U , or-dered by inclusion (thus the morphisms go from intersections to partial intersec-tions), and N ( U ) = N ( A ( U )) denotes the nerve of the category A ( U ), cf. Example6.6.The objects of A ( U ) make an open covering of X which is finer than U . Bychoosing for each non-empty finite intersection of elements of U one of these ele-ments, we obtain a map from A ( U ) to U , that we denote π ; it is a projection inthe sense of Eilenberg-Steenrod, see Example 6.9. In what follows we will alwaysassume that for U ∈ U , π ( U ) = U . The map π induces a simplicial map π ∗ fromthe simplicial set N ( U ) to the simplicial set K ( U ) (which is the usual nerve of thecovering U in the sense of Example 6.4); it maps the sequence V → · · · → V n ofelements of A ( U ) to the sequence ( π ( V ) , ..., π ( V n )) of elements of U .Given a presheaf F of abelian groups over X , we have defined the cosimpliciallocal system of ˇCech F ∨ over K ( U ) (cf. Example 6.13). To define a local system over N ( U ), we restrict F to a presheaf on A ( U ) and we take the lower cosimplicial localsystem F ∗ over N ( U ), as in Example 6.14. Given an element v = ( V → · · · → V n )of N n ( U ), remark that V = T ni =0 V i ⊆ T ni =0 π ( V i ), hence there is a well-definedrestriction map from F ∨ ( π ∗ v ) to F ∗ ( v ). This defines a lift ˜ π of π ∗ from F ∨ to F ∗ in the sense of Definition 6.22, hence a morphism(6.38) π ∗ : C • ( K ( U ); F ∨ ) → C • ( N ( U ); F ∗ ) Theorem 6.31.
The map π ∗ is an homotopy equivalence between C • ( K ( U )) and C • ( N ( U )) . Before presenting the proof, let us see how this implies the isomorphism be-tween topos cohomology and ˇCech cohomology in the case of abelian presheaveson a poset, provided it is a conditional meet semilattice i.e. that products existsconditionally. Let G be a contravariant functor of abelian groups on a poset A ,and let b G be the induced sheaf on the upper A-space X A . We have seen that thetopos cohomology of G ∈ PSh( A ) is isomorphic to the cohomology of the cochaincomplex ( C • ( N ( A ) , G ∗ ) , δ ), whereas the ˇCech cohomology of b G is the cohomologyof the complex ( C • ( K ( U A ) , G ∨ ) , δ ).The space X A has a finest canonical open covering U A made by the upper sets U α ; α ∈ A . An inclusion U α ⊆ U β corresponds to an arrow α → β , then thenatural inclusion A ֒ → A ( U A ) is a covariant functor, and induces an injectivesimplicial covariant functor ι : N ( A ) ֒ → N ( U A ).We have a diagram of simplicial sets(6.39) N ( A ) ι → N ( U A ) π ∗ → K ( U A )where the last arrow is induced by any projection π : A ( U A ) → U A that is theidentity on U A . Let us denote by j the simplicial map π ∗ ◦ ι . Given a = ( α →· · · → α n ) in N ( A ), we have(6.40) G ( α ) = G ∗ ( a ) = G ∨ ( j ( a )) = b G ( ∩ ni =0 U α i ) = b G ( U α ) , XTRA-FINE SHEAVES AND INTERACTION DECOMPOSITIONS 33 then there is a lift of j from G ∨ to G ∗ (cf. Definition 6.22) given by identities.Thus we deduce a morphism of chain complexes(6.41) j ∗ : C • ( K ( A ); G ∨ ) → C • ( N ( A ); G ∗ )that induces a morphism in cohomology. Corollary 6.32.
If products exist conditionally in A , the chain map j ∗ is a chainequivalence up to homotopy, thus induces an isomorphism in cohomology.Proof. Under the hypothesis, one has A ( U A ) = U A , since every intersection U α ∩· · · ∩ U α n equals U α ∧···∧ α n . Hence N ( A ) ∼ = N ( U A ). The map π ∗ is induced by π = id U A . The claim then follows from Theorem 6.31. (cid:3) The equivalence above is natural in the category of posets with presheaves up tohomotopy.We close this section with the proof of Theorem 6.31, inspired by a classicalargument of Eilenberg and Steenrod: starting with a simpicial complex K , theyassociated to it the poset A ( K ), whose elements are the faces (simplicies) of K ;the nerve N of this poset is naturally isomorphic to the barycentric subdivisionof K (cf. [33]). In [12, pp. 177-178], the authors proved that there exists anhomotopy equivalence between N and K . The following proof is an adaptation oftheir argument to this more general setting. Proof of Theorem 6.31.
1) first we construct by recurrence over n a linear applica-tion Sd n from C n ( N ( U )) to C n ( K ( U )), having the two following properties:( i ) (locality) for any c ∈ C n ( N ( U )) and any collection u = ( U , ..., U n ), the value( Sd n c )( u ) in F ( U u ) depends only of the values of c on the descendent of the opensets U i ; i = 0 , ..., n , i.e. the values c ( v ) ∈ F ( V n ) for the sequences v = ( V , .., V n ),where each V i ; i = 0 , ..., n is included in a U j ; j = 0 , ..., n ;( ii ) (morphism of cochain complex) d ◦ Sd ∗ = Sd ∗ ◦ d .For n = 0, and U ∈ U = K ( U ), we take Sd ( c )( U ) = c ( U ), this is allowedbecause U is also an element of N ( U ). The condition ( i ) is evidently satisfied,and ( ii ) is empty in this degree.For n = 1, c ∈ C ( N ( U )) and u = ( U , U ), we pose Sd c ( U , U ) = c ( U , U u ) − c ( U , U u ), where U u = U ∩ U . This is local, and for c ∈ C ( N ( U )):(6.42) Sd ( dc )( U , U ) = ( c ( U ) − c ( U u )) − ( c ( U ) − c ( U u ))= dc ( U , U ) = d ◦ Sd ( c )( U , U ) . Then take n ≥
2, and suppose that a map Sd q is constructed for every q ≤ n − i ) and ( ii ). Take a cochain c in C n ( N ( U )), and consider an element u = ( U , ..., U n ) of K n ( U ); remind we note U u the intersection (necessary non-empty) of the U i , i = 0 , ..., n . We define an element c u ∈ C n − ( N ( U )) by taking onevery decreasing sequence v = ( V , ..., V n − ),(6.43) c u ( v ) = c ( V , ..., V n − , U u ) , if V n − contains U u and taking c u ( V , ..., V n − ) = 0 in the opposite case.Then we define(6.44) Sd n ( c )( u ) = n X i =0 ( − n − i Sd n − ( c u )( U , ..., c U i , ..., U n ) | U u . The locality ( i ) follows from the recurrence hypothesis: the definition of c u depends only of U u which is a descendent of u , and this is the same for the restrictionto U u , moreover the value of Sd n − ( c u ) on ( U , ..., c U i , ..., U n ) depends only of thevalues of c u on the sequences of descendent of ( U , ..., c U i , ..., U n ).For ( ii ), we have to compute Sd n ( dc )( u ) for a cochain c ∈ C n − ( N ( U ); F ). Fora decreasing sequence V , ..., V n − , then, by writing V n = U u , we have( dc ) u ( V , ..., V n − ) = dc ( V , ..., V n − , U u )= n X j =0 ( − j c ( V , ..., c V j , ..., V n ) | U u = d ( c u )( V , ..., V n − ) | U u + ( − n c ( V , ..., V n − ) | U u ;where c u is also defined by c u ( V , ..., V n − ) = c ( V , ..., V n − , U u ) if V n − contains U u and c u ( V , ..., V n − ) = 0 in the opposite case. Which gives for reference, when c belongs to C n − ( N ( U ); F ):(6.45) d ( c u ) = ( dc ) u + ( − n − c. It follows from the recurrence hypothesis that Sd n ( dc )( u ) = n X i =0 ( − n − i Sd n − (( dc ) u )( U , ..., c U i , ..., U n ) | U u = n X i =0 ( − n − i ( Sd n − d ( c u ))( U , ..., c U i , ..., U n ) | U u + n X i =0 ( − i ( Sd n − c )( U , ..., c U i , ..., U n ) | U u = n X i =0 ( − n − i ( d ◦ Sd n − ( c u ))( U , ..., c U i , ..., U n ) | U u + n X i =0 ( − i ( Sd n − c )( U , ..., c U i , ..., U n ) | U u = ( − n ( d ◦ d ) Sd n ( c u )( u ) + d ( Sd n − ( c ))( u )= d ◦ Sd n − ( c )( u ) . Therefore Sd n verifies ( ii ).2) let us prove that the composition Sd ∗ ◦ π ∗ is homotopic to the identity of C • ( K ( U ); F ).For that purpose we construct a sequence of homomorphisms,(6.46) D n +1 K : C n +1 ( K ( U ); F ) → C n ( K ( U ); F ) , for n ≥
0, by recurrence over the integer n , such that(6.47) Id − Sd n ◦ π n = d ◦ D nK + D n +1 K ◦ d. For n = 0, and c ∈ C ( K ( U ); F ), we simply take D K ( c )( U ) = 0. This worksbecause, if c is a 0-cochain of K ( U ) , F ,(6.48) ( Sd ◦ π c )( U ) = c ( U ) . XTRA-FINE SHEAVES AND INTERACTION DECOMPOSITIONS 35
For n = 1, and c ∈ C ( K ( U ); F ), take(6.49) D K c ( U , U ) = c ( U , U , π ( U ∩ U ) | U ∩ U . This gives for c ′ ∈ C ( K ( U ); F ):(6.50) D K ( dc ′ )( U , U ) = c ′ ( U , π ( U u )) | U u − c ′ ( U , π ( U u )) | U u + c ′ ( U , U );where as usual we have denoted U ∩ U by the symbol U u . On the other side,( Sd ◦ π c ′ )( U , U ) = − ( π c ′ ) u ( U ) | U u + ( π c ′ ) u ( U ) | U u = − ( π c ′ )( U , U u ) | U u + ( π c ′ )( U , U u ) | U u = − c ′ ( U , π ( U u )) | U u + c ′ ( U , π ( U u )) | U u . Then Id − Sd ◦ π = D K ◦ d + d ◦ D K , as we expected.More generally, for any n , consider consider a n -cochain c of K ( U ) with respectto the local system F . For a sequence u ′ = ( U ′ , ..., U ′ n − ) in U , let us define(6.51) c πu ( U ′ , ..., U ′ n − ) = c ( U ′ , ..., U ′ n − , π ( U u ))if U u ′ ⊇ π ( U u ), and c πu ( U ′ , ..., U ′ n − ) = 0 if not.Then consider a decreasing sequence v = ( V , ..., V n − ) in A U . If U u ⊆ U π ( v ) = T n − i =0 π ( V i ), ( π n c ) u ( V , ..., V n − ) = π n c ( V , ..., V n − , U u )= c ( π ( V ) , ..., π ( V n − ) , π ( U u ))= c πu ( π ( V ) , ..., π ( V n − ))= π n − ( c πu )( V , ..., V n − ) . If U u * U π ( v ) , we have ( π n c n ) u ( v ) = 0 = ( π n − ( c πu ))( v ). Therefore, in all cases(6.52) ( π n c ) u ( v ) = π n − ( c πu )( v ) . Now assume that D q +1 K is defined for q ≤ n , satisfying the homotopy relation for Id − Sd q ◦ π q , and consider a n -cochain c of K ( U ) with respect to the local system F ; for every sequence u = ( U , ..., U n ) in U , the chosen definition of Sd n gives(6.53) ( Sd n ◦ π n ( c ))( U , ..., U n ) = n X i =0 ( − n − i ( Sd n − ( π n c ) u )( U , ..., c U i , ..., U n ) | U u . Thus, applying (6.52) we get(6.54)( Sd n ◦ π n ( c ))( U , ..., U n )) = n X i =0 ( − n − i Sd n − ◦ π n − ( c πu )( U , ..., c U i , ..., U n ) | U u . By applying the hypothesis of recurrence, we get(6.55) ( Sd n ◦ π n ( c ))( U , ..., U n )) = ( − n n X i =0 ( − i c πu ( U , ..., c U i , ..., U n ) | U u + n X i =0 ( − n +1 − i D nK ◦ d ( c πu )( U , ..., c U i , ..., U n ) | U u + n X i =0 ( − n +1 − i d ( D n − K ( c πu ))( U , ..., c U i , ..., U n ) | U u . The last sum is zero due to d ◦ d = 0, and the first sum is ( − n d ( c πu ), therefore(6.56) ( Sd n ◦ π n ( c ))( U , ..., U n ) = ( − n d ( c πu )( U , ..., U n )+ n X i =0 ( − n +1 − i D nK ◦ d ( c πu )( U , ..., c U i , ..., U n ) | U u As we obtained a formula for d ( c u ), we obtain a formula for d ( c πu ) | U u . In fact,writing U n +1 = π ( U u ),( dc ) πu ( U , ..., U n ) | U u = dc ( U , ..., U n , π ( U u )) | U u = n +1 X j =0 ( − j c ( U , ..., c U j , ..., U n ) | U u = d ( c πu )( U , ..., U n ) | U u + ( − n +1 c ( U , ..., U n ) | U u ;Then, replacing d ( c πu ) | U u by ( dc ) πu + ( − n c in the formula (6.55), we get(6.57) ( Sd n ◦ π n ( c ))( U , ..., U n ) = c ( U , ..., U n ) + ( − n ( dc ) πu ( U , ..., U n )+ ( − n +1 d ◦ D nK ◦ ( dc ) πu )( U , ..., c U i , ..., U n ) | U u − d ◦ D nk ( c )( U , ..., U n ) . Assuming that we have defined D nK on C n ( K ( U ); F ), we define D n +1 K on C n +1 ( K ( U ); F )by the following formula:(6.58) D n +1 K ( c ′ ) = ( − n +1 ( c ′ ) πu + ( − n +1 dD nK ( c ′ ) πu . This gives the awaited result.3) To finish the proof of the theorem, we have to demonstrate that the composition π ∗ ◦ Sd ∗ is homotopic to the identity of C • ( N ( U ); F ). For that, we construct asequence of homomorphisms,(6.59) D n +1 N : C n +1 ( N ( U ); F ) → C n ( N ( U ); F ) , by recurrence over the integer n ≥
0, such that(6.60) Id − π n ◦ Sd n = d ◦ D nN + D n +1 N ◦ d. For n = 0, and c ∈ C ( N ( U ); F ), we define D N ( c )( V ) = c ( π ( V ) , V ). Rememberthat if c is a zero cochain for N ( U ), Sd c ( U ) = c ( U ). Then(6.61) π Sd c ( V ) = c ( π ( V )) = c ( V ) + ( cπ ( V )) − c ( V )) = c ( V ) − D N ( dc ))( V );which gives c − π Sd c = D N ( dc )) as desired.Now assume the recurrence hypothesis, that there exist operators D q +1 K for q ≤ n ,satisfying the homotopy relation for Id − π q ◦ Sd q , and consider a n -cochain c of N ( U )with respect to the local system F ; for every decreasing sequence v = ( V , ..., V n )in A U , we have( π n ◦ Sd n ( c ))( v ) = ( Sd n c )( π ( V ) , ..., π ( V n )) | V n = n X i =0 ( − n − i Sd n − ( c ) π ( v ) ( π ( V ) , ..., [ π ( V i ) , ..., π ( V n )) | V n = n X i =0 ( − n − i π n − ( Sd n − ( c ) π ( v ) )( V , ..., b V i , ..., V n ) | V n ; XTRA-FINE SHEAVES AND INTERACTION DECOMPOSITIONS 37 which gives by applying the hypothesis of recurrence:(6.62) ( π n ◦ Sd n ( c ))( V , ..., V n ) = n X i =0 ( − n − i c π ( v ) ( V , ..., b V i , ..., V n ) | V n + n X i =0 ( − n +1 − i D nN ( dc π ( v ) )( V , ..., b V i , ..., V n ) | V n + n X i =0 ( − n +1 − i d ◦ D n − N ( c π ( v ) )( V , ..., b V i , ..., V n ) | V n . The last sum is zero due to d ◦ d = 0, the first one is equal to ( − n d ( c π ( v ) )( v ), andthe second one to ( − n +1 d ( D nN ( dc π ( v ) )( v ), that is(6.63) ( π n ◦ Sd n ( c ))( v ) = ( − n d ( c π ( v ) )( v ) + ( − n +1 d ( D nN ( dc π ( v ) )( v ) . But the relation (6.45) tells that(6.64) d ( c π ( v ) )( v ) = ( dc ) π ( v ) ( v ) + ( − n c ( v ) . Thus by substituting, we get(6.65) ( π n ◦ Sd n ( c ))( v ) = c ( v ) + ( − n ( dc ) π ( v ) )( v ) − d ( D nN ( c )( v ) − ( − n d ( D nN ( d ( c π ( v ) ))( v );which gives the expected result,(6.66) c ( v ) − ( π n ◦ Sd n ( c ))( v ) = d ( D nN ( c ))( v ) + D n +1 ( dc )( v ) π ( v ) )( v );if we define, for any c ′ ∈ C n +1 ( N ( U ); F ) and any v in N n ( U ):(6.67) D n +1 N ( c ′ )( v ) = ( − n +1 c ′ π ( v ) ( v ) + ( − n dD nN ( c ′ π ( v ) ( v ) . This ends the proof. (cid:3)
The constructions made in the proof show that the homotopy equivalence isnatural in the category of open covering of topological spaces and morphisms oflocal systems.
Appendix A. Topology and sheaves
Remind that a topological space is a set X , equipped with a subset T of the setof parts P ( X )—named its topology —that is supposed to contain X and the emptyset ∅ , and to be closed under union and finite intersection. A map f : X → Y between topological spaces is said continuous if the inverse image of an open setis an open set. A topology T is said finer than a topology T ′ if the identity iscontinuous from X T to X T ′ . It is equivalent to ask that T ′ ⊆ T as elements of P ( P ( X )).An open covering of an open set V ∈ T is a subset U ⊆ T such that V = S U ∈U U .A topology T can be seen as a category, whose objects are the open sets of X (i.e. the elements of T ); whenever U ⊆ V , there is one U → V . The resultingcategory T is a poset (see Section 3).A presheaf F over a topological space X is a contravariant functor from T to thecategory of sets E , i.e. a family of sets { F ( U ) = F U } U ∈T , and maps { π V U } ( U → V ) ∈T such that π UU = Id F ( U ) and π W V ◦ π V U = π W U when W ⊆ V ⊆ U . Frequently we will note π V U ( s ) = s | V , as a restriction. Sometimes, the elements s of F U arenamed sections of F over U .A sheaf is a presheaf which satisfies the two following axioms:(1) For every V ∈ T and every open covering U ⊆ T of V , if s, t are twoelements of F V such that for any U ∈ U we have s | U = t | U , then t = s .(2) For every V ∈ T and every open covering U ⊆ T of V , if a family ( s U ) U ∈U ∈ Q U ∈U F U is such that for all U, U ′ ∈ U , s U | ( U ∩ U ′ ) = s U ′ | ( U ∩ U ′ ), thenthere exists s ∈ F V such that for all U ∈ U , s | U = s U .The notion of presheaf extends to any category C in place of E : just take acontravariant functor from T to C . However the definition of sheaf requires a priorithat C is a sub-category of E .One of the main theorems in sheaf theory is the existence of a canonical sheaf F ∼ associated to a presheaf F on ( X, T ), built as follows [22, Sec. II.5]. Onesays s ∈ F U and t ∈ F V have the same germ at x if there exists W ⊆ U ∩ V such that s | W = t | W . Having the same germ at x is an equivalence relation andone denotes germ x s the corresponding equivalence class. More precisely, one candescribe the set of all germs as a colimit lim −→ x ∈ U F U over all open neighborhoodsof U ; the resulting set F x is called the stalk of F at x . Set Λ F = Q x ∈ X F x , andintroduce the obvious projection p : Λ F → X . Any s ∈ F U determines a map˙ s : U → Λ P , x ( x, germ x s ), which is a section of p . The set Λ F is topologizedintroducing { ˙ s ( U ) | U ∈ T , s ∈ F U } as a basis of open sets. Then F ∼ is definedas the sheaf of (continuous) sections of Λ P over the opens of X . This means thatan element of F ∼ ( U ) is a family ( s x ) ∈ Q x ∈ U F x which is locally a germ of F : forall y ∈ U , there exist V ∈ T and t ∈ F V such that y ∈ V ⊆ U and for all x ∈ V ,germ x t = s x . The map s ˙ s defines a natural transformation F → F ∼ , which isan isomorphism when F is a sheaf.We consider now the functoriality of sheaves. Let f : X → Y be a continuousmap; it induces a functor f − : T Y → T X between the topologies (seen as categories)of Y and X , respectively.(1) If F is a presheaf over X , the direct image f ∗ F is defined on Y by theformula: f ∗ F ( V ) = F ( f − ( V )). If F is a sheaf, this is also the case for f ∗ F [22]. (In fact, f ∗ F is also the pullback ( f − ) ∗ F of F under the functor f − according to [3, Sec. I.5].)(2) If G is a presheaf over Y , the inverse image f − G is defined on X by theformula: f − G ( U ) = lim −→ V ⊇ f ( U ) G ( V ), where the limit is taken over thedirected family of opens subsets V of Y which contain f ( U ). Even if F isa sheaf, in general f − G is not a sheaf. We make use of the sheafification,and define the pullback of G by f ∗ G = ( f − G ) ∼ .The functors f ∗ , f − between the corresponding categories of presheaves areadjoint i.e. for any presheaves F on X and G on Y , there exist natural bijections(A.1) Hom X ( f − G , F )) ∼ = Hom Y ( G , f ∗ F ) . Similarly, f ∗ is left adjoint to f ∗ in the categories of sheaves. Definition A.1.
A map of presheaves (resp. sheaves) from ( X, F ) to Y, G ) is apair ( f, ϕ ), where f : X → Y is continuous, and ϕ is a morphism from G to f ∗ F ,or equivalently a morphism ϕ ∗ from f − G (resp. f ∗ G ) to F . XTRA-FINE SHEAVES AND INTERACTION DECOMPOSITIONS 39
Appendix B. ˇCech cohomology We summarize some facts concerning ˇCech cohomology.B.1.
Limit over coverings. A preorder is a set P with a binary relation that istransitive and reflexive. Equivalently, is a small category P where there exists atmost one arrow between two objects. The preorder P is called directed if for anyobjects a and b of P , there exists an object c such that a → c and b → c .As we saw in Section 6.2, a covering U is called a refinement of another covering U ′ when every set of U is contained in some set of U ′ . In that case, there exists amap λ : U → U ′ , called projection , such that for every U ∈ U one has U ⊆ λ ( U ). Itis also said that U is finer than U ′ [11].This notion of refinement does not give in general a partial ordering amongcoverings, but only a pre-order. So it is unlike the notion of finer topology, whichcorresponds to the natural partial ordering by inclusion of subsets. This can beillustrated with two coverings of R , such that U = { ] n, ∞ [ | n even } and U ′ = { ] n, ∞ [ | n odd } . Lemma B.1.
The category of open coverings of X , such that U → U ′ if U ′ refines U , is a directed set .Proof. If U and V are open coverings of X , the set of non-empty intersections U ∩ V ,for U ∈ U and V ∈ V is a refinement of both U and V . (cid:3) Given a directed set P , a directed system of sets (associated to P ) is a covariantfunctor from P to a category C , i.e. a family of objects E a for a ∈ P , and afamily f ab of morphisms E a → E b , associated to ordered pairs a (cid:22) b , such that ∀ a, f aa = 1 E a and ∀ a, b, c, a (cid:22) b (cid:22) c ⇒ f ac = f bc ◦ f ab .By definition a direct limit of such direct system in the category C is an object E with a set of morphisms E a → E, a ∈ P , such that for any a (cid:22) b ϕ b ◦ f ab = ϕ a ,which is initial, i.e. for any object Y and set ψ a : E a → Y verifying the same rulethere exist a morphism h : E → Y making all evident diagrams commutative. Ifsuch a limit exists it is unique up to unique isomorphism, and denoted lim −→ E a .When C is the category of sets E , the direct limit always exists, it is a the quotientof the union of the disjoint sets b E a = E a × { a } by the equivalence relation e a ≈ e b if there exists c ∈ C , with a (cid:22) c , b (cid:22) c and f ac ( e a ) = f bc ( e b ), i.e. asymptoticequality. If the category C is the subcategory of E made by abelian groups andtheir morphisms, the direct limit is an abelian group. Definition B.2.
For all n ∈ N , H n ( X ; F ) = lim −→ H n ( U ; F ), the direct limit beingassociated to the directed set of open coverings of X .B.2. Functoriality.
Suppose given a map of presheaves ( f, ϕ ) : ( X, F ) → ( Y, G ),and two open coverings U , V of X and Y respectively, such that U is a refinementof f − ( V ).We can choose a projection map λ from U to V , i.e. ∀ U ∈ U , U ⊆ f − ( λ ( V )).From Proposition 6.10, two such maps are homotopic in the simplicial sense. Thisinduces a natural application of chain complexes:(B.1) ( f, ϕ, λ ) ∗ : C • ( V ; G ) → C • ( U ; F ) , which commutes with the coboundary operators.Consider the particular case of an inclusion J : X ֒ → Y . A covering V of Y induce a covering U of X , made of the (non-empty) intersections V ∩ X for V ∈ V ; there is an evident projection λ from U to V . Hypothesis : the map ϕ is surjective, i.e. for any open set V in Y the map ϕ V : G ( V ) → F ( V ∩ X ) is surjective.In particular this happens if G = J ∗ ( F ) over Y .If c ∈ C n ( U ; F ), there exists e c ∈ C n ( V ; G ) such that, for any family V , ..., V n ofelements of V , we have(B.2) ϕ ( e c ( V , ..., V n )) = c ( V ∩ X, ..., V n ∩ X ) ∈ F ( n \ i =0 ( V i ∩ X ) = ϕ ( G ( n \ i =0 ( V i )) . This gives f f λ ∗ ( e c ) = c , then the map f f λ ∗ = ( f, ϕ, λ ) ∗ is surjective.Let us define(B.3) C • ( V , U ; G , F ) = ker(( f, ϕ, λ ) ∗ ) . By the snake’s lemma, we obtain a natural long exact sequence in cohomology:(B.4) ... → H q ( V ; G ) → H q ( U ; F ) → H q +1 ( V , U ; G , F ) → H q +1 ( V ; G ) → H q +1 ( U ; F ) → ... This sequence survive to the direct limits over coverings and gives an exact ofˇCech cohomology of the pair ( F , G ) over the pair ( X, Y ). Appendix C. Finite probability functors
This is a continuation of Section 5 on free sheaves. Our aim is to give a proof ofthe Theorem 5.16.We introduce now the hypothesis that the { E i } i ∈ I are finite sets, of respectivecardinality N i .If N α denotes the cardinality of E α , we have N α = Q i ∈ α N i .If we suppose that A satisfies the strong intersection property, the sheaf V hasan interaction decomposition:(C.1) ∀ α ∈ A , V α = M β ⊆ α S β , Let us denote by D α the dimension of S α , for α ∈ A . We have N α = P α → β D β .Then the M¨obius inversion formula gives(C.2) ∀ α ∈ A , D α = X α → β µ α,β N β ;where the integral numbers µ αβ are the M¨obius coefficents of A .Let us remind what are these coefficients [32], Black 2015. For any locally finiteposet A , they are defined by the two following equations:(C.3) ∀ α, γ, δ α = γ = X β | α → β → γ µ α,β = X α → β → γ µ β,γ . (C.4) ∀ α, β, α β ⇒ µ α,β = 0 . This gives a function from
A × A to Z , which is named the M¨obius function ofthe poset. The M¨obius function of A op is given by µ ∗ β,α = µ α,β . XTRA-FINE SHEAVES AND INTERACTION DECOMPOSITIONS 41
For example, if A is the full set of parts of a finite set I , including the empty setor not, we have, for β ⊆ α :(C.5) µ α,β = ( − | α |−| β | , where | α | denotes the cardinality of α , for any α ∈ A . This formula is called theinclusion-exclusion principle. When β = ∅ , the above formula holds true if we pose |∅| = − α ⊇ ω are two elements of A , and if A ( α, ω ) is the sub-poset of A made bythe elements β such that α → β → ω , the restriction of the M¨obius function of A to A ( α, ω ) coincides with the M¨obius function of A ( α, ω ).The formula (C.5) extends to the poset associated to any simplicial complex.This follows from the preceding assertion, because in the case of a manifold, forevery pair of elements α, ω of A such that ω ⊆ α , the elements β between α and ω are the same in A or in the simplex defined by α .If A verifies the strong intersection property, for each α ∈ A , the dimension of H ( U A ; S α ) is D α , then Theorem 5.7 and Proposition 5.8 imply: Proposition C.1.
If the poset A is finite and satisfies the strong intersectionproperty, and if the { E i } i ∈ I are finite sets, (C.6) dim K H ( U A ; V ) = X α,β ∈A µ αβ N β . In particular for the full simplex ∆( n −
1) = P ( J ), if J has cardinality n , and if N i = N for any vertex, the dimension of H ( A ; V ) is N n . Proof.
Since we include the empty set, with V ∅ = S ∅ of dimension 1, we get: X α,β ∈A µ αβ N β = n X k =0 C kn k X l =0 C lk ( − k − l N l = n X k =0 ( − k C kn (1 − N ) k = n X k =0 C kn ( N − k = ( N − n = N n . (cid:3) Remark
C.2 . In this case, if we remove the empty set, and compute the expressionwe get the same result X α,β ∈A µ αβ N β = n X k =1 C kn k X l =1 C lk ( − k − l N l = n X k =1 ( − k C kn ((1 − N ) k − n X k =1 C kn ( N − k − n X k =1 C kn ( − k = (( N − n − − ((1 − n −
1) = N n . Let us now delete the maximal face α = I , then the poset A becomes theboundary ∂ ∆( n −
1) of the ( n − A , and compute the dimension of H ; we obtain X α,β ∈A × + µ αβ N β = n − X k =0 C kn k X l =0 C lk ( − k − l N l = n − X k =0 ( − k C kn (1 − N ) k = n − X k =0 C kn ( N − k = ( N − n − ( N − n = N n − ( N − n . Remark
C.3 . Now the expression P α,β ∈A µ αβ N β is not the same if we exclude ∅ ,because in this case, we have X α,β ∈A × µ αβ N β = n − X k =1 C kn k X l =1 C lk ( − k − l N l = n − X k =1 ( − k C kn ((1 − N ) k −
1) = n − X k =1 C kn ( N − k − n − X k =1 C kn ( − k = (( N − n − − ( N − n ) − ((1 − n − − ( − n )= N n − ( N − n + ( − n . We will see just below why there is a difference for the boundary ∂ ∆( n −
1) andnot for the simplex ∆( n − A satisfies the weak intersection property, then(C.7) ∀ α ∈ A , V α = M β ⊆ α S β . Proposition C.4.
If the poset A is finite and satisfies the weak intersection prop-erty, and if the { E i } i ∈ I are finite sets, (C.8) dim K H ( U A ; V ) = X α,β ∈A µ αβ ( N β − . Proof.
We apply Proposition 5.9 as we applied Proposition 5.8 to prove PropositionC.1. (cid:3)
In disguise, this result is known under the name of the Marginal Theorem ofH.G. Kellerer [19] (see also F. Mat´uˇs [24]).
Definition C.5.
Let A be a finite poset, the Euler characteristic of A is definedby(C.9) χ ( A ) = X α,β ∈A µ αβ . In fact, the Euler characteristic was defined by Rota [32], when A contains amaximal element I and a minimal element ∅ , by the formula(C.10) E ( A ) = 1 + µ I, ∅ . But take any finite poset A , and add formally to A a maximal element 1 and aminimal element 0, obtaining a poset A + . Then, for any α ∈ A ,(C.11) 0 = µ ( α,
0) + X β ∈A ,β ⊆ α µ ( α, β ) , XTRA-FINE SHEAVES AND INTERACTION DECOMPOSITIONS 43 and(C.12) 0 = µ (1 ,
0) + µ (0 ,
0) + X α ∈A µ ( α, . Consequently(C.13) χ ( A ) = µ (1 ,
0) + µ (0 ,
0) = E ( A + ) . Therefore the two definitions accord. See also the categorical extension of theseideas by Tom Leinster [21].The
Hall formula (cf. [32]), tells that(C.14) E ( A ) = r − r + r − ... ;where each r k is the number of non degenerate chains of length k in A . Thisnumber is the Euler characteristic of the nerve N ( A ) of the category A , therefore χ ( A ) coincides with the Euler characteristics of N ( A ). But we have seen in Section6 that the ˇCech cohomology of the (lower) Hausdorff space A with coefficients in Z , is isomorphic to the simplicial cohomology of N ( A ). Then χ ( A ) also coincideswith the Euler- ˇCech characteristic of the (lower) Hausdorff space A . By duality ofthe M¨obius function, this is also true for the upper topology.From the inclusion-exclusion formula, it is easy to show that for the poset of asimplicial complex, the number χ ( A ) is the alternate sum of the numbers of facesof each dimension:(C.15) χ ( A ) = a − a + ... as in the original definition by Euler.Now consider A (finite) as a topological subspace A t of the simplex P ( I ); itsclosure A is a simplicial complex. Moreover, if A satisfies the weak intersectionproperty, the inclusion of A t in A is an equivalence of homotopy; therefore, in thiscase, χ ( A ) is also the usual Euler characteristic of the metric space A .Consequently, Proposition C.4 can be rephrased by the following formula(C.16) dim K H ( U A ; V ) + χ ( A ) = X α,β ∈A µ αβ N β . Applying Theorem 5.7, we get the following result:
Theorem 5.16.
If the poset A is finite and satisfies the weak intersection property,and if the { E i } i ∈ I are finite sets, then (C.17) χ ( A ; V ) = ∞ X k =0 ( − k dim K H k ( U A ; V ) = X α,β ∈A µ αβ N β . Remark that we also have(C.18) χ ( A ; V ) = dim K H ( U A ; V ) + χ ( A ) . In the example of ∆( n − χ ( A ) = 1, and when A = ∂ ∆( n −
1) wehave χ ( A ) = 1 + ( − n , therefore, with all the N i equals to N , this explains theresults obtained in the previous remarks.The standard marginal problem: when compatible measures of sum 1 over theposet ∂ ∆( n −
1) come from a global measure, corresponds to Proposition C.4, thenthe measure always exists but it depends on ( N − n degrees of freedom. Moreover,in general none of these measures is positive. References
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