Foliations-Webs-Hessian Geometry-Information Geometry-Entropy and Cohomology †

Abstract: Let us begin by considering two book titles: A provocative title, What Is a Statistical Model? McCullagh (2002) and an alternative title, In a Search for Structure. The Fisher Information. Gromov (2012). It is the richness in open problems and the links with other research domains that make a research topic exciting. Information geometry has both properties. Differential information geometry is the differential geometry of statistical models. The topology of information is the topology of statistical models. This highlights the importance of both questions raised by Peter McCullagh and Misha Gromov. The title of this paper looks like a list of key words. However, the aim is to emphasize the links between those topics. The theory of homology of Koszul-Vinberg algebroids and their modules (KV homology in short) is a useful key for exploring those links. In Part A we overview three constructions of the KV homology. The first construction is based on the pioneering brute formula of the coboundary operator. The second construction is based on the theory of semi-simplicial objects. The third construction is based on the anomaly functions of abstract algebras and their abstract modules. We use the KV homology for investigating links between differential information geometry and differential topology. For instance, “dualistic relation of Amari” and “Riemannian or symplectic Foliations”; “Koszul geometry” and “linearization of webs”; “KV homology” and “complexity of models”. Regarding the complexity of a model, the challenge is to measure how far from being an exponential family is a given model. In Part A we deal with the classical theory of models. Part B is devoted to answering both questions raised by McCullagh and B Gromov. A few criticisms and examples are used to support our criticisms and to motivate a new approach. In a given category an outstanding challenge is to find an invariant which encodes the points of a moduli space. In Part B we face four challenges. (1) The introduction of a new theory of statistical models. This re-establishment must answer both questions of McCullagh and Gromov; (2) The search for an characteristic invariant which encodes the points of the moduli space of isomorphism class of models; (3) The introduction of the theory of homological statistical models. This is a pioneering notion. We address its links with Hessian geometry; (4) We emphasize the links between the classical theory of models, the new theory and Vanishing Theorems in the theory of homological statistical models. Subsequently, the differential information geometry has a homological nature. That is another notable feature of our approach. This paper is dedicated to our friend and colleague Alexander Grothendieck.


The Notation
Throughout the paper we use tha following notation. N is the set of non negative integers, Z is the ring of integers, R is the field of real numbers, C ∞ (M) is the associative commutative algebra of real valued smooth functions in a smooth manifold M. Let ∇ be a Koszul connection in a manifold M, R ∇ is the curvature tensor of ∇. It is defined by T ∇ is the torsion tensor of ∇. It is defined by Let X be a smooth vector field in M. L X ∇ is the Lie derivative of ∇ in the direction X · ι(X)R ∇ is the inner product by X. To a pair of Koszul connections (∇, ∇ * ) we assign three differential operators. They are denoted by D ∇∇ * , D ∇ and D ∇ .
(A.1) D ∇∇ * is a first order differential operator. It is defined in the vector bundle Hom(TM, TM).
Its values belong to the vector bundle Hom(TM ⊗2 , TM). (A.2) D ∇ and D ∇ are 2nd order differential operators. They are defined in the vector bundle TM.
Their values belong to the vector bundle Hom(TM ⊗2 , TM). Let X be a section of TM and let ψ be a section of T * M ⊗ TM. The differential operators just mentioned are defined by D ∇ (X) = ∇ 2 (X).
Part A of this paper is partially devoted to the global analysis of the differential equation The solutions to FE(∇∇ * ) are useful for addressing the links between the KV homology, the differential topology and the information geometry.
The purpose of a forthcoming paper is the study of the differential equations In the Appendix A to this paper we overview the role played by the solutions to FE * * (∇) in some still open problems.

Some Explicit Formulas
Let x = (x 1 , ..., x m ) be a system of local coordinate functions of M. In those coordinates the Christoffel symbols of both ∇ and ∇ * are denoted by Γ ij:k and Γ * ij:k respectively. We use those coordinate functions for presenting an element ψ ∈ M(∇∇ * ) as a matrix [ψ ij ]. Thus by setting ∂i = ∂ ∂x i one has ∇ ∂i ∂j = Γ ij:k ∂k.
We focus on FE(∇∇ * ) and of FE * * (∇). They are equivalent to the following system of partial differential equations [Θ k ij (X)] : In Part A we address the links between the following topics DTO, HGE, IGE and ENT. Those topics are presented as vertices of a square whose centre is denoted by KV H.
(1) DTO stands for Differential TOpology. In DTO, FWE stands for Foliations and WEbs.
(2) HGE stands for Hessian GEometry. Its sources are the geometry of bounded domains, the topology of bounded domains, the analysis in bounded domains. Among the notable references are [1][2][3]. Hessian geometry has significant impacts on thermodynamics, see [4,5], About the impacts on other related topics the readers are referred to [6][7][8][9][10][11][12]. (3) IGE stands for Information GEometry. That is the geometry of statistical models. More generally its concern is the differential geometry of statistical manifolds. The range of the information geometry is large [13]. Currently, the interest in information geometry is increasing. This comes from the links with many major research domains [14][15][16]. We address some significant aspects of those links. Non-specialist readers are referred to some fundamental references such as [17,18]. See also [4,[19][20][21][22][23]. The information geometry also provides a unifying approach to many problems in differential geometry, see [21,24,25]. The information geometry has a large scope of applications, e.g., physics, chemistry, biology and finance. (4) ENT stands for ENTropy. The notion of entropy appears in many mathematical topics, in Physics, in thermodynamics and in mechanics. Recent interest in the entropy function arises from its topological nature [14]. In Part B we introduce the entropy flow of a pair of vector fields. The Fisher information is then defined as the Hessian of the entropy flow. (5) KVH stands for KV Homology. The theory of KV homology was developed in [9]. The motivation was the conjecture of M. Gerstenhaber in the category of locally flat manifolds. In this paper we emphasize other notable roles played by the theory of KV homology. It is also useful for discussing a problem raised by John Milnor in [26].
The conjecture of Gerstenhaber is the following claim.
Every restricted theory of deformation generates its proper cohomology theory [27].
Loosely speaking, in a restricted theory of deformation one has the notion of both infinitesimal deformation and trivial deformation. The challenge is the search for a cochain complex admitting infinitesimal deformations as cocycles. In the present paper, KV H is useful for emphasizing the links between the vertices DTO, HGE, IGE and ENT. That is our reason for devoting a section to KV H.

Warning.
We propose to overview the structure of this paper. The readers are advised to read this paper as through it were a wander around the vertices of the square "DTO-HGE-IGE-ENT". Thus, depending on his interests and his concerns a reader could walk several times across the same vertex. For instance the information geometry appears in many sections, depending on the purpose and on the aims.

The content of the Paper
This paper is divided into Part A and Part B. Part A: Sections 1-7. Section 1 is the Introduction. Section 2 is devoted to algebroids, modules of algebroids and the theory of KV homology of the Koszul-Vinberg algebroids. To introduce the KV cohomology we have adopted three approaches. Each approach is based on its specific machinery. However, the readers will face three cochain complexes which are pairwise quasi isomorphic. The KV cohomology is present throughout this paper. At the end of Part B the reader will see that the theory of statistical models is but a vanishing theorem in the theory of KV hohomology. The first approach is based on the pioneering fundamental brute formula of the coboundary operator. Historically, the brute formula is the first to have been constructed [9].
This first approach is used in many sections of this paper. Regarding the theory of deformation of the Koszul Geometry, the KV cohomology is the solution to the conjecture of Gerstenhaber. In the theory of modules of KV algebroids the role played by the KV cohomology is practically SIMILAR to the role played by the Hocshild cohomology in the category of associative alggebroids and their modules. This last remark holds for the role played by the Chevalley-Eilenberg cohomology in the category of Lie algebroids and their modules. Nevertheless, our comparison fails in the theory of Extension of modules over algebroids. In both categories of extensions of modules over associative algebroids and Lie algebroids the moduli space of equivalence class is encoded by cohomology classes of degree one. In the category of extensions of KV modules the moduli space is encoded by a spectral sequence. That was a unexpected feature in [9]. The pioneering coboundary operator of Nijenhuis [28] may be derived from the total brute coboundary operator introduced in [29].
The second approach is based on the notion of simplecial objects. The third approach is based on the theory of anomaly functions for abstract algebras and their abstract modules. The idea has emerged from recent correspondences with one of my former teachers. The KV anomaly function of a Koszul connection ∇ may be expressed in terms of the ∇-Hessian operators ∇ 2 , namely This approach is a powerful for addressing the relationships between the global analysis, the differential topology and the information geometry. The approach by the anomaly functions suggests many conjectures. Among those conjectures is the following.
Conjecture. Every anomaly function of algebras and of modules yields a theory of cohomology of algebras and modules. Section 3. This section is devoted to the theory of KV (co)homology of Koszul-Vinberg algebroids. We focus on cohomological data which are used in the paper. Section 4. This section is devoted the KV algebroids which are defined by structures of locally flat manifold. The KV cohomology theory is used for highlighting the impacts on the differential topology of the information geometry and its methods. We make the most of some relationships between the KV cohomology and the global analysis of the differential equation FE * (∇∇ * ). We also sketch the global analysis of the differential equation This leads to the function LC ∇ → r b (∇) ∈ Z.
We explain how to interpret r b as a distance. (See the Appendix A to this paper ). For instance, the function r b gives rise to an numerical invariant r b (M) which measures how far from being an exponential family is a statistical model M. This result is a significant contribution to the information geometry, see [18,22,24]. Section 5. We are interested in how interact the information geometry, the KV cohomology and the geometry and Koszul. In particular we relate the notion of hyperbolicity and vanishing theorems in the KV cohomology. Section 6. This section is devoted to the homological version of the geometry of Koszul. Our approach involves the dualistic relation of Amari. The KV cohomology links the dualistic relation with the geometry of Koszul. Section 7. In this section summarize the highlighting features of Part A.
Section 8. This is the starting section of the second part B. This Part B is devoted to new insights in the theory of statistical models. On 2002 Peter McCullar raised the provocative question.

What Is a Statistical Model
Across the world (Australia, Canada, Europe, US) the McCullagh s paper became the object of many criticisms and questions by eminent theoretical and applied statisticians [30].
Part B is aimed at supplying some deficiencies in the current theory of statistical models. We address some criticisms which support the need of re-establishing the theory of statistical model for measurable sets. Those criticisms are used for highlight the lack of both Structure and Relations. Those criticisms also highlight the search of M. Gromov [15]. The need for structures and relations was the intuition of Peter McCullagh. Loosely speaking there is a lack of Intrinsic Geometry in the sense of Erlangen. Subsequently the lack of intrinsic geometry yields other things that are lacking. The problem of the moduli space of models is not studied, although this would be crucial for applied information geometry, and for applied statistics. That might be a key in reading some the controversy about [30]. Section 9. In this section we address the problem of moduli space of statistical models. The problem of moduli space in a category is a major question in Mathematic. It is generally a difficult problem that involves finding a characteristic invariant which encodes the point of the moduli space. Such an invariant is a crucial step toward the geometry and the topology of a moduli space. Among other needs, the problem of encoding the moduli space of models has motivated our need of a new approach, that is to say the need of a theory having nice mathematical structure and relations. In this Part B the problem of the moduli space is solved. To summarize the theorem describing the moduli spaces of statistical models we need the following notation.
The quantity E(q M )(∇) does not depend on the Koszul connection ∇. It is called the Fisher information of M.
This theorem emphasizes the Search for structure [16]. Section 10. This section is devoted to introduce the category of homological statistical models. This may be interpreted as a variant of the topology of the information. Another approach is to be found in Baudot-Bennequin [31].
The current theory (as in [17]) is called the classical (or local) theory. This means that a statistical model as in [17,18] is derived from the localization of a homological model. Loosely speaking such a model expresses a local vanishing theorem in the theory of homological statistical models. Section 11. This section is devoted to discussing the links between the geometry of Koszul and the theory of homological statistical models. Those investigations lead to this notable feature.
The Geometry of Koszul, the homological statistical models and the classical information geometry locally look alike. Section 12. Through Section 9 the framework is the category of equivariant locally trivial fibration. This assumption is weakened in Section 12. We recall the relationships between the Cech cohomology and the theory of locally trivial fiber bundle. We extend the scope of applications of the methods of the information geometry. Those extensions produce some interesting results. Here is an instance.

Theorem 2.
Let M be an oriented compact real analytic manifold and let C ω (M 2 ) be the space of real valued analytic functions in M 2 . There exists a non trivial map of C ω (M 2 ) in the family of (positive) stratified Riemannian foliation in M.
Sections 13. This Section 13 is a variant of Section 7. Section 14 is an appendix we have mentioned. It is devoted to overview a few new significant results. Those results are derived from the global analysis of the differential operators The solutions to a few open problems are announced.

Algebroids, Moduls of Algebroids, Anomaly Functions
The purpose of this section is to introduce basic notions in the algebraic topology of locally flat manifolds.

The Algebroids and Modules
Given a smooth fiber bundle B → M the set of smooth sections of B is denoted by Γ(B).

Definition 1. An (abstract) real algebra is a real vector space A endowed with a bilinear map
A × A → A Definition 2. An (abstract) real two-sided module of an (abstract) algebra A is a real vector space W with two bilinear mappings A × W → W,

Warning.
Here algebra means a multiplication a · b without any rule of calculations. So the product a · b · c is meaningless.
Throughout this paper, the smooth manifolds we deal with are connected and paracompact. In a smooth manifold M all geometrical objects we are interested in are smooth as well.
The vector space of smooth vector fields in a manifold M is denoted by X (M). It is a left module of the associative commutative algebra C ∞ (M).
Consider a real vector bundle E → M.
The real vector space of sections of E is denoted by Γ(E ).

Definition 3.
A real algebroid over a smooth manifold M is a real vector bundle whose vector space of sections is a real algebra.
So the vector space of sections of a real algebroid E is endowed with a R-bilinear map To simplify the multiplication of two sections is denoted s · s * .

Definition 4.
A two-sided module of an algebroid E is a vector bundle V → M whose vector space of sections is a two-sided module of the algebra Γ(E ).
Let s be section E and let v be a section of V. Both left action s on v and the right action of s on v are denoted by s · v and v · s.

Definition 5. An anchored vector bundle over M is a pair
formed by a real vector bundle E and a vector bundle homomorphism The homomorphism b is called the anchor map.

Anomaly Functions of Algebroids and of Modules
Let V be a two-sided module of an algebroid (E , b). 2,3]]. An anomaly function of an E -module V is

Definition 6. An anomaly function of an algebroid E is a 3-linear map
In this paper we are interested in some anomaly functions which have strong geometrical impacts. They are defined below.
The homogeneous vector sub-spaces are defined by Before pursuing we fix the following notation. Let and let a ∈ A, We are going to define the coboundary operator The coboundary operator is a linear map. It is defined by The operator δ KV satisfies the identity Therefore the pair (C * KV (A, W), δ KV ) is a cochain complex. Its cohomology space is denoted by

The conjecture of Gerstenhaber: Comments.
A KV algebra A is a two-sided module of itself. An infinitesimal deformations of A is a 1-cocycle of C KV (A, A) [9]. By the conjecture of Gerstenhaber the cohomology complex C KV (A, A) is generated by the theory of deformations in the category of KV algebras.
The theory of deformation of KV algebras is the algebraic version of the theory of deformation of locally flat manifolds [2]. Therefore, the complex C KV (A, ) is the solution to the conjecture of Muray Gerstenhaber in the category of locally flat manifolds [27].

Features.
(1) The 2nd cohomology space H 2 KV (A, A) is the space of non trivial deformations of A. The definition of KV algebra of a locally flat manifold will be given in the next section. Following [2] every hyperbolic locally flat manifold has non trivial deformations. Thus, if A is the KV algebra of a hyperbolic locally flat manifold then H 2 KV (A, A) = 0.
(2) Let W be a two-sided module of a KV algebra A. We consider W as a trivial KV algebra, viz w · w * = 0 ∀w, w * ∈ W.
Let EXT KV (A, W) be the set of equivalence classes of short exact sequences of KV algebras An interpretation of the 2nd cohomology space of C KV (A, W) is the identification Let W, W * be two-sided modules of A. Let EXT A (W * , W) be the set of equivalence classes of exact short sequences of two-sided A-modules In both the category of associative algebras and the category of Lie algebras we have Here HH(A, −) stands for Hochschild cohomology of an associative algebra A and H CE (A, −) stands for cohomology of Chevalley-Eilenberg of a Lie algebra A.
Unfortunately in the category of KV modules of KV algebras this interpretation of the first cohomology space fails. Loosely speaking in the category of KV algebras the set H 1 (A, Hom(W * , W)) is not canonically isomorphic to set EXT A (W * , W) [9].
The purpose is the total complex Its homogeneous vector subspaces are defined by The total coboundary operator is a linear map

The Theory of KV Cohomology-Version: the Semi-Simplicial Objects
Let V be a two-sided module of a KV algebra A. Our aim is the construction of semi simplicial A-modules whose derived cochain complex is quasi isomorphic to the KV cochin complex C KV (A, V).
With the multiplication we just defined, B is a real KV algebra. In other words we have Here In the A-module V we have a structure of left B-module which is defined by (s + λ)·v = s·v + λv ∀(s + λ) ∈ B, ∀v ∈ V.

Construction
LetB be the vector space spanned by A × R. Its elements are finite linear combinations of (s, λ), s ∈ A × R.
The tensor algebra ofB is denoted by T(B). It has a Z-grading. its homogeneous vector sub-spaces are defined by T q (B) =B ⊗q .
A monomial element is denoted by Here The KV algebra A is a two-sided ideal of the KV algebra B. Thereby, the vector spaceB is canonically a left module of A.
We define the natural two-sided action of R inB by setting λ · (s * , λ * ) = (λs, λλ * ), Thereby every vector subspace Before continuing we recall the (extended) action of A in tensor space T q (B), We recall a notation which has been used in the last subsections, The symbolx j means that x j is missing. Let 1 ∈ R be the unit element, then1 stands for (0, 1) ∈B. We are going to construct semi simplicial modules of B.

Notation-Definitions
Implicitly we use set isomorphism We go back to the Z-graded B-module Definition 12. Let j, q be two positive integers with j < q, let The linear maps d j : The right member of the last equality has the following meaning Structure. The maps d j and S j satisfy the following identities Definition 13. The system T q (B), d i , S i is called the canonical semi simplicial module of B.

The KV Chain Complex
From the canonical simplicial B-module we derive the chain complex C * (B). it has a Z-grading which is defined by Now one defines the ( linear) boundary operator By the virtue of (5a) we have d 2 = 0.

The V-Valued KV Homology
We keep the notation used in the preceding sub-subsection. So the vector spaces A, B and V are the same as in the preceding subsubsection.
We consider the Z-graded vector space Its homogeneous sub-spaces are defined by Every homogeneous vector subspace C q (B, V) is a left module of the KV algebra B. The left action is defined by Let j and q be two positive integers such that j < q.
To define the linear map Henceforth one defines the boundary operator So we obtain a chain complex whose homology space of degree q is denoted by H q (B, V).

Definition 14. The graded vector space
is called the total homology of B with coefficients in V.

Two Cochain Complexes
We are going to define two cochain complexes. They are denoted by C KV (B, V) and by C τ (B, V) respectively.
Let us set Let (j, q) be a pair of non negative integers such that j < q. We are going to define the linear map The family of linear mappings d j has property S · 1, viz We use these data for constructing two cochain complexes. They are denoted by (C * KV , d KV ) and by (C * τ , d τ ) respectively. The underlying graded vector spaces are defined by Their coboundary operators are defined by The simplicial formula (5a) yields the identities is called the V-valued KV cohomology of B. The cohomology space is called the V-valued total KV cohomology of B.
The algebra A is a two-sided ideal of the KV algebra B. Mutatis mutandis our construction gives the cohomology spaces H KV (A, V) and H τ (A, V). They are called the V-valued KV cohomology and the V-valued total KV cohomology of A.

Comments.
Though the spectral sequences are not the purpose of this paper we recall that the pair (A ⊂ B) gives rise to a spectral sequences E ij r [32][33][34]. The term E ij 0 is nothing other than H KV (A, V) [29]. In other words one has

Residual Cohomology
Before pursuing we introduce the notion of residual cohomology. It will be used in the section be devoted the homological statistical models.
The machinery we are going to introduce is similar to the machinery of Eilenberg [35]. In particular we introduce the residual cohomology. Our construction leads to an exact cohomology sequence which links the residual cohomology with the equivariant cohomology. We restrict the attention to the category of left modules of KV algebroids. We keep our previous notation.
We recall that for every positive integer q > 0 the vector space C q (B, V) is a left module of B. The left action of s ∈ B is defined by A straightforward consequence of this definition is that a left invariant cochain is a cocycle of both C * KV and C * τ . The vector subspace of left invariant q-cochains of B is denoted by H q e (B, V). It is easy to see that

Definition 16.
A KV cochain of degree q whose coboundary is left invariant invariant is called a residual KV cocycles.
(1) The vector subspace of residual KV cocycles of degree q is denoted by Z q KVres . (2) The vector subspace of residual coboundaries of degree q is defined by B The residual KV cohomology space of degree q is the quotient vector space. The definitions above lead to the cohomological exact sequences which is similar to those constructed by Eilenberg machinery [35]. We are going to pay a special attention to two cohomology exact sequences.
(1) At one side the operator d KV yields a canonical linear map (2) At another side every KV cocycle is a residual cocycle and every KV coboundary is a residual coboundary as well. Then one has a canonical linear map Those canonical linear mappings yield the following exact sequences (c.1): We replace the KV B by A. Then we obtain the exact sequences In the introduction we have stated a conjecture of M. Gerstenhaber, namely Every Restricted Theory of Deformation Generates Its Proper Theory of Cohomology.
From the viewpoint of this conjecture, the KV cohomology is the completion a long history [2,9,28]. Besides Koszul and Nijenhuis, other pioneering authors are Vinberg, Richardson, Gerstenhaber, Matsushima, Vey.
The challenge was the search for a theory of cohomology which might be generated by the theory of deformation of locally flat manifolds [8]. The expected theory is the now known KV theory of KV cohomolgy [9].

Example 2.
The total cohomology is close to both the pioneering Nijenhuis work [28,36]. In [29] we have constructed a spectral sequence which relates to [28,36].
From another viewpoint, the total KV cohomology is useful for exploring the relationships between the information geometry and the theory of Riemannian foliations. This purpose will be addressed in the next sections.

The Theory of KV Cohomology-Version the Anomaly Functions
This subsection is devoted to use the KV anomaly functions for introducing the theory of cohomology of KV algebroids and their modules.
This viewpoint leads to an unifying framework for introducing the theory of cohomology of abstract algebras and their abstract two-sided modules. Here are a few examples of cohomology theory which are based on the anomaly functions. Here (1) V is an (abstract) two sided module of an (abstract) algebra A.  Let A D be the category of (abstract) algebras and (abstract) modules whose structures are defined by the pair (A A , A AV ). So the rules of calculations in the category A are defined by the identities The challenge is the search of a particular family of linear maps Such a particular family d q must satisfy a condition that we call the property ∆.
Let us assume that a family d q is a solution to CH(D). Then the category A D admits a theory of cohomology with coefficients in modules.
The next is devoted to this challenge in the category of KV algebras and KV modules. The geometry version is the category of KV algebroids and KV modules of KV algebroids.

Challenge CH(D) for KV Algebras
Let W be a two-sided module of an abstract algebra A. We assume that the following bilinear mappings are non trivial applications Let f ∈ Hom(A ⊗q , W). We consider a monomial ξ ∈ A ⊗q+1 , so Our construction is divided into many STEPS.
Let (i < j) be a pair of positive integers with 1 ≤ i < j ≤ q. The linear the map In the right side member of Step 2.
For every pair (i, q Step 3. Let g ∈ Hom(A ⊗q+1 , W) and let Let i, j, k be three positive integers such that i < j < q + 2; k ≤ q + 2. We have already introduced the notation We define S k [i,j] (g) ∈ Hom(A ⊗q+2 , W) by setting Given a triple (i, j, k) with i < j < k < q + 2 we put The proof of the following statement is based on direct calculations.

Lemma 1.
( * * * * ) : ∑ Let f ∈ Hom(A q , W). In both the left side and the right side of the equality ( * * * * ) we replace g by ∑ i<j S [i,j] ( f ). Then we obtain a linear mapping Our aim is to evaluate ξ of E * * * * ( f ) at ξ ∈ A ⊗q+2 . Here To calculate [E * * * * ( f )](ξ) we take into account both STEP1 and STEP2. Then we obtain At the right side member Step 4.
We are in position to face CH(D).
Definition 17. Let f ∈ Hom(A ⊗q , W) and ξ = X 1 ⊗ ... ⊗ X q+1 ∈ A ⊗q+1 . We take into account Step 1, Step 2 and Step 3. Therefore, we define the linear map The following lemma is a straightforward consequence of the machinery in STEP3.

Lemma 2.
Lemma 2 tells us that ∂ 2 f (ξ) depends linearly on the values of the KV anomaly functions. The challenge CH(D) is won in the category of KV algebras and their two-sided KV modules. We replace the category of KV algebras and their two-sided modules by the category of KV algebroids and their bi-modules. Then we win the geometry version of CH(D.
We use Lemma 2 for introducing a theory of KV homology of KV algebras and their two-sided modules.

The KV Cohomology
Let W be a two sided KV module of a KV algebra A. We consider the graded vector space The homogeneous subspaces are defined by C q KV = 0 if q is a negative integer, C 0 KV = J(W), C q KV = Hom(A ⊗q , W) if q is a positive integer.
We define the linear map C q KV By Lemma 2 we obtain the following statement Theorem 3. For every two sided KV module W of a KV algebra A the pair (C * KV , ∂ KV ) is a cochain complex.

The Total Cohomology
Let W be a two-sided module of a KV algebra A. Our concern is the Z-graded vector space For our present purpose the maps S ij are not subject the requirement as in Step 2.
We define the coboundary operator ∂ τ by setting The quantity (∂ 2 τ f (ξ) depends linearly on the KV anomaly functions of the pair (A, W). Thus the pair (C * τ , ∂ τ ) is a cochain complex. Its cohomology is called the W-valued total KV cohomology of A. We denote it by H * τ (A, W).

The Residual Cohomology, Some Exact Sequences, Related Topics, DTO-HEG-IGE-ENT
In the next sections we will see that the links between the information geometry and the differential topology involve the real valued total KV cohomology of KV algebroids. Many relevant relationships are based on the exact sequences Now we are provided with cohomological tools which will be used in the next sections. We plan to perform KV cohomological methods for studying some links between the vertices of the square "DTO, IGE, ENT HGE" as in Figure 1. We recall basic notions.  DTO stands for Differential TOpology.
The purposes: Riemannian foliations and Riemannian webs. Symplectic foliations and symplectic webs. Linearization of webs.
Our aims: We use cohomological methods for constructing Riemannian foliations, Riemannian webs, linearizable webs.
Nowadays, there does not exist any criterion for deciding whether a manifold supports those differential topological objects. Our aim is to discuss sufficient conditions for a manifold admitting those structures. Our approach leads to notable results. The key tools are the KV cohomology and the dualistic relation of Amari. Both the KV cohomology and the dualistic relation product remarkable split exact sequences. Notable results are based on those exact sequences. HGE stands for Hessian GEometry.
The purposes: Hessian structures, geometry of Koszul, hyperbolicity, cohomological vanishing theorems. Our aims: The geometry of Koszul is a cohomological vanishing theorem. Statistical geometry and vanishing theorem, the solution to a hold question of Alexander K Guts (announced).
Theorem 3 as in [2] may be rephrased in the framework of the theory of KV homology. For a compact locally flat manifold (M, ∇) being hyperbolic it is necessary and sufficient that C 2 KV (A, C ∞ (M)) contains a positive definite EXACT cocycle. To be hyperbolic is a geometrical-topological property of the developing map of locally flat manifolds. To be hyperbolicitic means that the image of the developing is a convex domain not containing any straight line. This formulation is far from being a homological statement. So the Hessian GEOmetry is a link between the theory of KV homology and the Riemannian Riemannian geometry.
The geometry of Koszul, the geometry of homogeneous bounded domains and related topics have been studied by Vinberg, Piatecci-Shapiro and many other mathematicians [3]. The geometry of Siegel domains belongs to that galaxy [7,12]. Almost all of those studies are closely related to the Hessian geometry.
Among the open problems in the Hessian geometry are two questions we are concerned with. The first is to know whether the metric tensor g of a Riemannian manifold is a Hessian metric. Alexander K. Guts raised this question in a mail (to me) forty years ago. The second question is to know whether a locally flat manifold admits a Hessian tensor metric. The solutions to those two problems are announced in the Appendix A to this paper.

IGE stands for Information GEometry.
The purposes: The differential geometry of statistical models, the complexity of statistical models, ramifications of the information geometry.
Our aims: We revisit the classical theory of statistical models, requests of McCullagh and Gromov. A search of a characteristic invariant. The moduli space of models. The homological nature of the information geometry.
The information geometry is the differential geometry in statistical models for measurable sets. In both the theoretical statistics and the applied statistics the exponential families and their generalizations are optimal statistical models. There are many references, e.g., [17,18,22,37]. Here Murray-Rice 1.15 means Murray-Rice Chapter 1, Section 15. A major problem is to know whether a given statistical model is isomorphic to an exponential model. That is what we call the complexity problem of statistical models. This challenge is a still open problem. It explicitly arises from the purposes which are discussed in [22] here, see also [30]. In the appendix to this paper we present a recently discovered invariant which measures how far from being an exponential family is a given model. That invariant is useful for exploring the differential topology of statistical models. That is particularly important when models are singular, viz models whose Fisher information is not inversible.
ENT stands for ENTropy.

Pierre Baudot and Daniel
Bennequin recently discovered that the entropy function has a homological nature [31]. We recall that in 2002 Peter McCullagh raised a fundamental geometric-topological question in the theory of information: What Is a Statistical Model? [30] A few years after Misha Gromov raised a similar request: The Search of Structure. Fisher Information [15,16]. Those two titles are two formulations of the same need.
The paper of McCullagh became the subject of controversy. It gave rise to questions, discussions, criticisms, see [30].
In Part B of this paper we will be addressing this fundamental problem. A reading of the McCullagh paper would be useful for drawing a comparison between our approach and [15,16,30].

The Total Cohomology and Riemannian Foliations
In this section we focus on the KV algebroids which are defined by structures of locally flat manifolds. To facilitate a continuous reading of this paper we recall fundamental notions which are needed.

Definition 18.
A locally flat manifold is a pair (M, D). Here D is a torsion free Koszul connection whose curvature tensor R D vanishes identically.

The pair (M, D) defines a Koszul-Vinberg algebroid
The anchor map is the identity map of TM. The multiplication of sections is defined by D, viz The KV algebra of (M, D) is the algebra In the right hand member of the equality above d(θ(Y)) is the exterior derivative of the real valued function θ(Y).
Let S 2 (T * M) be the vector bundle of symmetric bi-linear forms in M.
We put The T * M-valued cohomology of the KV algebroid (TM, D, 1) is but the cohomology of A with coefficients in Ω 1 (M). The KV cohomology and the total cohomology are denoted by

Warning.
We observe that elements of S 2 (M) may be regarded as 1-cochains of A with coefficients in its left module Ω 1 (M). By [29] we have At another side we have the cohomolgy exact sequence By Equations (8) and (9) we obtain the inclusion maps Mutatis mutandis one also has Remark 1 (Important Remarks). We give some subtle consequences of (1).

By (R.1) and (R.3) we obtain the inclusion map
Let H 2 dR (M) be the second cohomology space of the de Rham complex of M. The following theorem is useful for relating the total KV cohomology and the differential topology.

Theorem 4. [29] There exists a canonical linear injection of H
The theorem above highlights a fruitful link between the total KV cohomology and the differential topology. We are particularly interested in D-geodesic Riemannian foliations in a locally flat manifold (M, D).

Throughout this paper a Riemannian metric tensor in a manifold M is a non-degenerate symmetric bilinear form in M.
A positive metric tensor is a positive definite metric tensor.
In the next we use the following definition of Riemannian foliation and symplectic foliation.

Definition 19.
A Riemannian foliation is an element g ∈ S 2 (M) which has the following properties

Warning.
When g is positive semi-definite our definition is equivalent to the classical definition of Riemannian foliation [38][39][40].
The complete integrability of Ker(g) and the conditions to be satisfied by the holonomy of leaves are equivalent to the Property (2.2).
The set of Riemannian foliations in a manifold M is denoted by RF (M). The last theorem above yields the inclusion map We often use the notion of affine coordinates functions in a locally flat manifold. For non specialists we recall two definitions and the link between them.
Definition 20. An m-dimensional affinely flat manifold is an m-dimensional smooth manifold M admitting a complete atlas (U j , φ j ) whose local coordinate changes coincide with affine transformations of the affine space R m .
We denoted an affine atlas by

The General Linearization Problem of Webs
In the framework RF (M) the inclusion may be rewritten as the exact sequence Let (M, ∇) be a locally flat manifold whose KV algebra is denoted by A. Every finite family in There does not exist any criterion to know whether a manifold supports Riemannian foliations. The exact cohomology sequences we have been performing provide us with a cohomological method for constructing Riemannian foliations in the category of locally flat manifolds. This is an impact of the theory of KV homology on DTO.
In the next section we will introduce other new ingredients which highlight the impacts on DTO of the information geometry.
Further we will see that those new machineries from the information geometry have a homological nature.
Another major problems in the differential topology is the linearization of webs. Among references are [41][42][43].

A Comment.
The distributions belonging to a web may have different dimensions. An example of problem is the symplectic linearization of lagrangian 2-webs.
Let (D j , j := 1, 2) be a lagrangian 2-web in a 2n-dimensional symplectic manifold (M, ω). The challenge is the search of special local Darboux coordinate functions (x, y) = (x 1 , ..., x n , y 1 , ..., y n ). Definition 24. An affine web in an affine space is a web whose leaves are affine subspaces.

Definition 25.
A web in a m-dimensional manifold is linearizable if it is locally diffeomorphic to an affine web in a m-dimensional affine space. Example 1. In the symplectic manifold (R 2 , e xy dx ∧ dy) one considers the lagrangian 2-web which is defined by This lagrangian 2-web is not symplectic linearizable. Example 2. We keep (L 1 , L 2 ) as in example.1. It is symplectic linearizable in (R 2 , (e x + e y )dx ∧ dy). The linearization problem for lagrangian 2-webs is closely related to the locally flat geometry [10,44,45].
Up to today the question as to whether it is linearizable is subject to controversies, see [42] and references therein.

The Total KV Cohomology and the Differential Topology Continued
We implement the KV cohomology to address some open problems in the differential topology. For our purpose we recall a few classical notions which are needed.

Definition 26. A metric vector bundle over a manifold M is a vector bundle
The notions of transversally Riemannian foliation and transversally symplectic foliation are weaker than the notion of Riemannian foliation and symplectic foliation. However if ∇ a torsion free Koszul connection every ∇geodesic transversally Riemannian foliation is a Riemannian foliation. Every ∇-geodesic transversally symplectic foliation is a symplectic foliation.
For the general theory of Riemannian foliations the readers are referred to [39,40,46], see also the monograph [38] and the references therein.
We have pointed out that criterions for deciding whether a smooth manifold admits Riemannian foliations (respectively symplectic foliations ) are missing. Our purpose is to address this existence problem in the category SLC whose objects are symmetric gauge structures. Such an object is a pair (M, ∇) where ∇ is a torsion free Koszul connection in M. The category of locally flat structure LF is a subcategory of SLC. The theory of KV homology is useful for discussing geodesic Riemannian foliations in the category LF . In a locally flat manifold (M, D) we have been dealing with the decomposition Here A is the KV algebra of (M, D). Let b 2 (M) be the second Betti number of M. We define the numerical geometric invariant r(D) by ).
We consider the exact sequences Thus r(D) is formally the dimension of S A 2 (M). The present approach leads to the following statement Proposition 1. If r(D) > 0 then M admits non trivial D-geodesic Riemannian foliations.
Proof. Let B be a non zero element of S A 2 (M) and let D be the kernel of B.
Then (M, B) is a Riemannian manifold the Levi-Civita connection of which is D. Therefore, the proposition holds.
Before proceeding we define three numerical invariants The non negative integers r(M) and s(M) are global geometric invariants. They connect the total KV cohomology to geodesic Riemannian foliations. By this viewpoint the proposition has an interesting corollary.

Corollary 1.
In an m-dimensional manifold M suppose that the following inequalities are satisfied Then the manifold M admits a locally flat structure (M, D * ) which supports a non trivial D * -geodesic Riemannian foliation.
The integer s(M) is a local characteristic invariant of some class of 2-webs in Hessian manifolds. Let (M, D) be a locally flat manifold whose KV algebra is denoted by A. we recall that a Hessian metric tensor in (M, D) is a inversible cocycle g ∈ Z 2 KV (A, R). Proof. The proof is based on methods of the information geometry. Let A and A * be the KV algebras of (M, D) and of (M * , D * ) respectively. By the hypothesis there exists a pair of geosic Riemannian foliations Their KV algebras are denoted byÃ andÃ * .
Step a There exists a 1-cocycle By the definition ofD we have Further im(ψ) isD-geodesic and Ker(B) is D-geodesic. Therefore, the pair is a 2-web in M.
By the choice of B and B * we have rank(Ker(ψ)) = rank(Ker(ψ * )) = m − s. Thereby those leaves are locally isomorphic to affine sub-spaces.
Step b The distribution im(ψ) isD-geodesic. Therefore, near the same point p 0 ∈ (M,D) we linearize im(ψ by choosing appropriate local affine coordiante functions The leaves of im(ψ) are defined by Thus near p 0 the foliation defined by m(ψ) is isomorphic to an linear foliation.
Step c By both step a and step b we choose a neighbourhood of p 0 which is the domain of systems of appropriate local coordinate functions (x, y) and (x * , y * ). From those data we pick the local coordinate functions (x, y * ) = (x 1 , ..., x m−s , y * 1 , ..., y * s ).
In final, near the p * O ∈ M * the web (Ker(ψ * ), im(ψ * )) is diffeomorphic to the affine web whose leaves are parallel to a decomposition Here V m−s and V s are vector subspaces of R m . Their dimensions are m − s and s.

Conclusion.
There exists a unique linear transformation φ of R m such that The differential of Φ is denoted by Φ * . We express the properties above by This ends the sketch of proof of Theorem.
In the next we use the following definitions.
is in general position if the distributions Ker(B j ), j ∈ J are in general position.
The following statement is a straight corollary of the theorem we just demonstrated.

Proposition 2.
In a locally flat manifold (M, D) with r(D) > 0 every finite family in general position define a linearizable Riemannian web.

The KV Cohomology and Differential Topology Continued
We have seen how the total cohomology and linearizable Riemannian webs are related. More precisely the theory of KV cohomology provides sufficient conditions for a locally flat manifold admitting linearizable Riemannian webs. That approach is based on the split exact sequence

Kernels of 2-Cocycles and Foliations
Not all locally flat manifolds admit locally flat foliations. The existence of locally flat foliations is related to the linear holomnomy representation, viz the linear component of the affine holonomy representation of the fundamental group. Via the developing map the affine holonomy representation is conjugate to the natural action of the fundamental group in the universal covering. The KV homology is useful for investigating the existence of locally flat foliations. To simplify we work in the analytic category. So our purposes include singular foliations.
For those purposes we focus on an elementary item which has a notable impacts on our request. Let (M, D) be a locally flat manifold whose KV algebra is denoted by A. Let g ∈ C 2 (A, C ∞ (M)). The left kernel and the right kernel of g are denoted by Ker(g) and K 0 er(g) respectively.
Ker(g) is defined by The scalar KV 2-cocycles have elementary relevant properties (1) The left kernel of every KV 2-cocycle is closed under the Poisson bracket of vector fields.
(2) The right kernel of every KV 2-cocycle is a KV subalgebra of the KV algebra A.
We translate those elementary properties in term of the differential topology Theorem 7. In an analytic locally flat manifold (M, D) maps the set of analytic 2-cocycles in the category of analytic stratified foliations M, maps the set of analytic 2-cocycles in the category of stratified locally flat foliations, (3) If a 2-cocycle g is a symmetric form then Ker(g) is a stratified locally flat transversally Riemannian foliation.
The vector subspace of symmetric 2-cocycles the kernels of which are D-geodesic is denoted byZ 2 KV (A). The corresponding cohomology vector subspace is denoted bỹ By the exact sequence

The Information Geometry, Gauge Homomorphisms and the Differential Topology
We combine the dualistic relation with gauge homomorphisms to relate the total cohomology and two problems.
(i) The first is the existence problem for Riemannian foliations.
(ii) The second is the linearization of webs.
Those relationships highlight other roles played by the total KV cohomology. Through this section we use the brute coboundary operator.

The Dualistic Relation
We are interest in the foliation counterpart of the reduction in statistical models. The statistical reduction theorem is Theorem 3.5 as in [18]. We recall the notions which are needed.

Definition 31.
A dual pair is a quadruple (M, g, D, D * ) where (M, g) is a Riemannian manifold, D and D * are Koszul connections in M which are related to the metric tensor g by We recall that a Riemannian tensor is a non degenerate symmetric bilinear 2-form. The dualistic relation between linear connections plays a central role in the information geometry [17,18,47,48].  (1) Both D and D * are torsion free.
(2) D is torsion free and A is symmetric, viz (3) D * is torsion free and the metric tensor g a is KV cocycle of the KV algebra A * of the locally flat manifold (M, D * ). Proof. Let us prove that 1 implies (2) If both T D and T D * vanish identically then A is symmetric, viz A(X, Y) = A(Y, X).
Since D is a flat connection, (2) implies that both the torsion tensor and the curvature tensor of D vanish identically. Then (M, D) is a locally flat manifold whose KV complex is denoted by (C * (A, R), δ KV ). Using the dualistic relation of the pair (M, g, D, D * ) one obtains the identity Let us prove that (3) implies (4).
The assertion (3) implies that (M, D * ) is a locally flat manifold. Since g is δ KV -closed D is torsion free. Thereby (M, g, D, D * ) is a dually flat pair.
This implication derives directly from the definition of dually flat pair.

A Comment.
From the proposition just proved arises a relationship between the dually flatness and the KV cohomology. Indeed let (M, D 0 ) be a fixed locally flat manifold whose KV algebra is denoted by A 0 . Let C * KV (A 0 , R) be the KV complex ofR-valued cochains of the KV algebroid (TM, D 0 , 1). We know that every g ∈ Rie(M) yields a flat pair (M, g, D 0 , D g ).
Here D g is the flat Koszul connection defined by Proposition 4. The following assertions are equivalent.
The scalar KV cohomology of a fixed locally flat manifold (M, D 0 ) provides a way of constructing new locally flat structures in M. Indeed let us set So the dualistic relation leads to the map We recall that a gauge map in TM is a vector bundle morphism of TM in TM which projects on the identity map of M. The readers interested in others topological studies involving connections and gauge transformations are referred to [49].
Given two symmetric cocycles g, g * ∈ Hes(M, D 0 ) there is a unique gauge transformation The following properties are equivalent We fix a metric tensor g * ∈ Hes(M, Every g-symmetric gauge transformation φ defines the metric tensor This gives rise to the flat pair (M, g φ , D 0 , D g φ ).
To simplify we set We note Sym(g) the subset of g-symmetric gauge transformations φ such that the following assertions are equivalent The Lie group of D 0 -preserving gauge transformations of TM is denoted by G 0 . It is easy to see that for every φ ∈ Sym(g) the following assertions are equivalent Henceforth we deal with a fixed g * ∈ Hes(M, D 0 . The triple (M, g * , D 0 ) leads to the dually flat pair (M, g, D 0 , D g * ). We set The tangent bundle TM is regarded as a left KV module of the KV algebroid (TM, D * , 1). The KV algebras of (M, D 0 ) and of (M, D * ) are denoted by A 0 and by A * respectively. Their coboundary operators are noted δ 0 and δ * respectively.
We focus on the role played by the total KV cohomology of the algebroid (M, D * , 1). Let φ be a g * -symmetric gauge transformation. Then φ gives rise to the metric tensor g φ which is defined by Lemma 3. The following assertions are equivalent,

Hint.
Use the following formula Following the pioneering definition as in [2] a hyperbolic locally flat manifold is a positive exact Hessian manifold (M, D, δ KV θ). We extend the notion of hyperbolicity by deleting the condition that δ KV θ is positive. Now denote by Hyp(M, D 0 ) the set of exact Hessian structures in (M, D 0 ).
A hyperbolic structure is defined by a triple (M, D, θ) where (M, D) is a locally flat manifold and θ is a de Rham closed differential 1-form such that the symmetric bilinear δ KV θ is definite.
The following statement is a straightforward consequence of Lemma 3.

Corollary 2.
The following statements are equivalent.
Proof of Corollary. By (1) there exists a (de Rham) closed differential 1-form θ such that Let ξ be the unique vector field such that Since the quadruple (M, g * , D 0 , D * ) is a dually flat pair one has the identity Thus we get the expected conclusion, viz Conversely let us assume that there exists a vector ξ satisfying the identity That leads to the identity In other words one has g φ ∈ Hyp(M, D 0 ).
This ends the proof of Corollary 2.
The set of g * -symmetric gauge transformation is denoted by Σ(g * ).
We have the canonical isomorphism Now we define the setsZ

Reminder.
We recall that a hyperbolic manifold (or a Koszul manifold) is δ KV -exact Hessian manifold (M, g, D). It is easily seen that the set of positive hyperbolic structures in a locally flat manifold (M, D) is a convex subset of Hes(M, D).
So show the Koszul geometry is a vanishing theorem in the theory of KV homology of KV algebroids. The theory of homological statistical model (to be introduced in Part B) is another impact on the information geometry of the KV cohomology.
At the present step we have the relations Another outstanding result of Koszul is the non rigidity of compact positive hyperbolic manifolds [2]. The non rigidity means that every open neighborhood of a positive Hyperbolic locally flat manifold (M, D, δ KV θ) contain another positive hyperbolic locally flat structure which is not isomorphic to (M, D). This non rigidity property may be expressed with the Maurer-Cartan polynomial function P A MC of (M, D) ( see the local convexity theorem in [29]. In the next sub-subsection we revisit the notion of dual pair of foliations as in [18].

Statistcal Reductions
The statistical reduction theorem is the following statement. The foliation counterpart of the reduction theorem is of great interest in the differential topology of statistical models see [18]. In the preceding sections we have addressed a cohomological aspect of this purpose. The matter will be more extensively studied in a forthcoming paper (See the Appendix A).
In mathematical physics a principal connection 1-form is called a gauge field.
In the differential geometry a principal connection 1-form in a bundle of linear frames is called a linear connection.
In the category of vector bundle Koszul connections are algebroid counterpart of principal connection 1-forms.
In a tangent bundle TM, depending on concerns and needs Koszul connections may called linear connections or linear gauges.

A Uselful Complex
In this subsubsection we fix a dually flat pair (M, g, D, D * ) whose KV algebras are denoted by A and by A * . The tangent bundle TM is endowed the structure left module of the anchored KV algebroids (TM, D, 1) and (TM, D * , 1). This means that each of the KV algebras A or A * is regarded as a left module of itself.
We consider the tensor product We endow C with the Z bi-grading.
. We recall that C * (A, R) stands for C * (A, C ∞ (M)). For every non negative integer q we set We defines the linear map So we obtain a linear map Therefore, we consider the bi-graded differential vector space That is a bi-graded cochain complex whose q th cohomology is denoted by H q (C). The cohomology inherits the bi-grading Here In the next subsubsection we shall discuss the impacts of this cohomology.

Remark 2.
The pair (C * * , δ * * ) generates a spectral sequence [34]. That spectral sequence is a useful tool for simultaneously computing both the KV cohomology and the total KV cohomology of KV algebroids. Those matters are not the purpose of this paper.

The Homological Nature of Gauge Homomorphisms
Giving a dually flat pair (M, g, D, D * ) one considers the linear map Here the symmetric 2-form q ψ is defined by To relate the bi-complex (C * * , δ * * ) and the space of gauge homomorphisms we use the following statement.
Suppose that ψ ∈ M(D, D * ). Then we have Since both D and D * are torsion free one has the identity Thus ψ is a (1,0)-cocycle of the total KV complex (C * * , δ * * ). At another side the relation D * X • ψ = ψ • D X leads to the identity So q ψ is a (0,2)-cocycle of complex (C * * , δ * * ). We conclude that (2) implies (1). We recall the formula By this formula Thus the statement (2) is equivalent to the system δ τ ψ = 0, To continue the proof we perform the following lemma.

Lemma 4 ([29]
). For every symmetric cochain B ∈ C 0,2 , viz B(X, Y) = B(Y, X) the following identities are equivalent By Lemma 4 the bilinear form q ψ is D-parallel. Thereby we get the identity To usefully interpret this identity we involve the dualistic relation This expression leads to the identity A highlighting consequence is the identity To every vector field X we assign the linear map Then we rewrite Equations (14) and (15) as We consider the last identities in the framework of the Sternberg geometry [50,51].
is C ∞ (M)-bi-linear it belongs to the first Kuranishi-Spencer prolongation of the orthogonal Lie algebra so(g). Thereby S X (Y) vanishes identically. In other words we have This ends the proof of Theorem

A Comment.
The Sternberg geometry is the algebraic counterpart of the global analysis on manifolds. It has been introduced by Shlomo Sternberg and Victor Guillemin. It is an algebraic model for transitive differential geometry [50]. In that approach the Riemannian geometry is a geometry of type one. All of its Kuranishi-Spencer prolongations are trivial. The unique relevant geometrical invariant of the Riemnnian geometry is the curvature tensor of the Levi-Civita connection. Except the connection of Levi-Civita the other metric connections have been of few interest. Really other metric connections may have outstanding impacts on the differential topology. I shall address this purpose in a forthcoming paper.

The Homological Nature of the Equation FE ∇∇ *
Before proceeding we plan to discuss some homological ingredients which are connected to the differential equation Let us consider a dually flat pair (M, g * , D, D * ) and the KV complex Lemma 4 yields the following corollary.
Corollary 3. We keep the notation used the preceding sub-subsection. Given a gauge morphism ψ the following statements are equivalent.
In final we get So we obtain This end the proof of the corollary.
From the mapping we deduce the canonical linear map We define another map Here the differential 2-form ω is defined by This yields a linear map Now let (M, g, ∇, ∇ * ) be a dually flat pair whose KV algebras are denoted by A and A * . We identify the vector space Γ(Hom(TM, TM)) with the space C 1 τ (A * , A * ). We keep the notation D ∇∇ * , C ij , δ ij and q ψ . Therefore, we can rephrase Lemma 4 as it follows. (1) : D ∇∇ * (ψ) = 0, Here is an interesting feature. In a dually flat pair (M, g, ∇, ∇ * ) we combine the double complex That allow us to replace the differential equation by the homological equation That is a relevant impact on the global analysis of combinations of the KV cohomological methods with methods in the information geometry.

Computational Relations. Riemannian Foliations. Symplectic Foliations: Continued
We continue to relate the vector space of gauge homomorphisms and the differential topology. The tools we use are the KV cohomology and the Amari dualistic relation.
Let (M, g, D, D * ) be a dual pair. The vector subspace of g-preserving elements of M(D, D * ) is denoted by M(g, D, D * ). Thus every ψ ∈ M(g, D, D * ) satisfies the identity g(ψ(X), Y) + g(X, ψ(Y)) = 0. Now we fix a Koszul connection D 0 and we define the map We define the non negative integers Lemma 5. The integer n(D 0 ) does not depend on the choice of g ∈ Rie(M).

An Idea.
We fix a metric tensor g. For every g * ∈ Rie(M) there is a unique g-symmetric vector bundle morphism Therefore, we have Now one defines the numerical invariant n(M). Given a Koszul connection ∇ the vector space of ∇-parallel differential 2-forms is denoted by Ω ∇ 2 (M). Every dual pair (M, g, ∇, ∇ * ) gives rise to the linear isomorphisms The isomorphism (1) derives from the linear map The isomorphism (2) is defined by Reminder.
According our previous notation elements of Ω ∇ 2 (M) are ∇-geodesic symplectic foliations. Those of S ∇ 2 (M) are ∇-geodesic Riemannian foliations. Thus we apply methods of the information geometry to relate the gauge geometry and the differential topolgy.

Digressions.
Our construction may open to new developments. Here are some unexplored perspectives.
(b) Suppose that the manifold M is compact and suppose that g ∈ S ∇ 2 (M) is a positive Riemannian foliation, viz g(X, X) ≥ 0 ∀X.
Then the theory of Molino may be applied to study g [38]. Therefore, the structure theorem of Molino tells that g gives rise to a Lie foliation whoses leaves are the adherences leaves ofḡ [39]. (c) In the principal bundle of first order linear frames of M the analog of a Koszul connection ∇ is a principal connection 1-form ω whose curvature form is denoted by Ω. The curvature form is involved in constructing characteristic classes of M, (the formalism of Chern-Weill.) At another side ∇-geodesic Riemannian foliations and ∇-geodesic symplectic foliations are Lie algebroids. They have their extrinsic algebraic topology. In particular the theory of integrable systems may be performed in every leaf of ω ∈ Ω ∇ 2 (M). If one considers the α-connections in a statistical model those new insights may be of interest.
Here is an interpretation of the numerical invaraint n(∇).

Theorem 10.
We assume there exists ∇ ∈ SLC(M) whose linear holonomy group H(∇) is neither an orthogonal subgroup nor a symplectic subgroup. If n(∇) > 0 then the manifold M admits a couple (F r , F s ) formed by a ∇-geodesic Riemannian foliation F r and a ∇-geodesic symplectic foliation F s .

Proof. Let g be a Riemannian metric tensor in M.
Since for all x ∈ M there exists ψ ∈ M(∇, ∇(g) such that The assumption that the holonomy group H(∇) is neither orthogonal nor symlectic implies Both distributions Ker(q ψ ) and Ker(ω ψ ) are ∇-geodesic. Since ∇ is torsion free those distributions are completely integrable.
The theorem is proved.

A Useful Comment.
Let (M, D) be a locally flat manifold whose KV algebra is denoted by A. To every dual pair (M, g, D, D g ) we assign the short split exact sequence which is canonically isomorphic to the short exact sequence We have already defined the geometric invariant We observe that the integer n(D) is a byproduct of methods of the information geometry while r(D) is a byproduct of homological methods. However the split short exact sequence (****) leads to the equality n(D) = r(D).
Here is a straight consequence of the theorem we just proved.

Proposition 7. Every odd-dimensional manifold M with n(M) > 0 admits a geodesic symplectic foliation.
The dualistic relation of Amari has another significant impact on the differential topology.

A Digression.
We recall that a statistical manifold is a torsion free dual pair (M, g, ∇, ∇ * ). If the vector space M(∇, ∇ * ) is non-trivial then it plays an outstanding role in the differential topology of M. We define a canonical map of M(∇, ∇ * ) in the category of 2-webs by M(∇, ∇ * ) ψ → Ker(ψ + ) ⊕ im(ψ + ).
Thus one may regard elements of M(∇, ∇ * ) as orthogonal 2-webs in M. We keep our previous notation. The we have Now suppose that (M, g, ∇, ∇ * ) is a dually flat pair whose KV algebras are noted A and A * . We take into account the inclusion M(∇, ∇ * ) ⊂ Z 1 τ (A * , A * ).
We have a map of M(∇, ∇ * ) in the space of de Rham 2-cocyles which is defined by Assume that the cocycle ψ ∈ M(∇, ∇ * ) is exact. Then there exists ξ ∈ A * such that By the dualistic relation one easily sees that Therefore one gets a canonical linear map The next subsubsection is devoted to a few consequences of items we just discussed.

Riemannian Webs-Symplectic Webs in Statistical Manifolds
We introduce Riemannian webs and symplectic webs and we discuss their impacts on the topology of statistical manifolds. We recall that a Riemannian foliation is a symmetric bilinear form g ∈ S 2 (M) with the following properties (a) rank(g) = constant, (b) L X g = 0∀X ∈ Γ(Ker(g)).
To avoid confusions the pair (D, g) stands for the Riemannian foliation g.

Definition 36.
A Riemannian k-web is a family of k Riemannian foliations in general position (D j , g j ), j := 1, ..., k. A symplectic k-web is a family of k symplectic foliations in general position (D j , ω j ); j := 1, ..., k).
Let (M, g, D, D * ) be a dually flat pair whose KV algebras are denoted by A and A * . We recall the inclusion M(D, D * ) ⊂ Z 1 τ (A * , A * ). Consider a statistical manifold (M, g, ∇, ∇ * ). By the classical theorem of Frobenius every ∇-parallel differential system in M is completely integrable.
The distributions D j = Ker(g j ) are in general position. We consider the family Ψ + j ∈ Σ(g) defined by We get the family of g-orthogonal 2-webs defined by
There is family of g-skew symmetric gauge morphisms Ψ − j defined by Since ∇ and ∇ * are torsion free Ker(Ψ − j ) and im(Ψ − j ) are completely integrable. Since g is positive definite we get the 2-web Further every leaf of im(Ψ − j ) is a symplectic manifold.
The differential 2-form ω is called simple if the foliation Ker(Ψ − ) is simple. The considerations we just discussed may have remarkable impacts on the topological-geometrical structure of statistical manifolds.
From our brief discussion we conclude Theorem 12. Consider a statistical manifold (M, g, ∇, ∇ * ). Every non trivial simple symplectic foliation ω ∈ Ω ∇ 2 (M) is defined by a Riemannian submersion on a symplectic manifold.

The Hessian Information Geometry, Continued
In [52] Shima pointed out that the Fisher informations of many classical statistical models are Hessian metric tensors.
At another side the exponential models (or exponential family) may be considered as optimal Statistical models.
As already mentioned there does not exists any criterion for knowing whether a given statistical model is isomorphic to an exponential model [22], [13] In the category of regular models, viz models whose Fisher information is a Riemannian metric, it is known that the Fisher information of an exponential model is a Hessian Riemannian metric [18,52].
In this subsection we address the general situation. We give a cohomological characterization of exponential models. We also introduce a new numerical invariant r b which measures how far from being an exponential family is a given statistical model. See the Appendix A to this paper.
We recall that the metric tensor g of a Hessian structure (M, D, g) is a 2-cocycle of the KV complex To non specialists we go to recall some constructions in the geometry of Koszul [2,52], see also [53]. Let ((M, x * ), D) be a pointed locally flat manifold whose universal covering is denoted by (M,D). The parallel transport along a loop γ(t) yields a linear action of π 1 (x * ) in T x * M which is denoted by f ([γ]). Let [γ], [γ * ] ∈ π 1 (x * ). The composition of paths leads to the formula

Definition 38 ([2]
). An m-dimensional locally flat structure (M, D) is called hyperbolic ifQ(M) is a convex domain not containing any straight line in T x * M.

Definition 39. A locally flat manifold (M, D) is complete if the mapQ is a diffeomorphism onto T x * M.
Among the major open problems in the theory of space groups is the conjecture of Markus. Conjecure of Markus: a compact locally flat manifold (M, D) whose linear holonomy group is unimodular is complete.
Before pursuing we recall KV cohomological version of Theorem 3 as in [2].

Theorem 13 ([2]).
A necessary condition for a locally flat manifold (M, D) being hyperbolic is that (M, D) carries a positive Hessian structure whose metric tensor is exact in the KV complex of (M, D). This condition is sufficient if M is compact.
We have already mentioned a notable consequence of this theorem of Koszul. In the category of compact locally flat manifolds the subcategory of hyperbolic locally flat structures is the same thing as the category of positive exact Hessian structures. So The geometry of compact hyperbolic local flat manifolds is an appropriate vanishing theorem.
In the preceding sections the family of Hessian metrics in a locally flat manifold (M, D) is denoted by Hes(M, D). Therefore, H + es(M) stands for the sub-family of positive Hessian metric tensors. It is a convex cone in Rie(M).
We have already used the KV complex for expressing the dually flatness. More precisely let (M, D) be a fixed locally flat manifold whose KV algebra is noted A. A dual pair (M, g, D, D(g)) is dually flat if and only if g ∈ Z 2 KV (A, R). Therefore, every dually flat pair (M, g, D, D * ) yields two cohomology classes Thereby, we can use methods of the information geometry for rephrasing Theorem 3 as in [2].

If M is compact this ( vanishing ) condition is sufficient.
About the geometry of Koszul the non specialists are referred to [2,7,8,52] and bibliography therein [12].
About applications of the geometry of Koszul the readers are refereed to [12,13,54,55]. About relationships between the theory of deformation and the theory of cohomology, the readers are referred to [9,27,56].

The α-Connetions of Chentsov
Still, nowadays, the information geometry deals with models (Θ, P) whose underlying m-dimensional manifold Θ is an open subset of the euclidean space R m . Further the Fisher information g is assumed to be regular, viz (Θ, g) is a Riemannian manifold. In this paper this classical information geometry is called the local information geometry. This "local nature" will be explained in Part B of this paper.
At the moment we plan to investigate other topological properties of the local statistical models. Let (Θ, P) be an m-dimensional local statistical model for a measurable set (Ξ, Ω). The manifold Θ is a domain in the Euclidean space R m . The function P is non negative. It is defined in Θ × Ξ. We recall the requirements P is subject to. is the symmetric bi-linear form g which is defined by Here d θ stands for the differentiation with respect to the argument θ ∈ Θ. (5) The Fisher information is positive definite.
Remark 3. The Fisher information g can be defined using any Koszul connection ∇ according to the following formula The right member of the last equality does not depend on the choice of the Koszul connection ∇. From now on, we deal with a generic statistical model. This means that we do not assume the Fisher information g is definite.
Let θ = (θ 1 , ..., θ m ) be a system of Euclidean coordinate functions in R m . To every real number α is assigned the torsion free Koszul connection ∇ α whose Christoffel symbols in the coordinate (θ j ) are This definition agrees with any affine coordinate change. We put Now we assume a model (Θ, P) is regular. Then the Christoffel symbols and the Fisher information are related by the formula Γ α ij,k = g(∇ α ∂ i ∂ j , ∂ k ). Further every quadruple (Θ, g, ∇ α , ∇ −α ) is a statistical manifold [18,48]. Thus we have a family of splitting short exact sequences So the machinery we have developed in the preceding sections can be performed to explore the differential topology of regular local statistical models. For that purpose the crucial tool is the family of vector space We consider the abstract trivial bundle The machinery in the preceding subsection leads to the following proposition.
We have used some gauge morphisms to construct Riemannian submersions of statistical manifolds over symplectic manifolds. The notions we just introduced lead to similar situations. Theorem 15. Let (Θ, P) be a regular statistical model whose Fisher information is denoted by g. Every simple non zero singular section defines an α-family of Riemannian submersions of (Θ, g) onto symplectic manifolds. Proposition 9. We assume a model (Θ, P) is regular. For every nonzero real number α one has The Sketch of Proof. The proof is based on the short exact sequences  At one side, by the virtue of the reduction theorem as in [18] every leaf F of Ker(B) ⊥ inherits a dual pair (F, g F , ∇ α F , ∇ −α F ). At another side, B gives rise to the Riemannian structure (F, B). Furthermore both ∇ α F and ∇ −α F are torsion free metric connections in (F, B). Thereby one gets The last equality holds if and only if α = 0. This contradicts our assumption. The proposition is proved.
The proposition above is a separation criterion for α-connections in the following sense. For every nonzero real number α the vector subspace S ∇ α 2 (Θ) is transverse to S ∇ −α 2 (Θ) in the vector space S 2 (Θ) of symmetric forms in Θ.

The Exponential Models and the Hyperbolicity
A challenge is the search of a criterion for deciding whether a model (Θ, P) is an exponential family. That is the challenge in [22]. Still, nowadays, this problem is open.
The Fisher information of a regular exponential model is a Hessian Riemannian metric. We are going to demonstrate that the converse is globally true. Our proof is based on cohomological arguments.
In the Appendix A to this paper we introduce a new numerical invariant r b (Θ, P) which measures how far from being an exponential family is a model (Θ, P).
The invariant r b derives from the global analysis of differential operators Now we are going to provide a cohomological characterization of exponential models. Before pursuing we recall a definition. and a smooth function ..., θ m )).
Theorem 16. Let (Ξ, Ω) be a measurable set and let (Θ, P) be an m-dimensional statistical model for (Ξ, Ω). The Fisher information of (Θ, P) is denoted by g. The following statements are equivalent.
By the virtue of (2) we have Here ψ ∈ C ∞ (Θ) and (C, F)(ξ) = (C(ξ), F 1 (ξ), ..., F m (ξ)) ∈ R m+1 . Therefore, one has Thereby one can write This shows that we have We use a strategy similar to that used in [52]. However our arguments do not depend on rank(g). Let ∇ ∈ LF(Θ) whose KV algebra is denoted by A. We assume Thus we have In (Θ, ∇) we fix a system of local affine coordinate functions {θ 1 , ..., θ m } whose domain is a convex open subset U. We write the matrix of Fisher information g in the basis ∂ ∂θ j , namely g = ∑ g ij dθ i dθ j . Here ).
The assumption for all i, j, k. We use a notation which is used in [52]. We consider the differential 1-forms Every differential 1-form h j is a de Rham cocycle. By the Lemma of Poincaré the convex open set U supports smooth functions φ j , j =: 1, ..., m which have the following property We put Thus the differential 1-form ∑ j φ j dθ j is de Rham closed. Since U is convex it supports a local smooth function Ψ such that dΨ = ∑ φ j dθ j .
Since the exponential function is injective this local property of (Θ, P) is a global property, in other words the model is globally an exponential model. In final assertion (1) implies assertion (2). This ends the demonstration of the theorem Some Comments.
(i) It must be noticed that the demonstration above is independent of the rank of the Fisher information g.
Therefore, the theorem holds in singular statistical models. (ii) In regular statistical models the theorem above leads to the notion of e-m-flatness as in [18]. (iii) When the Fisher information g is semi-definite the dualistic relation is meaningless. However data (Θ, g, ∇, ∇ * ) may be regarded as data depending on the transversal structure of the distribution Ker(g). (iv) In the analytic category the Fisher information is a Riemannian foliation. Therefore, both the information geometry and the topology of information are transversal concepts. This may be called the transversal geometry and the transversal topology of Fisher-Riemannian foliations. (v) The theorem above does not solve the question as how far from being an exponential family is a given model.
It only tells us that exponential families are objects of the Hessian geometry.
The framework for addressing the challenge just mentioned is the theory of invariants. That is the purpose of a forth going work. Some new results are anounced in the Appendix A to this paper.

The Similarity Structure and the Hyperbolicity
We consider a dually flat pair (M, g, ∇, ∇ * ). Both (M, ∇, g) and (M, g, ∇ * ) are locally hyperbolic in the sense of [2]. So they locally support the geometry of Koszul. That is a consequence of the classical Lemma of Poincare.
Every point of M has an open neighborhood U supporting a local de Rham closed differential 1-forms and subject to the following requirements g|U = δω, By the virtue of Theorem 3 as in [2], for both (M, g, ∇) and (M, g, ∇ * ) being globally hyperbolic it is necessary that Every choice of local differential 1-forms ω and ω * gives rise to a unique pair of local similarity vector fields (H, H * ), viz ∇ X H = X, for all vector fields X. The vector fields H and H * are Riemannian gradients of ω * and of ω respectively. This means that those differential 1-forms are defined by Here ι H stands for the inner product by H, viz ι H g(X) = g(H, X).
This short discussions lead to the following statement

Some Highlighting Conclusions
In this Part A our aim has been to address various purposes involving the theory of KV homology. Doing that we have pointed significant relationships between some major topics in mathematics and the local information geometry. Those relationships might be sources of new investigations.
We summarize some relevant relationships we have been concerned with.

The Total KV Cohomology and the Differential Topology
We have addressed the existence problem for a few major objects of the differential topology. Riemannian foliations and symplectic foliations. Riemannian webs and their linearization problem.
To those questions we have obtained substantial solutions in the category of locally flat manifolds. The cohomological methods we have used are based on the split exact short cohomology sequence

The KV Cohomology and the Geometry of Koszul
The Hessian Geometry is a byproduct local vanishing Theorems in the theory of KV cohomology. The geometry of Koszul is a byproduct of global vanishing Theorem in the same sitting.

The KV Cohomology and the Information Geometry
The category of finite dimensional statistical models for a measurable set (Ξ, Ω) contains the subcategory of finite dimensional Hessian manifolds. From this viewpoint the Hessian information geometry is nothing but the exponential information geometry (i.e., the geometry of exponential families and their genelarizations). The framework for those purposes is closely related to vanishing Theorems in the theory of KV cohomology.
At another side cotangent bundles of Hessian manifolds are Kaehlerian manifolds. This aspect has been discussed by many authors, see [52] and the bibliography ibidem.

The Differential Topology and the Information Geometry
A lot of outstanding links between the differential topology and the information geometry are based on the dualistic relation of Amari. This approach leads to significant results in the category of statistical manifolds. In a statistical manifold (M, g, ∇, ∇ * ) we have introduced the splitting short exact sequence Here (i) Ω ∇ 2 (M) is the space of ∇-geodesic symplectic foliations in M; (ii) S ∇ 2 (M) is the space of ∇-geodesic Riemannian foliations in M.
The numerical invariant n(∇) has outstanding impacts on the differential topology of M. See our results on orthogonal 2-webs and on Riemannian submersions on symplectic manifolds.

The KV Cohomology and the Linearization Problem for Webs
In a locally flat pair (M, g, ∇, ∇ * ) we consider the short exact sequence The linearization of webs of is a difficult outstanding problem in the differential topology.
G k [Ω ∇ 2 (M)] stands for the family formed by We have introduced the double complex It gives rise to spectral sequences which may be useful for computing the KV cohomologies H * τ (A * , A * ) and H * KV (A, C ∞ (M)). That is not the purpose of this paper. However this double complex is useful for replacing the first order differential equation D ∇∇ * (ψ) = 0 by the homological equation We have proved the homological nature of the space of gauge homomorphisms M(∇, ∇ * ). This is useful for relating the image of M(∇, ∇ * ) in H 1 τ (A * , A * ) and the pair H 2 dR (M), H 1,2 (C).

B. The Theory of StatisticaL Models
In the introduction of this paper we have recalled the problem raised by Peter McCullagh.
What is a statistical model [30]? By the way we have recalled a variant request of Misha Gromov.
In a Search for a Structure. The Fisher Information [15,16].
McCullagh and Gromov choose the same framework for addressing their purpose, The theory of category. This Part B is devoted to the same purpose.
Further the moduli space of isomorphism class of objects of a category C is denoted by [Ca]. In general it is difficult to find an invariant inv a which encodes [Ca]. Subsequently to the questions raised by McCullagh and by Gromov the moduli space of isomorphism class of statistical models is discussed in this Part B. Nowadays, there exists a well established theory of statistical models. The classical references are Amari [17], Amari and Nagaoka [18]. Other remarkable references are Barndorff-Nielsen (Indian Journal of Mathematics 29, Ramanujan Centenary Volume) [21,24], Kass and Vos [37], Murray-Rice (Chapter 1, Section 15 in [22]). In Part A of this paper we have been dealing with this current theory. It has been called the local theory. We suggest reading the attempt by McCullagh to establish a conceptually consistent theory of statistical models [30]. In its time, the paper of McCullagh had been the object of controversy and questions.
We are aimed at re-establishing the theory of statistical models. Our motivations have emerged from some criticisms.
The current theory presents some deficiencies that we plan outlining. (i) A weakness of the current theory is its lacking in geometry; (ii) In the literature on the information geometry many references define an m-dimensional statistical model as an open subset of an Euclidean space R m . Though this definition may be useful for dealing with coordinate functions, it is topologically and geometrically useless. Let Γ be the group of measurable isomorphisms of a measurable set (Ξ, Ω). The Fisher information of widely used models are Hessian metrics [52]. This observation is relevant. However the Fisher information is not a characteristic invariant.
We intend to face the following challenges. Challenge 1. Revisit the theory of geometric statistical models for measurable sets. Challenge 2. The Search for a geometric characteristic invariant for statistical models. We recall that such an invariant will encode the points the moduli space of models. Before continuing we recall some definitions.

Definition 42.
A geometric invariant of a model for (Ξ, Ω) is a datum which is invariant under the action of the symmetry group Aut(Ξ, Ω).
The framework which is useful for re-establishing the theory of statistical models is the category of locally trivial fiber bundles.
As we have mentioned the need for introducing a new theory of statistical model emerges from some criticisms. We recall the definition a statistical model [18,22,24].

The First Critique
The first critique arises from requirement (5). From the viewpoint of fiber bundles the requirement (5) is useless. Consider the Cartesian product That is the same thing as the trivial fiber bundle Therefore P θ is the restriction to the fiber E θ of the function P.

The Second Critique
The second critique emerges from the requirement (1). This requirement (1) is too restrictive. It excludes many interesting compact manifolds such as flat tori, euclidean sphere, compact Lie groups.

The Third Critique
From the viewpoint of the differential topology the requirement (6) may be damage to the topology of Θ. When the Fisher information g is singular its kernel is in involution. Thus the topological-geometrical information that are contained in g are transverse to the distribution Ker(g). If Ker(g) is completely integrable then topological and geometrical informations which are contained in g are transversal to the foliation Ker(g). See Part A of this paper. This ends the criticisms.
To motivate for deleting the requirement (1) we construct a compact statistical model which satisfies all of the requirements except the requirement (1).
Let E be the tangent bundle of the circle S 1 . E is the trivial line bundle We consider the fonctions f , F and P defined by The function P(θ, t) has the following properties These properties show that there is a one to one correspondence between the circle S 1 and a subset of probability densities in R. Thus S 1 is a compact 1-dimensional manifold of probabilities in the measurable set (R, β(R)). Here β(R) is the family of Borel subsets of R.

Another Construction.
For every positive integer m we consider positive real numbers α 1 < α 2 < ... < α m and the real functions which are defined by Now we consider the tangent bundle of the m-dimensional flat torus TT m , We put The function P(θ, t) satisfies the following requirements (1) If θ = θ * there exists t * * ∈ R m such that P(θ, t * * ) = P(θ * , t * * ), We deduce that the pair (T m , P) is an m-dimensional manifold of probability densities in the measurable set (R m , β(R m )).
The image of every local chart of T m is a local statistical model in the classical sense [17,18,22]. This ends the Digression.
We are motivated for introducing a new theory of statistical models whose localization yields the current theory. The theory we introduce is an answer to McCullagh and to Gromov.

The Preliminaries
In this Part B we face three major challenges. Challenge 1. Taking into account the criticisms we have raised our aim is to introduce a new theory of statistical model whose localization leads to the classical theory of statistical models. Challenge 2. The second challenge is the Search for an invariant which encodes the point of the moduli space of isomorphism class of statistical models. Challenge 3. We introduce the theory of homological statistical model and we explore the links between this theory and the challenge 2. Challenge 4. The fourth challenge is to explore the relationships between "challenge 1, challenge 2, challenge 3" and "Vanishing Theorems in the theory of KV homology".
The theory of KV cohomology and the geometry of Koszul play important roles. We introduce the needed definitions.
Let (Ξ, Ω) be a measurable set. Let Aut(Ξ, Ω) be the group of measurable isomorphisms of Ξ. Let (M, D) be a locally flat manifold whose KV algebra is denoted by A.
We keep the notation used in Part A of this paper. For instance S 2 (M) is the vector space of differentiable symmetric bi-linear forms in M.

Definition 44. A random Hessian metric in (M, D) is a map
which has the following properties (1) for every vector field X the real number Q(x, ξ)[X, X] is non negative, furthermore ∀v for every ξ ∈ Ξ, the random KV cochain is a random cocycle of the KV complex [C * KV (A, C ∞ (M)), δ KV ].
Let (Ξ, Ω, p) be a probability space. A random Hessian metric Q generates a Hessian structure (M, g Q , D) whose tensor metric g Q is defined by The group Aut(Ξ, Ω) of measurable isomorphisms of (Ξ, Ω) is denoted by Γ.

Warning.
When Ξ is a topological space elements of Γ are continuous maps. When Ξ is a differentiable manifold elements of Γ are differentiable maps. Let P(Ξ) be the Boolean algebra of all subsets of Ξ. The abstract group Aut(Ξ, P(Ξ)) is a subgroup of the group Isom(Ξ) of isomorphisms of the set Ξ.

Definition 45.
A measurable set (Ξ, Ω) is called homogeneous if the natural action of Γ in Ξ is transitive.
Throughout this paper we will be dealing with homogeneous measurable sets. Below we introduce the framework of the theory of statistical models. (1) M is a connected m-dimensional smooth manifold. The map π : E → M is a locally trivial fiber bundle whose fibers E x are isomorphic to the set Ξ.
(4) Every point x ∈ M has an open neighborhood U which is the domain of a local fiber chart The local charts are subject to the following compatibility relation is an affine local chart of the locally flat manifold (M, D), • φ U (π(e)) = p 1 (Φ U (e)).

Comments. Requirements
Both requirements (4) and (5) yield the following remarks: the following action is differentiable the following action is an affine action both the left side member and the right side member of (5) have the following meaning.
Consequently (5) implies that for all x ∈ U ∩ U * one has Therefore we get Suppose that U, U * and U * * are domains of local chart with The requirement (3) means that the fibration π is Γ equivariant. The Figure  We recall that the group Γ acts in both E and M. Figure 2 expresses that the projection π of E on M is Γ-equivariant.  The Figure 3 represents the properties (m.1) and (m.2). We are going to define the category of statistical model for (Ξ, Ω). The framework is the category F B(Γ, Ξ). At one side we recall that the group Γ also acts in R m × Ξ. At another side the localizations are made coherent thanks to Cech cocycles γ UU * . Figure 3 tells two informations. Firstly localizations are Γ-equivariant, secondly thanks to Cech cocycles localizations are coherent.

The Category GM(Ξ, Ω)
We keep the notation used in the previous subsections. Our purpose is the category of statistical models GM(Ξ, Ω). which has the following properties (ρ * ).
[ρ 1 ]: For every local chart (U, supports a non negative real valued function P U subject to the following requirements.
[ρ 1.1 ]: For every fixed ξ ∈ Ξ the function is a probability space. Further the operation of integration Ξ commutes with the operation of differentiation If U ∩ U * = ∅ then P U , P U * and γ UU * are related by the formula [ρ 1.4 ]: Let U ⊂ M be an open subset and let γ ∈ Γ. Let us assume that both U and γ · U are domains of local charts We assume that those local charts satisfy ρ 1.1 , ρ 1.2 and ρ 1.3 . Then the relations implies the equality Actually, ([ρ 1.3 ]) has the following meaning: ∀e ∈ E U∩U * . The Figure 4 represents This ends the comment.
The family U j is an open covering of M. The pair E j × U j is the domain of the local chart (Φ j × φ j ).
In particular A = (U j , φ j ) is an affine atlas of the locally flat manifold (M, D). We have , Ω]. The atlas A Φ satisfies requirements (ρ 1.1 ), (ρ 1.2 ) and (ρ 1.3 ). In E U j the local function p j is defined by By the virtue of of [ρ 1.3 ] one has p i (e) = p j (e) for all e ∈ E U i ∩U j . Thereby there exists a unique function E e → p(e) ∈ R whose restriction to E j coincides with p j . The restriction to E x is denoted by p x . The triple is a probability space.

Definition 50.
The function E e → p(e) ∈ R is called the probability density of the model [E , π, M, D].

Comments.
(i) We take into account the global probability density p. Then an object of the category GM(Ξ, Ω) is denoted by (ii) The function p is Γ-equivariant. THIS IS THE GEOMETRY in the sense of Erlangen program.
(iii) We have not used any argument depending the dimension of manifolds.
The Figure 5 expresses coherence to local probability densities We are in position to define the morphisms of the category GM(Ξ, Ω). In Figure 5 one sees that modulo the dynamics of the group Γ in R m × Ξ all localizations look alike. Figure 6 show that local probability densities {p i } are but localizations of a global probability density p is a statistical model in the classical sense [18,22]. So we have taken up the challenge 1.

Two Alternative Definitions
We introduce two other presentations of the category GM(Ξ, Ω). Those presentations highlight the connection with the searches of McCullagh and Gromov. Those presentation is useful in both the theoretical statistics and the applied statistics [17,18,21,24,55,57,58].
We consider the category MSE whose objects are a probability spaces (Ξ, Ω, p).

Remark 4.
A morphism as in the last definition has a statistical nature. An isomorphisms of (Ξ, Ω, p) on (Ξ * , Ω * , p * ) is an sufficient statistic. The category MSE is useful for introducing two variant descriptions of the category GM(Ξ, Ω).
Definition 54. We use the previous notation.
(1) A statistical model is a locally trivial fiber bundle over a locally flat manifold The fibers of π are probability spaces. At the present step it is clear that the information geometry is structured.

Fisher Information in GM(Ξ, Ω)
We consider a MSE-fibration The Fisher information to be defined is an elementg of Γ(S 2 (T * M)).
We recall that every MSE-fiber M x , x ∈ M has a structure of probability space Let X, Y be local vector fields which are defined in a open neighbourhood of x ∈ M.
Definition 56. The Fisher information at x is defined by We recall that the horizontal differentiation commutes with the integration along the MSE-fibers, viz So the Fisher information g is well defined. It has the following properties (1) g is positive semi-definite, (2) g is an invariant of the Γ-geometry in [E , π, M, D, p].

Exponential Models
Let [E , π, M, D, p] be an object of GM(Ξ, Ω). We recall that data which are defined in E are called random data in the base manifold M. The operation of integration along the MSE-fibers is denoted by F . Thus a random datum µ is called smooth if its image F (µ) is smooth.
Conversely every datum θ * which is point-wise defined in M is the image of the random datum So we get Thus at every x ∈ M one has A random affine function is a function E e → a(e) ∈ R subject to the requirement is called an exponential model if the following conditions are satisfied.
(1) The base manifold M supports a locally flat structure (M, ∇) and a real valued function ψ ∈ C ∞ (M).
(2) The total space E supports a real valued random function a.
Remark 5. At one side Localizations of exact Hessian homological statistical models yield the classical Koszul Information Geometry [55]. That is but the classical Hessian Information Geometry. At another side the KV homology learns that the Hessian information Geometry is the same think as the geometry of exponential famillies see Part A Section 5, Theorem 16.

Reminder.
In the Appendix A to this paper the reader will find a new invariant r b (p) measuring how far from being an exponential model is an MSE-fibration By the virtue of results in Part A, to be an exponential model depends on homological conditions.

The Entropy Flow
We are going to introduce the notion of local entropy flow. Subsequently we will show that the Fisher information of a model [E , π, M, D, p] is the Hessian of the local entropy flow.
To start we consider a MSE-fibration That is another presentation of the statistical model [E , π, M, D, p]. Let [U j , Φ j × φ j , γ ij , P j ] be an atlas of [E , π, M, D, p]. We put Then every [Θ j , P j ] is a local statistical model for (Ξ, Ω). Le X, Y be two vector fields defined in U j and let ψ X (t) and ψ Y (s) be their local flows defined in U j . Then we set Φ j (e) = [θ j (e), ξ j (e)] = [φ j (π(e)), ξ j (e)], e ∈ E j , , ξ j (e)). To pursue we use the compatibility of local charts of the atlas[U j , Φ j × φ , , γ ij , P j ] . If φ j (π(e)) = γ ij · φ i (π(e)).
We recall thatψ(t) andψ(s) are defined bỹ Those reminders are useful for concluding that whenever . So the local entropy flow does not depend on local charts. If both X and Y are complete vector fields then their entropy flow is globally defined. A notable consequence is the following statement.

Direct calculations yield
We know that ∂ 2 ∂s∂t commutes with Ξ . Therby we conclude that be an m-dimensional statistical model for a measurable set (Ξ, Ω). To define the family of α-connections we work in a local chart (Φ U , Φ U ). We set In the base manifold (M, D) the local chart [U, φ u ) yields a system of local affine coordinate functions θ = (θ 1 , ..., θ m ).
We use the notation as in [18]. Given a real number α we define the α-connection ∇ α by its Christoffel symbols in the local coordinate functions θ. Those Christoffel are denoted by Γ α i j : k. We proceed as it follows.
Step 1: In the open subset Θ U ⊂ R m we put This local definition ofΓ α ij:k agrees with affine coordinate change in Θ U .
Step 2: In the open subset U Γ α,U ij:k is defined by Since the definition ofΓ α ij:k agrees with an affine coordinate change we can use an atlas for constructing a Koszul connection ∇ α (A). Since the construction of ∇ α (A) agrees with affine coordinate change the connection ∇ α (A) is independent from the choice of A. Every α-connection is torsion free. So an MSE-fibration gives rise to a ma R α → ∇ α ∈ SLC(M).
If the Fisher information g is definite then (M, g, ∇ α , ∇ −α ) is a dual pair [17,48]. By the virtue of the definition of the Fisher information g a local section of section of Ker(g) is a local vector field X ∈ X (M) such that X · p = 0.
Therefore, it is easily seen that L X g = 0.
So if data are analytic then g is a stratified Riemannian foliation. We recall that a random differential q-form in E is a mapping E e → ω(e) ∈ ∧ q (T * π(e) )M such F ω ∈ Ω q (M).
The vector space of random differential q-forms is denoted by Ω q (E). Let Φ U × φ U be a local chart of M. We set We recal that in Φ U (E U ) the partial differentiation ∂ ∂θ is called the horizontal differentiation in E U . Therefore we use the relation for setting the de Rham complex of random differential forms. Namely The complex Ω(E) is a complex of Γ-modules. Here Then the cohomology space H * (Γ, Ω(E)) is bigraded, The probability density p is Γ-invariant. It is an element of H 0,0 (Γ, Ω(E)).

Another Homological Nature of Entropy
One of main purpose of [14] is the homological nature of the entropy. The classical entropy function of a statistical model [E, π, M, D, p] is defined by E(π(e)) = E π(e) p(e * )log(p(e * )).
Thus the entropy E(π(e)) is Γ-equivariant. Therefore, it defines an equivariant cohomology class This is another topological nature of the entropy. For another viewpoint see [16,31]. Our purpose is to show the theory of statistical models has a homological nature in the category F B(Γ, Ξ). Further the to every fiber p|E x is a probability density.

A Comment.
Let (U, Φ × φ) be a local chart of [E, π, M, D] and let x * ∈ U. We set The last definition above says that we obtain the probability space

The Moduli Space of the Statistical Models
We are going to face another major open problem. The challenge is the search for an invariant which encodes the points of the orbit space That is what is called the problem of moduli space. This problem of moduli space is a major challenge in both the differential geometry and the algebraic geometry (see the theory of Teichmuller). The problem is rather confusedly addressed in [30]. Subsequently it provoked controversies and criticisms.

The Hessian Functor
We consider the category BF whose objects are pairs {M, B} formed by a manifold M equipped bilinear forms B ∈ Γ(T * ⊗2M ).
In Part A we have defined the Hessian differential operator of a Koszul connection ∇, namely Those operators are useful for addressing the problem of moduli spaces. For our purpose four categories are involved, (1) The category LC whose objects are gauge structures (M, ∇), (2) The category GM whose objects are statistical models for measurable sets, (3) the category BF whose objects are manifolds equipped bilinear forms,

Reminder.
We recall that for vector fields X, Y the bilinear form q M [∇](X, Y) is defined by The functor q M is called the Hessian functor of the model Our aim is to demonstrate the following claim. Up to isomorphism a statistical model M is defined by its Hessian functor q M . The functor q M is an significant contribution to the information geometry.
We We assume that Thus in every local trivialization Θ U × Ξ one has the identity belongs to R Ξ + . This ends the idea.
To simlplify we use the following notation.
In the category F B(Γ, Ξ) we consider an isomorphism (1) Let ψ * be the differential of ψ.
for all vector fields X * , Y * ∈ X (M 2 ). (2) It is clear that the datum [E 1 , π, M 1 , D 1 , p 2 • Ψ] is an object of the category GM(X, Ω). Then for vector fields X, Y in M 1 we calculate (at X, Y) the right hand member of the following equality Direct calculations yield Thus for all ∇ ∈ LC(M 1 ) we have q [p 2 •Ψ] (∇) = q p 2 (ψ * (∇)).
We summarize the calculations just carried out as it follows Lemma 8. Keeping the notation we just used namely p 2 and Ψ × ψ we have the following equality We are in position to face the problem of moduli space in the category GM(Ξ, Ω).

Demonstration.
The demonstration is based on Lemmas 7 and 8.
According to our construction one has the following functor This functor q Ψ is the Hessian functor of the model Further Ψ × ψ is an isomorphism of M Ψ onto M 2 . Let us prove that assertion (2) implies assertion (1). By of the definition of morphism of models, the pair Ψ × ψ is an isomorphism of M 1 onto M 2 if and only if Here we set the explicit formulas. Let ∇ ∈ LC(M 1 ). For all vector fields X, Y in M 2 we have now we observe that Therefore (2) impies the equality This shows the implication (2) → (1).
Let us prove that assertion (1) implies assertion (2). Now we assume that that (1) holds, viz Then both M 1 and M Ψ have the same Hessian functor. By the virtue of Lemma 8 above we deduce that This ends the demonstration.
Reminder. They are called MSE-fibrations.
Those categories are canonically equivalent. Further the actions the group G in those categories lead to the same moduli space We rephrase the theorem on moduli space.
parametrizes the moduli space m.
This ends the challenge 2.

The Homological Statistical Models
In this section we introduce the theory of homological statistical models (HSM in short). We address the links between this theory and the local theory as in [17].
The theory of homological statistical models is useful for strengthening the central role played by the theory of KV homology in the information geometry and in the topology of the information [14,[16][17][18]22,30,37,60,61].
We introduce the theory of localization of homological models. We use it to highlight the role played by local cohomological vanishing theorems as well as the role played by global cohomological vanishing theorems.
The framework is the category F B(Γ, Ξ).
Let [E, π, M, D] be an m-dimensional object of the category F B(Γ, Ξ), viz m = dim(M). The KV algebra of (M, D) is denoted by A. The smooth manifold R m supports a sheaf of KV algebrasÃ. This sheaf is locally isomorphic to A. The vector space C ∞ (R m ) is a left module ofÃ. The affine action of Γ in R m isÃ-preserving.
Let (U, Φ U × φ U ) be a local chart of [E, π, M, D]. We recall that dφ U is the differential of φ U . We have dφ U (A) =Ã(φ U (U)).

Comments.
The equality has the following meaning. For v, w ∈ T φ U (e) Θ U one has Morphisms of homological models are defined by replacing the probability P U by the random cocycle Q U . Definition 62. The category of homological statistical models for a measurable set (Ξ, Ω) is denoted by HSM(Ξ, Ω).

The Cohomology Mapping of HSM(Ξ, Ω)
We consider an m-dimensional object of HSM(Ξ, Ω) which is defined by a complete atlas The underlying object of the atlas A is denoted by [E, π, M, D]. We set We are not making any difference between (U j , A) and (Θ j ,Ã). We put set If we assume that We put q j (e) = Q j (Φ j (e)) ∀e ∈ E j . Here , ξ i (e)) = γ ij (θ i (e), ξ i (e)) ∀e ∈ E ij . Therefore we get q i (e) = q j (e) ∀e ∈ E ij .
Therefore q j is the restriction to E j of a (globally defined) map E e → Q(e) ∈ Z 2 KV (A, R).

It is clear that
The action of Γ in E is Q-preserving. Thus a homological statistical model is a quintuplet [E, π, M, D, Q].
Here Q is a map E e → Q(e) ∈ Z 2 KV (A, R).

An Interpretation of the Equivariant Class [Q]
We intend to interpret the cohomology class [Q] as an obstruction class. If [E, π, M, D, Q] is exact then there exists a random differential 1-form θ such that That useful for seing that Ker(Q) is in involution.

Definition 65.
We are interested in random functions defined in R m × Ξ.
(2) A random closed differential 1-form θ has the property p * − EXP if every x ∈ R m has an open neighbourhood U satisfying the following conditions, U × Ξ support a random function f subject to two requirements: • θ = d f , • f has the property p * − Exp.
(3) An exact homological statistical model [E, π, M, D, Q] has property p * − EXP if there exists a random differential 1-form θ satisfying the following conditions • θ has the property p * − EXP,

Localization
Our purpose is to explore the relationships between the theory of homological statistical models and the theory of local statistical model as in [18,22], Barndorff-Nielsen 1987 Our aim is to show that the current (local) theory is a byproduct of the localization of homological models. The notion of localization of homological models is but the notion of local vanishing theorem. (2) If the [E, π, M, D, Q] has the property p * − EXP then [E, π, M, D, Q] is locally isomorphic to a classical statistical model (Θ, P) as in [18].
The Sketch of Proof of (1). Let (U, Φ × φ) be a local chart of [E, π, M, Q]. We set We assume that Θ U is an open convex subset of R m . Θ supports a system of affine coordinate functions θ = (θ 1 , ..., θ m ).
Since Q(θ, ξ) is a random KV cocycle ofÃ we have The last equality is equivalent to the following system We fix ξ ∈ Ξ. For every j the random differential 1-form β j is defined by Every β j (θ, ξ) is a cocycle of the de Rham complex of Θ U .
By the virtue of the Lemma of Poincaré there exists a local function h j (θ, ξ) such that β j = dh j . Now the differential 1-formθ is defined bỹ Direct calculations lead to the following equality This ends the proof of (1).
The proof of (2). We assume that M = [E, π, M, D, Q] has the property p * − Exp. We keep the notation we just used. The random differential 1-formθ is a (de Rham) cocycle. Therefore Θ × Ξ supports a random function h(θ, ξ) such thatθ = dh.
Thus the pair (Θ U , P Q ) is a local statistical model for (Ξ, Ω). This ends the proof of (2). The theorem is demonstrated.
The pair (Θ U , P Q ) is called a localization of M. Let [E, π, M, D, p] be an object of the category GM(Ξ, Ω). We set Q p = D 2 log(p).
Thefore we get the exact homological statistical model M p = [E, π, M, D, Q p ] So the notion of vanishing theorem has significant impacts on the information geometry. To simplify an exact models which having the property p * − Exp (for some probabily space) are is called Exp − models. Theorem 23. The notation is that used previously.

Reminder: New Insigts.
(1.1) The Information GEometry is the geometry of statistical models. (1. 2) The Information topology is the topology of statistical models.

The Homological Statistical Models and the Geometry of Koszul
Our purpose is to to relate the category of homological statistical models and the geometry of Koszul. This relationship is based on the localization of homological statistical models. Proposition 12. EX PHEHSM(Ξ, Ω) stands for the subcategory whose objects are Hessian Exp-models. (2) If M is compact then the subcategory of exact Hessian homological Exp-models EX PHYHSM(Ξ, Ω) is sent in the category of hyperbolic structure in in (M, D).

Examples
This section is devoted to a few examples. The construction involves some basic notions of the differential topology. Let f ∈ C ∞ (M) and x ∈ M. The function Let L ∞ H (M) be the set of f * ∈ C ∞ (M) such that The function P f * (x, h) is defined by .
The pair (M, P f * ) is a probability density in H. Now set Therefore the datum [M × H, p 1 , M, ∇, P f * ] is a statistical model for (H, P(H)). Here P(M) is the boolean algebra of subsets of H and p 1 is the trivial fibration of M × H over M.

Example 2: Geometry
We focus on an example which plays a significant role in global analysis (and geometry) in some type of bounded domains [2,3]. This example relates the geometry of Koszul and Souriau Lie groups thermodynamics [4] and bibliography therein.
Let C ⊂ R m be a convex cone and let C * be its dual. The characteristic function of C is defined by This gives rise to the following function C × C * (v, v * ) → P(v, v * ) = exp(− < v, v * >) C * exp(− < v, w * >)dw * So (C, P) is a statistical model for (C * , dw * ). Here dw * is the standard Borel measure.
From (2) one easily deduces that Ker(g) is in involution. By the virtue of Theorem of Frobenius (1) and (2) imply that Ker(g) is completely integrable.
In the category of differentiable manifolds, not all involutive singular distributions are completely integrable. Nevertheless, that is true in the category of analytic manifolds [62].

This subsection is mainly devoted to examples of stratified Riemannian foliations in analytic manifolds.
For more details about those object the readers are referred to [46,63].
Theorem 24. Let M be an orientable compact analytic manifold. Let C ω (M 2 ) be the space real valued analytic functions defined in M 2 . There exists a canonical map of C ω (M × M) in the family of analytic stratified Riemannian foliations in M.

The Idea of Construction.
Let dv be an analytic volume element in M. In M we fix an analytic torsion free Koszul connection ∇.
The analytic bilinear form g f ∈ S 2 (M) is defined by The form g f has the following properties.
(a) g f does not depend on the choice of ∇, (b) g f is symmetric and positive semi-definite, (c) If X is a section of Ker(g f ) then L X g f = 0.

Conclusion.
If rank(g f ) = constant then g f is a Riemannian foliation as in [38][39][40]46]. If rank(g f ) is not constant we apply [62]. Thereby g f is an analytic stratified Riemannian foliation.

Reminder.
The idea of the strafication of g f .
Step 0 The open subset U 0 ⊂ M is defined by rank(g f (x * )).
The closed analytic submanifold F 1 ⊂ M is defined by Step 1 In the pair (F 1 , g f ) the open subset U 1 ⊂ F 1 is defined by rank(g f (x * )).
Step 2 We iterate this construction. Then we have get a filtration of M This filtration has the following properties (1) F j−1 \ F j is a analytic submanifold of M.
(2) g f defines a regular Riemannian foliation in (F j−1 \ F j , g f ), Remark 6. The extrinsic geometry of submanifolds is a particular case of the geometry of singular foliation [25].

Criticisms
In Part B we have raised some criticisms. We have constructed structures of statistical models in flat tori. An m-dimensional flat torus is not homeomorphic to an open subset of R m . The second criticism is the lack of dynamics. Subsequently, the problem of moduli space is absent from the classical theory. That deficiency is filled in by the characteristic functor M = [E, π, M, D, p] → q M .
The current theory requires a model to be identifiable. From the viewpoint of locally trivial fiber bundles, that requirement is useless.

Complexity
In both the theoretical information geometry and the applied information the exponential models and their generalizations play notable roles. What we call the complexity of a model [E, π, M, D, p] is its distance from the category of exponential models. Up to today there does not exist any INVARIANT which measures how far from being an exponential is a given model. This problem has a homological nature. We have produced a function r b which fills in that lack. (See the Appendix A below).

KV Homology and Localization
We have introduced the theory of homological model. Among the notable notions that we have studied is the localization of homological statistical models. It links the theory of homological models and the current theory as in [22]. It may be interpreted as a functor from the theory of homological models to the classical theory of statistical models. On another side, the notion of Vanishing Theorem is useful in linking HSM(Ξ, Ω) with both GM(Ξ, Ω) and LM(Ξ, Ω).

Homological Models and Hessian Geometry
In the category HSM(Ξ, Ω) the Hessian functor is the functor from HEHSM(Ξ, Ω) to the category of randon Hessian manifolds.
Furthermore, every structure of probability space (Ξ, Ω, p * ) gives rise to a canonical functor from HEHSM(Ξ, Ω) to the category of Hessian manifolds. The canonical functor is defined by [E, π, M, D, Q] → F p * Q