Abstract
This is the first of two papers that introduce a deformation theoretic framework to explain and broaden a link between homotopy algebra and probability theory. In this paper, cumulants are proved to coincide with morphisms of homotopy algebras. The sequel paper outlines how the framework presented here can assist in the development of homotopy probability theory, allowing the principles of derived mathematics to participate in classical and noncommutative probability theory.
Introduction
There is a surprising coincidence between homotopy morphisms in operadic algebra and cumulant functions in probability theory. In an attempt to develop this coincidence into a meaninful relationship between homotopy algebra and probability theory, this paper presents a deformation theoretic framework to explain the coincidence. This framework is used in a sequel paper [1] to introduce a homotopy theory of probability.
In probability theory one considers an algebra of random variables and a linear map \(e\) from this algebra to the complex numbers \(\mathbb {C}\). Typically, the map \(e\) does not respect the product structure; that is for random variables \(X\) and \(Y\),
In fact, the failure of \(e\) to be an algebra map measures important correlations between random variables. For example, the bilinear map defined on the space of random variables by
defines the covariance. The map \(\kappa _2\) fits into an infinite hierarchy of multilinear maps \(\{\kappa _n\}\) called cumulants. The important notion that a set of random variables \(\{X_1, \ldots , X_n\}\) be independent is defined using the vanishing of the cumulants.
This paper presents a mathematical framework for studying linear maps between algebras that do not respect the products. The framework is a manifestation of the following idea:
the failure of a map to respect structure has structure, if you know where to look.
Specifically, Sect. 2 contains a construction, which in a simple case has the following input and output:

Input

chain complexes \(C=(V,d)\) and \(C'=(V',d')\)

degree zero bilinear maps \(V\times V \rightarrow V\) and \(V'\times V' \rightarrow V'\)

a chain map \(f:C\rightarrow C'\)

Output

a sequence of degree zero multlinear maps \(\{f_n:V^{\times n} \rightarrow V'\}_{n=1}^\infty \).
For the input, the differentials and the chain map are not assumed to have any compatibility with the bilinear maps. The sequence of maps \(\{f_n\}\) in the output constitute an \(A_\infty \) algebra morphism between two \(A_\infty \) algebras which arise during the construction.
A probability space provides an example of the input data for the construction. The chain complexes \(C\) and \(C'\) are, respectively, \((V,0)\) and \((\mathbb {C},0)\), the space of random variables and the complex numbers both with zero differential. The bilinear maps are given by the products of random variables and complex numbers. The chain map \(C\rightarrow C'\) is the expectation \(e:V\rightarrow \mathbb {C}\). Thus, the construction can be applied producing an \(A_\infty \) morphism
as output. The main result of this paper is that this \(A_\infty \) morphism coincides with the Boolean cumulants (see Definition 7 and Remark 3): \(\kappa _n=e_n\) for all \(n\).
The paper proceeds as follows. Section 2 contains the details of the construction of the output \(A_\infty \) morphism. While \(A_\infty \) structures have been studied in topology since the middle of the last century [7], Sect. 2 also includes a brief overview of \(A_\infty \) algebras and morphisms. This may be helpful to readers unfamiliar with \(A_\infty \) structures, and it also highlights the notion of equivalence in their moduli spaces, which is an important tool in our approach. Section 3 contains the definition of the cumulants and the main proposition. There is nothing novel in the proof of the main proposition, which is a formal verification. The novelty is the statement, which connects the two previously unlinked concepts of homotopy morphisms and cumulants.
The authors would like to thank Tyler Bryson, Joseph Hirsh, Tom LeGatta, and Bruno Vallette for many helpful discussions.
\(A_\infty \) algebras and morphisms
The book [3] is good reference for \(A_\infty \) algebras.
Definitions
Let \(V\) be a graded vector space. Let \(V^{\otimes n}\) denote the \(n\)th tensor power of \(V\) and \(TV=\oplus _{n=1}^\infty V^{\otimes n}\). As a direct sum, linear maps from \(TV\) to a vector space \(W\) correspond to collections of linear maps \(\{V^{\otimes n} \rightarrow W\}_{n=1}^\infty \). Also, \(TV\) is a coalgebra, free in a certain sense, so that coalgebra maps from a coalgebra \(\mathcal {C}\) to \(TV\) correspond to linear maps \(\mathcal {C}\rightarrow V\). This freeness also implies that coderivations from a coalgebra \(\mathcal {C}\) to \(TV\) correspond to linear maps \(\mathcal {C}\rightarrow V\). All these correspondences are onetoone: any linear map \(C\rightarrow V\) can be lifted uniquely to a coderivation \(\mathcal {C} \rightarrow TV\), or lifted uniquely to a coalgebra map \(\mathcal {C} \rightarrow TV\).
Definition 1
An \(A_\infty \) algebra is a pair \((V,D)\) where \(V\) is a graded vector space and \(D:TV \rightarrow TV\) is a degree one^{Footnote 1} coderivation satisfying \(D^2=0\). An \(A_\infty \) morphism between two \(A_\infty \) algebras \((V,D)\) and \((V',D')\) is a differential coalgebra map \(F:(TV,D)\rightarrow (TV',D')\). In other words, an \(A_\infty \) map from \((V,D)\) to \((V',D')\) is a degree zero coalgebra map \(F:TV \rightarrow TV'\) satisfying \(FD=D'F\).
The identification
can be used to give the data of an \(A_\infty \) algebra. That is, a coderivation \(D:TV \rightarrow TV\) can be given by a sequence \(\{d_n\}\) of degree one linear maps \(d_n:V^{\otimes n} \rightarrow V\) on the graded vector space \(V\). The condition that \(D^2=0\) implies an infinite number of relations satisfied by various compositions among the \(\{d_n\}\). Likewise, the identification
can be used to give the data of an \(A_\infty \) morphism between \((V,D)\) and \((V',D')\). That is, a coalgebra map \(F:TV \rightarrow TV'\) can be given by a sequence \(\{f_n\}\) of degree zero maps \(f_n:V^{\otimes } \rightarrow V'\). The condition that \(FD=D' F\) encodes an infinite number of relations among the \(\{f_n\}\), the \(\{d_n\}\) and the \(\{d'_n\}\). So, the identifications in Eqs. (1) and (2) provide two equivalent ways of describing \(A_\infty \) algebras and morphisms and it is convenient to move between the two ways since certain notions or computations are easier to describe in one or the the other description of the equivalent data. For example, it is straightforward to compose two differential coalgebra maps \(F:(TV,D)\rightarrow (TV',D')\) and \(G:(TV',D')\rightarrow (TV'',D'')\) as \(GF:(TV,D)\rightarrow (TV'',D'')\) and thus define the composition of \(A_\infty \) morphisms, but it is more involved to express the components \((gf)_n:V^{\otimes n} \rightarrow V''\) in terms of the \(f_k:V^{\otimes k} \rightarrow V'\) and \(g_m:V'^{\otimes m} \rightarrow V''\).
Note that \(A_\infty \) structures can be transported via isomorphisms. In particular, if \((V,D)\) is an \(A_\infty \) algebra and \(G:TV \rightarrow TW\) is any degree zero coalgebra isomorphism, then for
the pair \((W,D^G)\) is again an \(A_\infty \) algebra.
Morphisms can be transported as well. If \(F\) is an \(A_\infty \) morphism between \((V,D)\) and \((V', D')\) and \(G:TV \rightarrow TW\) and \(H:TV' \rightarrow TW'\) are coalgebra isomorphisms, then
is an \(A_\infty \) morphism between the \(A_\infty \) algebras with the transported structures \((W,D^G)\) and \((W',(D')^H)\).
Definition 2
Two \(A_\infty \) algebras \((V,D)\) and \((V',D')\) are equivalent if there exists a coalgebra isomorphism \(G:TV \rightarrow TV'\) so that \(D'=D^G.\)
Spaces of \(A_\infty \) algebras
Consider an \(A_\infty \) algebra \((V,D)\). The condition that \(D\) has degree one and that \(D^2=0\) imply that the first component \(d_1:V \rightarrow V\) of \(D\) has degree one and satisfies \(d_1\circ d_1=0\). So, the pair \((V,d_1)\) is a chain complex. Often it is appropriate to view the chain complex \((V,d_1)\) as a fundamental object and to consider the remaining components \(d_2, d_3, \ldots \) of \(D\) as structure on the chain complex \((V,d_1)\).
Definition 3
Let \(C=(V,d)\) be a chain complex. An \(A_\infty \) structure on \(C\) is an \(A_\infty \) algebra \((V,D)\) with \(d_1=d.\) Let \(\mathcal {M}_C\) denote the set of \(A_\infty \) structures on \(C\).
Definition 4
Let \(C=(V,d)\) be a chain complex. The gauge group \(\mathcal {G}_C\) is the subgroup of \(\hbox {GL}(TV)\) consisting of degree zero coalgebra automorphisms with first component \(\mathrm{id }:V\rightarrow V\). The gauge group \(\mathcal {G}_C\) acts, on the right, by conjugation on \(\mathcal {M}_C\):
The gauge group \(\mathcal {G}_C\)
The gauge group \(\mathcal {G}_C\) as defined in Definition 4 is a Lie subgroup of \(\mathrm {GL}(TV)\). The Lie algebra of \(\mathcal {G}_C\) is the Lie subalgebra of \(\mathrm {gl}(TV)\) consisting of all degree zero coderivations \(TV \rightarrow TV\) with first component \(0:V\rightarrow V\). This Lie subalgebra can be identified via Eq. (1) with the vector space of degree zero maps \(\prod _{k=2}^\infty \hom (V^{\otimes k},V)\). Any map \(a:V^{\otimes k}\rightarrow V\) with \(k>1\) can be lifted to a coderivation \(A:TV\rightarrow TV\) and then exponentiated to obtain a gauge group element \(\mathbf {a}\in \mathcal {M}_C\) where
The orbits of the one parameter subgroup \(t\mapsto \exp (t A)\) of the gauge group \(\mathcal {G}_C\) are curves in the space \(\mathcal {M}_C\). Each of these curves connects an \(A_\infty \) structure \(D\) at \(t=0\) to an equivalent \(A_\infty \) structure \(D^{\mathbf {a}}\) at \(t=1\) (Fig. 1).
Remark 1
In classical deformation theory, equivalent structures are identified to form a quotient moduli set. Rather than identifying equivalent structures, a simplicial moduli space can be constructed. The points of the simplicial moduli space consist of all structures. The paths in the simplicial moduli space consist of equivalences between structures. Higher dimensional parts correspond to equivalences between equivalences. This paper involves \(A_\infty \) structures which are equivalent to trivial \(A_\infty \) structures. In order to see the application to probability theory, gauge equivalent structures should not be identified, so the relevant moduli space is the simplicial moduli space.
Spaces of \(A_\infty \) morphisms
Consider an \(A_\infty \) morphism \(F:TV\rightarrow TV'\) between \(A_\infty \) algebras \((V,D)\) and \((V',D')\). The conditions that \(F\) has degree zero and that \(FD=D' F\) imply that the first component \(f_1:V\rightarrow V'\) has degree zero and satisfies \(f_1d_1=d'_1 f_1\). Thus \(f_1:(V,d_1)\rightarrow (V',d'_1)\) is a chain map. Here, the chain map \(f_1:(V,d_1)\rightarrow (V',d'_1)\) is viewed as a fundamental object and the remaining components \(f_2, f_3, \ldots \) of \(F\) are viewed as a structure on the chain map \(f_1\).
Definition 5
Let \(C=(V,d)\) and \(C'=(V',d')\) be chain complexes and let \(f:C\rightarrow C'\) be a chain map. An \(A_\infty \) morphism on \(f\) is an \(A_\infty \) morphism \(F:(TV,D)\rightarrow (TV,D')\) between two \(A_\infty \) structures on \(C\) and \(C'\) with \(f_1=f.\) Let
be the set of \(A_\infty \) morphisms on \(f\) between \(A_\infty \) structures on \(C\) and \(C'\). That is, a triple \((D,F,D')\in \mathcal {M}_f\) consists of degree one, square zero coderivations \(D:TV\rightarrow TV\) and \(D':TV'\rightarrow TV'\) with \(d_1=d\) and \(d'_1=d'\) and a degree zero coalgebra map \(F:TV \rightarrow TV'\) satisfying \(FD=D'F\) with \(f_1=f\).
The product of gauge groups \(\mathcal {G}_C\times \mathcal {G}_{C'}\) acts, on the right, on \(\mathcal {M}_f\) by
The picture of morphisms equivalent to chain maps is similar to that of structures equivalent to chain complexes (Fig. 2).
Remark 2
A seemingly trivial situation will be important in the next section. An ungraded vector space \(V\) can be considered a chain complex \(C=(V,0)\) by setting the degree of every element of \(V\) to be zero and setting the differential \(d=0\). The gauge group \(\mathcal {G}_C\) acts trivially on \(\mathcal {M}_C\) when \(d=0\) since \(G^{1} 0 G = 0\) for any \(G\in \mathcal {M}_C\). Any linear map \(f:V\rightarrow V'\) between vector spaces \(V\) and \(V'\) is a chain map between \(C=(V,0)\) and \(C'=(V',0)\) when \(V\) and \(V'\) are regarded as chain complexes with zero differentials. If \(a:V^{\otimes k} \rightarrow V\) and \(a':V'^{\otimes m} \rightarrow V'\), then \(a, a'\) can be lifted to coderivations \(A, A'\) and exponentiated to obtain coalgebra automorphisms \(\mathbf {a}\in \mathcal {G}_C\) and \(\mathbf {a}' \in \mathcal {G}_{C'}\). The \(A_\infty \) structures \(D^{\mathbf {a}}\) and \(D'^{\mathbf {a}'}\) on \(C\) and \(C'\) are identically zero, but the morphism \(F^{\mathbf {a},\mathbf {a}'}: TV\rightarrow TV'\) is typically nonzero. That is, \(F^{\mathbf {a},\mathbf {a}'}\) is a nonzero \(A_\infty \) morphism between two zero \(A_\infty \) structures.
For example, a straightforward computation in the simple case that \(a:V^{\otimes 2} \rightarrow V\) and \(a':V'^{\otimes 2} \rightarrow V\) reveals the first two structure morphisms \(f_1^{\mathbf {a},\mathbf {a}'}:V\rightarrow V'\) and \(f_2^{\mathbf {a},\mathbf {a}'}:V^{\otimes 2} \rightarrow V'\) to be the maps defined by
Probability spaces and cumulants
Probability spaces
One modern approach to probability theory (see, for example, [8]) begins with the following definition:
Definition 6
A probability space is a triple \((V,e,a)\) where \(V\) is a complex vector space, \(e:V\rightarrow \mathbb {C}\) is a linear function, and \(a:V\times V \rightarrow V\) is an associative bilinear product. Elements of \(V\) are called random variables and the number \(e(X)\) is called the expected value of the random variable \(X\in V\). The notation \(X_1X_2\) is used for \(a(X_1,X_2)\). Multiplication of complex numbers is denoted by \(a'\). For \(n>1\), the notation \(\alpha _n\) (and \(\alpha '_n\)) is used for the linear maps \(V^{\otimes n}\rightarrow V\) (and \(\mathbb {C}^{\otimes n}\rightarrow \mathbb {C}\)) obtained by repeated multiplication; for \(n=1\), \(\alpha _1=\mathrm{id }_V\) (and \(\alpha '_1=\mathrm{id }_\mathbb {C}\)). The expectation values of products \(e(X_1\cdots X_n)\) are called joint moments.
The product on \(V\) is not assumed to be commutative. Elements of \(V\) are sometimes called observables when the product is not commutative, but here no special terminology is used to distinguish between commutative and noncommutative probability spaces.
Cumulants
Definition 7
Let \((V,e,a)\) be a probability space. The \(n\) th cumulant of \((V,e,a)\) is the linear map \(\kappa _n:V^{\otimes n} \rightarrow \mathbb {C}\) defined recursively by the following equation:
where \(P\) ranges over all ordered partitions \(P=(n_1,\ldots , n_k)\) with \(\sum \nolimits _{j=1}^k n_j = n\).
Remark 3
The cumulants defined above have been called Boolean cumulants [6] to distinguish these cumulants from the classical cumulants, which are defined in the special case when the product on \(V\) is commutative, and the free cumulants which are important in free probability theory [5]. For a survey of various kinds of cumulants and their combinatorics, see [2].
Equation (3) expresses the joint moment \(e(X_1\cdots X_n)\) in terms of products of cumulants. For the first few values of \(n\), this equation is given by:
One easily solves these equations for the cumulants expressed in terms of products of joint moments:
and in general finds
The sum in Eq. (4) above is over the same set: \(P\) ranges over all ordered partitions \(P=(n_1,\ldots , n_k)\) with \(\sum \nolimits _{j=1}^k n_j = n\).
Main Proposition
Hypotheses for the Main Proposition
Let \((V,e,a)\) be a probability space. Consider both \(V\) and the complex numbers \(\mathbb {C}\) as graded vector spaces concentrated in degree zero. Then \((V,0)\) and \((\mathbb {C},0)\) are \(A_\infty \) algebras and the map \(e:V\rightarrow \mathbb {C}\) defines an \(A_\infty \) morphism between these two \(A_\infty \) algebras. Denote by \(E:TV\rightarrow T\mathbb {C}\) the lift of \(e\) as a coalgebra map. Furthermore, following the notation of Sect. 2.3, let \(\mathbf {a}=\exp (A)\) where \(A:TV\rightarrow TV\) is the lift of \(a:V\otimes V \rightarrow V\) as a coderivation, and similarly let \(\mathbf {a'}=\exp (A')\), where \(A':T\mathbb {C}\rightarrow T\mathbb {C}\) is the lift of \(a':\mathbb {C}\otimes \mathbb {C}\rightarrow \mathbb {C}\) to a coderivation. Then, \(E^{\mathbf {a},\mathbf {a}'}:TV\rightarrow T\mathbb {C}\) is an \(A_\infty \) morphism between \((V,0^{\mathbf {a}})=(V,0)\) and \((\mathbb {C},0^{\mathbf {a}'})=(\mathbb {C},0)\). Let \(e_n:V^{\otimes n}\rightarrow \mathbb {C}\) denote the components of the \(A_\infty \) morphism \(E^{\mathbf {a},\mathbf {a}'}\).
Main Proposition
\(\kappa _n=e_n\) for all \(n\).
Proof
oth the collection \(\{\kappa _n:V^{\otimes n}\rightarrow \mathbb {C},n\ge 1\}\) and the single map \(e:V\rightarrow \mathbb {C}\) can be extended as coalgebra maps \(TV\rightarrow T\mathbb {C}\). Let \(K\) and \(E\), respectively, denote these extensions. The statement of the proposition is that these two coalgebra maps are related by means of the coalgebra isomorphisms \(\mathbf {a}\) and \(\mathbf {a}'\) as in the following diagram.
It suffices to check that the components
coincide when evaluated on a vector \(X_1 \otimes \cdots \otimes X_n \in V^{\otimes n}\).
The map \(TV \mathop {\rightarrow }\limits ^{E} T\mathbb {C}\rightarrow \mathbb {C}\) is zero except for the one to one component \(e:V\rightarrow \mathbb {C}\). Recall that \(A\) denotes the lift of \(a\) to a coderivation \(TV\rightarrow TV\) and \(\mathbf {a}=\exp (A)\). So the only components of \(\mathbf {a}\) that contribute to the composition in question are \(\frac{1}{(n1)!} A^{n1}:V^{\otimes n}\rightarrow V\).
The nonzero components of \(A\) are of the form \(a_n:V^{\otimes n}\rightarrow V^{\otimes n1}\) and are given by
Then the only nonzero component of \(\frac{1}{(n1)!}A^{n1}\) is the composition \(\frac{1}{(n1)!}a_2\circ \cdots \circ a_n\). The expression for \(a_n\) is a sum of \((n1)\) terms, so this composition has \((n1)!\) terms. Applied to \(X_1\otimes \cdots \otimes X_n\), this composition then yields \(X_1\cdots X_n\). So the composition along the left and bottom takes \(X_1\otimes \cdots \otimes X_n\) to
The map \(K:TV \rightarrow T\mathbb {C}\) evaluated on \((X_1\otimes \cdots \otimes X_n)\) breaks into the following sum.
where \(P\) ranges over all ordered partitions \(P=(n_1,\ldots , n_k)\) with \(\sum \nolimits _{j=1}^k n_j = n\).
As in the calculation above for \(\mathbf {a}\), the nonzero component of \(\mathbf {a}'\) mapping \(\mathbb {C}^{\otimes k}\rightarrow \mathbb {C}\) is \(\frac{1}{(k1)!}(A') ^{k1}\) which maps \(z_1\otimes \cdots \otimes z_n \mapsto z_1\cdots z_n\). Hence, the composition along the top and right yields
here the sum is over the same partitions as above. The coincidence of the two maps now follows from Definition 7 of the cumulants. \(\square \)
Remark 4
In classical probability theory, random variables are measurable \(\mathbb {C}\)valued functions on a measure space and the expectation value of a random variable is defined by integration. The product of measurable functions is measurable and defines the product of random variables. In this situation, the product is commutative and associative. One can define a classical probability space as a probability space \((V,e,a)\) for which \(a:V\times V \rightarrow V\) is commutative. The entire discussion in Sects. 2 and 3 of this paper can be symmetrized for a classical probability space. The requisite modifications and results are contained in Sect. 2 of [1].
Homotopy probability theory
The starting point of homotopy probability theory is to replace the space \(V\) of random variables with a chain complex \(C=(V,d)\) of random variables.
Definition 8
The data of a homotopy probability space consists of a chain complex \(C=(V,d)\), a chain map \(e:C \rightarrow \mathbb {C}\), and a degree zero associative product \(a:V^{\otimes 2} \rightarrow V\).
The expectation \(e\) and the differential \(d\) are not assumed to satisfy any properties with respect to \(a:V^{\otimes 2}\rightarrow V\).
The coincidence of the cumulants for a probability space and an \(A_\infty \) morphism on the expectation provides the guide for how to proceed for a homotopy probability space. Cumulants for a homotopy probability space are defined as the \(A_\infty \) morphism \(E^{\mathbf {a}, \mathbf {a}'}\) on the chain map \(e:(V,d)\rightarrow (\mathbb {C},0)\) associated to the product \(a:V^{\otimes 2}\rightarrow V\) and the product \(a':\mathbb {C}^{\otimes 2}\rightarrow \mathbb {C}\) of complex numbers. These cumulants are an \(A_\infty \) morphism between the \(A_\infty \) structure \((V,D^\mathbf {a})\) and the (zero) \(A_\infty \) structure \((\mathbb {C},D^{\mathbf {a}'})\).
These ideas are elaborated in Sects. 2 and 3 of [1] and illustrated with an example in Sect. 4 of [1].
Notes
Readers familiar with \(A_\infty \) algebras will be aware that the definition of an \(A_\infty \) algebra usually involves a shift of degree, but no degree shift is used in this paper.
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Acknowledgments
This material is based in part upon work supported by the National Science Foundation under Award No. DMS1004625. This work was supported by the IBS (CA130501). This work was supported by the Midcareer Researcher Program through NRF grant funded by the MEST (No. 20100000497). The last author would like to acknowledge the excellent working environment provided by the Simons Center for Geometry and Physics.
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Communicated by Jim Stasheff.
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DrummondCole, G.C., Park, JS. & Terilla, J. Homotopy probability theory I. J. Homotopy Relat. Struct. 10, 425–435 (2015). https://doi.org/10.1007/s400620130067y
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DOI: https://doi.org/10.1007/s400620130067y
Keywords
 Probability
 Cumulants
 Homotopy
Mathematics Subject Classification (2000)
 55U35
 46L53
 60Axx