Abstract
One of the core advantages topological methods for data analysis provide is that the language of (co)chains can be mapped onto the semantics of the data, providing a natural avenue for human understanding of the results. Here, we describe such a semantic structure on Chen’s classical iterated integral cochain model for paths in Euclidean space. Specifically, in the context of population time series data, we observe that iterated integrals provide a modelfree measure of pairwise influence that can be used for causality inference. Along the way, we survey recent results and applications, review the current standard methods for causality inference, and briefly provide our outlook on generalizations to go beyond time series data.
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D.L. is supported by the Office of the Assistant Secretary of Defense Research & Engineering through ONR N000141612010, and the Natural Sciences and Engineering Research Council of Canada (NSERC) PGSD3.
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Appendix: Path Space Cochains
Appendix: Path Space Cochains
Chen’s formulation of a cochain model begins by defining a de Rhamtype cochain complex A _{dR} on a general class of spaces called differentiable spaces, generalizing the usual differential forms defined on manifolds. Path spaces are examples of differentiable spaces, and thus are associated with such a de Rham cochain complex. By defining iterated integrals using higherdegree forms on \(\mathbb {R}^N\), rather than the 1forms used in Definition 1, we obtain forms on \(P\mathbb {R}^N\) rather than functions. Finally, he shows that the forms generated by iterated integrals form a subcomplex of A _{dR}, that is, in fact, quasiisomorphic to A _{dR}. A detailed account of this construction is found in [11], and a more modern treatment can be found in [17].
Smooth structures are defined on manifolds by using charts to exploit the welldefined notion of smoothness on Euclidean space. Charts can be viewed as probes into the local structure of a manifold. However, as homeomorphisms of some Euclidean space of fixed dimension, charts are a rather rigid way to view local structure as they are both maps into and out of a manifold. Differentiable spaces relax this homeomorphism condition, and only require its plots, the differentiable space analog of a chart, to map into the space. Baez and Hoffnung [3] further discuss these ideas, along with categorical properties of differentiable spaces.
Definition 9
A differentiable space is a set X equipped with, for every Euclidean convex set \(C \subseteq \mathbb {R}^n\) with nonempty interior and for any dimension n, a collection of functions ϕ : C → X called plots, satisfying the following:

1.
(Closure under pullback) If ϕ is a plot and f : C′→ C is a smooth, then ϕf is a plot.

2.
(Open cover condition) Suppose the collection of convex sets {C _{j}} form an open cover of the convex set C, with inclusions . If ϕi _{j} is a plot for all j, then ϕ is a plot.

3.
(Constant plots) Every map \(f:\mathbb {R}^0 \rightarrow X\) is a plot.
It is clear that any manifold is a differentiable space by taking all smooth maps ϕ : C → M to be plots. We obtain a canoncial differentiable space structure on PM by noting that, given any map \(\overline {\alpha }: C \rightarrow PM\), there is an associated adjoint mapα : I × C → M defined by \(\alpha (t, x) = \overline {\alpha }(x)(t)\). Consider the collection of all maps \(\overline {\alpha }: C \rightarrow M\) for which the adjoint α is a smooth map, which clearly satisfies the first and third conditions. To obtain a collection of plots on PM, we additionally include all maps \(\overline {\alpha }: C \rightarrow PM\) such that the hypothesis of the second condition is true.
Definition 10
A pformω on a differentiable space X is an assignment of a pform ω _{ϕ} on C to each plot ϕ : C → X such that if f : C′→ C is smooth, then ω _{ϕf} = f ^{∗}ω _{ϕ}. The collection of pforms on X is denoted \(A^p_{dR}(X)\), and the graded collection of all forms on X is A _{dR}(X).
Linearity, the wedge product, and the exterior derivative are all defined plotwise. Namely, given ω, ω _{1}, ω _{2} ∈ A _{dR}(X), \(\lambda \in \mathbb {R}\), and any plot ϕ : C → X,

(ω _{1} + λω _{2})_{ϕ} = (ω _{1})_{ϕ} + λ(ω _{2})_{ϕ},

(ω _{1} ∧ ω _{2})_{ϕ} = (ω _{1})_{ϕ} ∧ (ω _{2})_{ϕ}, and

(dω)_{ϕ} = dω _{ϕ}.
Therefore, A _{dR}(X) has the structure of a commutative differential graded algebra, and we may define the de Rham cohomology
of differentiable spaces.
From here forward, we will focus on the case of forms on \(P\mathbb {R}^N\), for which there is a special, easily understood class of forms defined using iterated integrals. Much of what we explicitly construct can be lifted to paths in manifolds of interest,or to more general mapping spaces, and will be discussed in forthcoming work by the authors.
Definition 11
Let ω _{1}, …, ω _{k} be forms on \(\mathbb {R}^N\) with \(\omega _i \in A^{q_i}_{dR}(\mathbb {R}^N)\). The iterated integral \(\int \omega _1 \ldots \omega _k\) is a \(\left ( (q_1 + \ldots + q_k)  k \right )\)form on \(P\mathbb {R}^N\) defined as follows. Let \(\overline {\alpha }:C \rightarrow P\mathbb {R}^N\) be a plot with adjoint \(\alpha :C \times I \rightarrow \mathbb {R}^N\). Decompose the pullback of ω _{i} along α on C × I as
where \(\omega _i^{\prime }, \omega _i^{\prime \prime }\) are q _{i}forms on C × I without a dt term. Then, the iterated integral is defined as
Consider the conceptual similarities between this definition, and the one given in Definition 1. In the language of our present formulation, S ^{I}( Γ), as given in Eq. (1), is the iterated integral where \(\omega _l = \mathrm {d} x_{i_l}\) viewed through the onepoint plot \(\overline {\alpha }_\Gamma : \{*\} \rightarrow P\mathbb {R}^N\) defined by \(\overline {\alpha }_\Gamma (*) = \Gamma \).
Definition 12
Let \(Chen(P\mathbb {R}^N)\) be the subvector space of forms on \(P\mathbb {R}^N\) generated by
where

\(\omega _i \in A_{dR}(\mathbb {R}^N)\), for i = 0, …, k + 1,

\(\int \omega _1 \ldots \omega _k\) is the iterated integral in the previous definition, and

\(\pi _0, \pi _1: P\mathbb {R}^N \rightarrow \mathbb {R}^N\) are the evaluation maps at 0 and 1 respectively.
Theorem 6 ([11])
The complex \(Chen(P\mathbb {R}^N)\)is a differential graded subalgebra of \(A_{dR}(P\mathbb {R}^N)\).
This theorem is proved by showing that the \(Chen(P\mathbb {R}^N)\) is closed under the differential and the wedge product. As we will not make use of the details, we refer the reader to [11] for further discussion of the differential, noting only that the additional forms \(\pi _0^*(\omega _0)\) and \(\pi _{1}^*(\omega _{k+1})\) are required for closure. The wedge product structure is analogous to the shuffle product identity in Theorem 3, and is proved in a similar manner. Note that the wedge product structure for 0cochains is exactly Theorem 3.
Given m forms \(\omega _i \in A^{q_i}_{dR}(\mathbb {R}^N)\) and σ a permutation of the set [m], we denote by \(\epsilon _{\sigma , (q_i)} \in \{1, 1\}\) the sign such that
As the notation suggests, \(\epsilon _{\sigma , (q_i)}\) depends on both the permutation and the ordered list of the degrees (q _{i}).
Lemma 2
Let \(\omega _i \in A^{q_i}_{dR}(R^N)\)for i = 1, …, k + l. We have the following product formula:
Theorem 6 and the following theorem show that the subcomplex of iterated integrals \(Chen(P\mathbb {R}^N)\) is a cochain model for \(P\mathbb {R}^N\).
Theorem 7
The two commutative differential graded algebras, \(A_{dR}(P\mathbb {R}^N)\)and \(Chen(P\mathbb {R}^N)\), have the same minimal model as \(\mathbb {R}^N\).
Returning our focus to iterated integrals as functions, we see that the S ^{I} are 0cochains in this model, constructed via pullback and integration. Indeed, consider the evaluation map \(\mathrm {ev}_k : \Delta ^k \times P\mathbb {R}^N \rightarrow (\mathbb {R}^N)^k\) defined by
Then, S ^{I} is the image of \(\otimes _{l=1}^k \mathrm {d} x_{i_l}\) under the composition
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Giusti, C., Lee, D. (2020). Iterated Integrals and Population Time Series Analysis. In: Baas, N., Carlsson, G., Quick, G., Szymik, M., Thaule, M. (eds) Topological Data Analysis. Abel Symposia, vol 15. Springer, Cham. https://doi.org/10.1007/9783030434083_9
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