Introduction
Short history of calculus
The origins of differential calculus lie in Newton's Method of Fluxions (1736) and Leibniz's Nova methodus pro maximis et minimis (1684). Both works already gave rules for products and composites. In modern operator notation, the first-order chain rule is the functorial identity
which replaces composition of nonlinear maps by composition of linear maps. At higher orders this simplicity breaks: repeated differentiation applies the product rule to the products of derivatives created by the chain rule. The result acquires lower-order correction terms and rapidly growing symbolic combinatorics. These terms were worked out degree by degree for a long time, until Faà di Bruno's papers of 1855 and 1857 settled the scalar combinatorics in general order [FaaDiBruno1855] [FaaDiBruno1857] [Johnson2002].
General formulas for higher-dimensional maps appeared remarkably late in the literature. Fraenkel gave an arbitrary-order formula for maps between Banach spaces in 1978 [Fraenkel1978]. He noted that Cartan and Dieudonné had described how to construct such formulas without displaying the result, and called the full finite-dimensional coordinate formula "monstrous." In modern notation, the compact coordinate-free form is the partition chain rule
In this formula, the combinatorics have been compressed into the partitions of \([k]\). It has one summand for every partition. Expanding it in partial-derivative coordinates introduces a further layer of multi-indices, multiplicities, and factorials; Constantine and Savits gave a systematic formula for that expansion only in 1996 [CS1996]. Our companion paper [FDB] derives both forms from one operation, Taylor composition.
Several schools of calculus had fully developed their theories before these formulas became available and had to cope with higher derivatives in different ways:
Classical differential geometry [KobayashiNomizu1963] handles higher order by choosing a connection. Raw second derivatives acquire lower-degree correction terms under changes of coordinates; the Christoffel symbols absorb these corrections, and covariant differentiation can then be iterated. This makes the connection, its coordinate coefficients, and the curvature produced by commutators explicit. Because the connection is additional, non-canonical data, the resulting theory does not express raw higher differentials as canonical functorial morphisms.
Jet theory handles higher order contravariantly, by retaining a function germ only up to a prescribed order. Two functions have the same \(k\)-jet at \(x\) when their difference lies in \(\mathfrak m_x^{k+1}\), so the algebra of \(k\)-jets is the quotient \(\mathcal O_x/\mathfrak m_x^{k+1}\) [Ehresmann1951] [Saunders1989] [KMS1993] [Cartan1971]. This makes finite-order dependence and functorial composition explicit: pullback preserves powers of the maximal ideal and therefore descends to the quotients, while prolongations carry higher-order differential equations and operators. The combinatorics of the higher chain rule remains implicit in the algebra of jet composition.
Distribution theory handles higher order on the covariant dual side [Schwartz1966] [Hormander1983]. In finite dimensions, distributions supported at a point are precisely finite-order differential operators evaluated at that point. They push forward by \((\phi_*u)(f):=u(f\circ\phi)\), and pushforward preserves point support. This makes duality with function germs, support, order, and functorial transport explicit. The coordinate expansion of that transport is not normally singled out from its definition by duality.
Discrete calculus handles higher order through repeated operations on values at separated points. Forward differences [Norlund1924] [Jordan1965] and divided differences [FL2007] make finite-step identities and interpolation explicit; difference quotients [BGN2004] provide an analytic calculus, while umbral operator calculus [RomanRota1978] organizes shift-invariant operators algebraically. Duarte and Torres [DT2008] derive recursive formulas for cubical differences indexed by partitions, with shifts of the basepoint. Because of these shifts, their formulas do not define functorial transport between fixed-basepoint cube spaces.
The four-operator calculus
The preceding schools fit into one picture with two independent directions. In one direction, observables pull back while probes and differential operators push forward. In the other, cubical probes record exact finite displacements while symmetric probes record infinitesimal data at one point. Crossing these directions gives the four operators \(\Delta^+\), \(\Delta_+\), \(D^+\), and \(D_+\).
The two ladders below show where these operators sit. Their outer maps are the usual pullback of functions and pushforward of measures or distributions; their middle spaces give concrete coordinates for local probes. The smooth ladder uses Taylor coefficients and point-supported distributions, while the discrete ladder uses Möbius coordinates and signed cube measures. In each ladder, the covariant and contravariant operators are paired. The weighted collapse developed in the body runs between the ladders.
This arrangement is the roadmap for the paper. After displaying both ladders, the body constructs the exact discrete pair, passes from cubes to symmetric probes by weighted collapse, and then develops the smooth pair. The body works in affine spaces. Functoriality also lets the smooth operators glue over manifolds; the globalization exhibit Globalization: Bundles over Manifolds 🔗 records that extension and its relation to Pohl's higher-order tangent bundles.
The smooth ladder
Let \(X\) and \(Y\) be affine spaces modeled on finite-dimensional real vector spaces \(E\) and \(F\). A \(C^\infty\) map \(\phi:X\to Y\) with \(y=\phi(x)\) induces the diagram
Here \(\mathcal{E}(X)\) is the space of smooth functions, \(\mathcal{E}'(X)\) the distributions, and \(\mathcal{E}'(X;x)\) those supported at \(x\). The completion \(\hat{\mathcal{E}}(X;x)\) is the formal jet space at \(x\). The unlabeled arrows are the natural inclusion and completion maps.
The symmetric tangent space through order \(k\) is \(ST_k(X;x):=\SYM_{\leq k}(T_xX)\), and \(ST_*(X;x)=\SYM(T_xX)\) contains all orders. Its cotangent counterpart \(\STH^*(X;x)\) is the completed symmetric algebra of \(T_x^*X\); the two are related by the permanent pairing. The map \(\delta(x;-)\) realizes symmetric tangent probes as point-supported distributions, while \(t(x;-)\) is full Taylor expansion.
The outer vertical maps are the usual pushforward of distributions and pullback of functions and jets. These outer columns require finite dimension; the two middle operators make sense for affine Banach spaces. Full definitions appear in (27) 🔗, Covariant differentials 🔗, and Contravariant differentials 🔗.
In degree zero, \(D^+\) is unital and \(D_+\) is counital; this records \(y=\phi(x)\). The degree-one component of each operator is induced by \(D(\phi;x)\), and its functoriality is the ordinary chain rule. The higher-degree terms give the Faà di Bruno corrections.
Covariantly, \(D_+(\phi;x):ST_*(X;x)\to ST_*(Y;y)\) is the unique coalgebra lift of the total differential. On symmetric monomials,
For a second map \(\psi:Y\to Z\), we obtain the following "higher chain rule" identity:
Evaluating this identity on a monomial recovers Fraenkel's partition form of the Faà di Bruno formula [Fraenkel1978]. Since a partition of \(k\) elements has at most \(k\) blocks, \(D_+\) maps degree \(k\) to degrees at most \(k\) and therefore acts on \(ST_*\) without completion. Its degree filtration has associated graded \(\gr_rD_+(\phi;x)=\SYM^r(D(\phi;x))\).
In finite dimensions, \(D_+\) is also realized as the pushforward of point-supported distributions: \(\phi_*\delta(x;\xi)=\delta(y;D_+(\phi;x;\xi))\). This is the \(\delta\)-square in the ladder. Thus coordinate change for a point-supported differential operator of arbitrary order is computed by one application of \(D_+\).
Contravariantly, \(D^+\) is the transpose of \(D_+\) under the permanent pairing. For \(\ell_1,\dots,\ell_k\in F^*\), write \(\phi^{(i)}:=\ell_i\circ(\phi-y)\). On symmetric monomials,
where \(\nu!:=\prod_i\nu_i!\). Since \(\phi^{(i)}(x)=0\), any term with some \(\nu_i=0\) vanishes, and substitution is defined on every truncation and hence on the completion. Thus \(D^+\) is an algebra morphism given by Taylor composition [JR1979] [FM2014]. This is the middle adjunction square in the ladder, while the \(t\)-square identifies \(D^+\) with jet pullback.
The discrete ladder
The discrete operators have analogous covariant and contravariant diagrams and require no regularity assumptions. An arbitrary map \(\phi\) induces
Here \(F(X):=\Map(X,\IR)\) is the space of functions on \(X\), and \(F'(X)\) is the space of finitely supported signed measures. The tangent cubes of order \(k\) are \(CT_k(X;x):=\Map(\KP_+(k),E)\), where \(\KP_+(k)\) consists of the nonempty subsets of \([k]\); these are Möbius coordinates, and \(CT_*(X;x)\) collects all orders. The co-cubes \(CT^k(X;x):=\Map(CT_k(X;x),\IR)\) and their product \(\CTH^*(X;x)\) are paired with cubes by evaluation.
The map \(\delta(x;-)\) realizes a tangent cube as its signed vertex measure, while \(\Delta(-;x)\) sends a function to its cubical jet. The outer maps are the usual pushforward of measures and pullback of functions. \(\Delta_+(\phi;x)\) transports the vertices through \(\phi\) and returns the result to Möbius coordinates; \(\Delta^+(\phi;x)\) is precomposition with this map. Full definitions appear in Covariant differences 🔗 and Contravariant differences 🔗.
There is no discrete analogue of the two point-local columns in the smooth diagram. On the function side there is no completion: the vanishing ideal of a point is idempotent, \(I_x^2 = I_x\), so the jet quotients \(F(X)/I_x^m\) collapse to \(\IR\) and nothing sits between the functions and the co-cubes. On the measure side the point-supported measures are the scalar multiples of \(\delta(x)\) and add no nontrivial local data. A cube measure is supported on its vertices rather than concentrated at \(x\). The discrete ladder thus carries no point-local layer, and the discrete jet is a prolongation of a function to cubical probes, not a quotient of it. The remaining three squares commute for arbitrary maps: the \(\delta\)-square is the pushforward of cube measures, the pairing square is the discrete adjunction, and the jet square is the cubical jet pullback. The pushforward \(\Delta_+(\phi; x)\) is the Möbius conjugate of the vertexwise pushforward, functorial with no regularity assumptions, and its coordinate expansion is the covering form of the Faà di Bruno formula [DFDB],
with covers in place of partitions; the order-two case is computed in the exhibit An Order-Two Cube under Pushforward and Collapse 🔗. The cubical pullback \(\Delta^+(\phi;x)\) is contravariantly functorial, an algebra morphism for the pointwise product of co-cubes, and compatible with the cubical jet, \(\Delta(f \circ \phi;\, x) = \Delta^+(\phi;\, x)\, \Delta(f;\, y)\). The interface \(\Delta(-;\, x)\) itself is not multiplicative, and its defect is precisely the discrete product rule, indexed by 2-covers: where the Taylor expansion turns products of functions into products of series, the finite difference of a product mixes the faces of the cube.
The collapse
The smooth and discrete diagrams are connected column by column. On the function side the smooth functions embed in all functions, \(\mathcal{E}(X) \mono F(X)\); on the measure side the finitely supported measures embed in the distributions, \(F'(X) \mono \mathcal{E}'(X)\); and in the middle the symbol map \(\sigma_k: CT_k(X; x) \to ST_*(X; x)\) sends a cube to a symmetric tensor. Its contravariant companion identifies Taylor coefficients with leading coefficients of co-cubes. Both maps arise from the same weighted collapse. The rescaling \((\lambda_t c)(A) = t^{|A|} c(A)\) collapses a cube onto its basepoint. For \(C^k\) maps, terms indexed by covers of excess weight converge to zero, leaving the partition terms. At the same time, cube measures converge to point-supported distributions:
through the symbol map \(\sigma_k\), whose partition sum has the same form as \(D_+\). Pairing the rescaled cube with an observable \(f\) that is \(C^k\) near \(x\) gives
The left side is a finite difference and the right side is the action of a point-supported differential operator. Contravariantly, the same collapse extracts Taylor coefficients from cubical jets. It therefore intertwines the discrete operators \(\Delta_+,\Delta^+\) with the smooth operators \(D_+,D^+\).
Both pushforwards admit closed inverse formulas. The inverse of \(D_+(\phi;x)\) is a finite Neumann sum involving the inverse of the first derivative, while \(\Delta_+(\phi;x)\) is inverted vertexwise. These formulas reduce coordinate changes of point operators to one evaluation Laplace Operator in an Arbitrary Chart 🔗.
Related work
The ingredients of the calculus occur in several established theories. Kolář, Michor, and Slovák [KMS1993] treat natural operations, jets, and Weil functors, including the spaces underlying the smooth diagram. Johnson [Johnson2002] surveys the classical Faà di Bruno formula, while Frabetti and Manchon [FM2014] treat its Hopf-algebraic formulation [JR1979] [FGB2005]. These accounts describe composition abstractly or on formal power series. Here the same composition is represented by adjoint operators on symmetric tangent and cotangent spaces. Synthetic differential geometry [Kock2006] also uses infinitesimal probes, but takes nilpotent extensions as primitive rather than obtaining them from collapsing finite probes. McCullagh's tensor calculus of moments and cumulants [McCullagh1987] contains the same partition transform as the symbol map [LS1959] [Speed1983].
The difference-quotient calculus of Bertram, Glöckner, and Neeb [BGN2004] [Bertram2013] is likewise functorial and relates discrete quotients to derivatives. Our cubical model instead retains all faces of a probe, producing the covering formula and the covariant--contravariant adjunction. Difference forms and their de Rham limit use the alternating part of cubical data; this article uses the symmetric part.
Automatic differentiation [GriewankWalther2008] gives an algorithmic counterpart. Higher-order forward-mode AD transports truncated Taylor coefficients through composites, the covariant direction, and higher-order reverse-mode transports covector coefficient arrays, the contravariant direction [Betancourt2018] [Sangha2025]. In differential linear logic, Clift and Murfet [CM2020] construct the cofree-coalgebra lift of Kleisli morphisms with the closed partition formula, using [Murfet2015], Thm. 2.22. Their coalgebra lift is the formal-algebraic construction underlying \(D_+\). Our treatment identifies it with pushforward of point-supported distributions and places it beside its contravariant adjoint and the two exact cubical operators.
The references above provide the individual algebraic, distributional, combinatorial, and computational components, each establishing part of the picture. We believe the following contributions are new. First, the exact discrete calculus: the cubical pushforward and pullback in Möbius coordinates, functorial and adjoint for arbitrary maps, with the covering Faà di Bruno formula as coordinate expansion. Second, the weighted collapse: the cover-weight filtration under which the exact calculus converges to the smooth one, cube measures converging to point-supported distributions through the symbol map. Third, the closed partition formula for the pushforward of point-supported distributions in symmetric tangent coordinates, established analytically and adjoint to Taylor pullback; the higher chain and product rules are obtained as collapse limits of the corresponding exact identities rather than by induction on the order. Fourth, the finite Neumann formula for the inverse pushforward, which transports point operators through arbitrary coordinates in closed form. The four-functor calculus, its adjunctions, and the collapse connecting the two sectors combine these elements into one functorial picture.
Setting
Throughout, \(X, Y, Z\) are affine spaces over real Banach spaces \(E, F, G\); points are denoted \(x \in X\), \(y \in Y\), \(z \in Z\). Maps of affine spaces are \(\phi: X \to Y\) and \(\psi: Y \to Z\) with \(\phi(x) = y\), \(\psi(y) = z\). We say \(\phi\) is \(C^k\) near \(x\) if it is \(k\) times Fréchet differentiable with continuous derivatives on a neighborhood of \(x\) [LangRFA]. The derivative \(D^r(\phi; x; u_1, \dots, u_r)\) is evaluated on the directions \(u_i\) and is symmetric and continuous multilinear.
Notation. We prefer explicit arguments over subscript and bracket notation, writing \(\Delta(f; x; c)\) instead of \(\Delta_c f(x)\) and \(D(f; x; v_1, \dots, v_k)\) instead of \(D^k f(x)[v_1, \dots, v_k]\). Semicolons and commas are both argument separators; we use semicolons as a visual hint when arguments are of different kinds. Three conventions keep the formulas light. Tail currying: dropping trailing arguments denotes the resulting map, as in \(D_+(\phi; x) = D_+(\phi; x; -)\). Slot currying: a dash in an argument slot denotes the map in that slot, as in \(\Delta(-; x):f\mapsto \Delta(f; x)\). Overloading: one symbol may carry several definitions, selected by the type of its argument. The direction slot of \(D\) accepts a tuple or a symmetric tensor, so \(D(f; x; v_1,v_2)\) and \(D(f; x; v_1v_2)\) denote the same value. The argument determines the order of differentiation, so no order superscripts appear.
For \(k \geq 1\) write \([k] = \set{1, \dots, k}\). For a finite set \(S\), \(\KP(S)\) is its power set and \(\KP_+(S) = \KP(S) \setminus \set{\emptyset}\) is the set of nonempty subsets. The set \(\Part(S)\) consists of the unordered partitions of \(S\) into nonempty blocks; for example, \(\set{\set{1,3}, \set{2}} \in \Part([3])\). We abbreviate \(\KP(k):=\KP([k])\), \(\KP_+(k):=\KP_+([k])\), and \(\Part(k):=\Part([k])\).
Covariant differences
We first construct the discrete covariant calculus. Cubes are represented in Möbius coordinates, maps act vertexwise, and Möbius inversion gives the cubical pushforward. We establish functoriality, adjunction, the product rule, and the covering formula for arbitrary maps.
(1) Definition (Functions and measures). We use the following spaces, maps, and pairing.
- \(F(X) := \Map(X, \IR)\) is the space of functions on \(X\), with the topology of pointwise convergence.
- \(F'(X) := \mathcal{L}(F(X), \IR) \isom \Map_{\mathrm{fin}}(X, \IR)\) is its continuous dual, the finitely supported signed measures \(\mu = \sum_i \lambda_i \delta(z_i)\).
- A map \(\phi: X \to Y\) acts by pullback \(\phi^* f := f \circ \phi\) on functions and by pushforward \(\langle \phi_* \mu, f \rangle := \langle \mu, \phi^* f \rangle\) on measures.
- \(\langle \mu, f \rangle := \sum_i \lambda_i f(z_i)\) is the duality pairing between \(F'(X)\) and \(F(X)\).
(2) Definition (Cubes and co-cubes). For \(k \geq 1\) define:
- \(CT_k(X; x) := \Map(\KP_+(k), E)\), the tangent cubes of order \(k\); its coordinates are indexed by the nonempty subsets of \([k]\).
- \(CT^k(X; x) := \Map(CT_k(X; x), \IR)\) is the space of co-cubes of order \(k\).
- \(CT_*(X; x) := \bigsqcup_{k \geq 0} CT_k(X; x)\), where \(CT_0(X; x) := \set{0}\).
- \(CT^0(X; x) := \Map(CT_0(X; x), \IR) \isom \IR\).
- \(\CTH^*(X; x) := \Map(CT_*(X; x), \IR) = \prod_{k \geq 0} CT^k(X; x)\).
- \(\Cube_k(X; x) := \set{q: \KP(k) \to X,\; q(\emptyset) = x}\) is the space of geometric cubes of order \(k\) based at \(x\).
- The values \(c(\set{i})\) are the legs of \(c \in CT_k(X; x)\); the values \(c(A)\) with \(|A| \geq 2\) are its Möbius defects.
- The affine cube \(c = \Aff(v_1, \dots, v_k)\) is defined by \(c(\set{i}) = v_i\) and \(c(A) = 0\) for \(|A| \geq 2\); a cube is curved if it is not affine.
- For \(T \subseteq [k]\), the restriction \(\del_T c := c|_{\KP_+(T)}\) is the face on \(T\), regarded as a cube of order \(|T|\) by increasing relabeling.
- \(\langle \omega, c \rangle := \omega(c)\) is the evaluation pairing of \(CT^k(X; x)\) with \(CT_k(X; x)\), and of \(\CTH^*(X; x)\) with \(CT_*(X; x)\).
(3) Remark (Iterated tangent spaces). The tangent cubes are literally the fibers of iterated tangent spaces. If \(X\) is open in an affine Banach space with model space \(E\), then \(TX\isom X\times E\), and iteration gives \(T^kX\isom X\times\Map(\KP_+(k),E)\). Thus \(T(TX)\) has coordinates \((x,v_{\set{1}},v_{\set{2}},v_{\set{1,2}})\), and each further tangent step adds a direction for every existing coordinate. The original basepoint remains in \(X\), while all other coordinates range over \(E\); fixing that basepoint at \(x\) leaves precisely \(CT_k(X;x)\). This viewpoint is presumably classical; we learned it from Duarte and Torres [DT2008].
Cubes and forward differences
(4) Proposition (Boolean Möbius inversion). Let \(S\) be a finite set, \(G\) an abelian group, and \(a:\KP(S)\to G\). Define
Then \(\mu\circ\zeta=\zeta\circ\mu=\id\) on \(\Map(\KP(S),G)\).
Proof. The coefficient of \(a(R)\) in \((\mu\zeta a)(T)\) is \(\sum_{R\subseteq U\subseteq T}(-1)^{|T|-|U|} =(1-1)^{|T\setminus R|}=[R=T]\). The computation for \(\zeta\mu\) is the same.
(5) Corollary (Boolean sieve). For \(R\subseteq S\), \(\sum_{R\subseteq U\subseteq S}(-1)^{|S|-|U|}=[R=S]\).
(6) Definition (Zeta and Möbius transforms). The geometric realization and the tangent logarithm
are maps \(\zeta_x: CT_k(X; x) \to \Cube_k(X; x)\) and \(\mu_x: \Cube_k(X; x) \to CT_k(X; x)\).
(7) Lemma (Zeta-Möbius inversion). \(\zeta_x\) and \(\mu_x\) are inverse bijections.
Proof. Apply Boolean Möbius inversion (4) 🔗 to the function \(T \mapsto q(T) - x\) on \(\KP(k)\), which vanishes at \(T = \emptyset\) for based cubes.
(8) Definition (Forward difference). For \(k \geq 1\) define:
- \(\Delta(f; x; c) := \sum_{T \subseteq [k]} (-1)^{k - |T|}\, f(x+\sum_{\emptyset\neq R\subseteq T}c(R)) \in G\) is the forward difference of \(f:X\to G\) along \(c\in CT_k(X;x)\).
- For a finite family \((w_j)_{j\in J}\), \(\Delta(f;x;(w_j)_{j\in J}):= \sum_{L\subseteq J}(-1)^{|J|-|L|}f(x+\sum_{j\in L}w_j)\).
- \(\Delta(f; x; w_1, \dots, w_k) := \Delta(f; x; \Aff(w_1, \dots, w_k))\) is the difference along the affine cube with legs \(w_1,\dots,w_k\).
- On the unique order-zero cube \(0 \in CT_0(X; x)\), set \(\Delta(f; x; 0) := f(x)\).
- \(\Delta(\phi; x; c) \in F\) for \(\phi: X \to Y\) and \(k \geq 1\) is defined by the same alternating sum in the translation space of \(Y\).
(9) Remark. On affine cubes, \(\Delta(f; x; v_1, \dots, v_r) = (\Delta_{v_1} \circ \cdots \circ \Delta_{v_r} f)(x)\) is the classical iterated forward difference, where \(\Delta_v f := f(\cdot + v) - f(\cdot)\); the operators \(\Delta_v\) commute.
(10) Proposition (Discrete Taylor duality). For \(f: X \to G\) and \(c \in CT_k(X; x)\), the functions \(T \mapsto \Delta(f; x; \del_T c)\) and \(T \mapsto f(\zeta_x(c)(T))\) on \(\KP(k)\) are Möbius and zeta transforms of each other:
For affine cubes this is Newton's formula \(f(x + \sum_{i \in T} v_i) = \sum_{R \subseteq T} \Delta(f; x; v_R)\).
Proof. \(\Delta(f; x; \del_T c)\) is by definition the Möbius transform of the vertex function; invert with (4) 🔗.
Cubical pushforward
(11) Definition (Cubical pushforward, cf. [DT2008] Sec. 3). For a map \(\phi: X \to Y\) define:
- \((\phi_* q)(T) := \phi(q(T))\) is the geometric pushforward \(\phi_*: \Cube_k(X; x) \to \Cube_k(Y; y)\).
- \(\Delta_+(\phi; x) := \mu_y \circ \phi_* \circ \zeta_x\) is the cubical pushforward \(\Delta_+(\phi; x): CT_k(X; x) \to CT_k(Y; y)\).
(12) Lemma (Coordinates of the pushforward). For \(\emptyset \neq T \subseteq [k]\),
and on affine cubes \(\Delta_+(\phi;\, x;\, \Aff(v_1, \dots, v_k))(T) = \Delta(\phi;\, x;\, v_T)\), where \(v_T = (v_i)_{i \in T}\).
Proof. \(\mu_y(\phi_* \zeta_x c)(T) = \sum_{R \subseteq T} (-1)^{|T| - |R|} (\phi(\zeta_x(c)(R)) - y)\); the terms \(-y\) cancel since the coefficients sum to zero, and \(\zeta_x(c)(R)\) for \(R \subseteq T\) depends only on \(\del_T c\).
(13) Theorem (Exact functoriality). For arbitrary maps \(\phi: X \to Y\), \(\psi: Y \to Z\) and every \(k \geq 1\),
Proof. \((\psi \circ \phi)_* = \psi_* \circ \phi_*\) holds vertexwise. Insert \(\zeta_y \circ \mu_y = \id\) (7) 🔗 between the two factors in \(\mu_z \circ \psi_* \circ \phi_* \circ \zeta_x\).
Adjunctions
The cube measure records the forward difference as a pairing in \(F'(X)\) (1) 🔗.
(14) Definition (Cube measure). The cube measure of \(c \in CT_k(X; x)\) is
The forward difference is the pairing of a cube measure against an observable:
(15) Theorem (Discrete adjunction). For arbitrary \(\phi: X \to Y\), \(f: Y \to G\), and \(c \in CT_k(X; x)\),
Proof. By (7) 🔗, \(\zeta_y(\Delta_+(\phi; x)c) = \phi_*(\zeta_x c)\); the first identity follows vertexwise, the second by pairing against \(f\).
Let \(A\) be a Banach algebra and \(f, g: X \to A\). The pointwise product \(fg\) is again a function \(X \to A\).
(16) Theorem (Discrete product rule). For arbitrary \(f, g: X \to A\) and \(c \in CT_k(X; x)\),
the sum over all ordered pairs \((I, J)\) of subsets with \(I \cup J = [k]\), not necessarily disjoint.
Proof. By Taylor duality (10) 🔗, \(f(\zeta_x(c)(T)) = \sum_{I \subseteq T} \Delta(f; x; \del_I c)\) and similarly for \(g\). Multiply, insert into the alternating sum defining \(\Delta(fg; x; c)\), and exchange sums: the pair \((I, J)\) occurs for the vertices \(T \supseteq I \cup J\), with total coefficient \(\sum_{I \cup J \subseteq T \subseteq [k]} (-1)^{k - |T|} = [I \cup J = [k]]\) by the Boolean sieve (5) 🔗.
Affine reconstruction and Faà di Bruno
Let \(\Cov(k) := \set{\KC \subseteq \KP_+(k) : \bigcup \KC = [k]}\) be the set of covers of \([k]\), and set \(\mathrm{wt}(\KC) := \sum_{A \in \KC} |A|\).
With respect to the difference pairing, a curved cube expands as a sum of affine cubes indexed by covers.
(17) Proposition (Affine reconstruction). For \(f: X \to G\) and \(c \in CT_k(X; x)\),
Proof. Each vertex is a vertex of the affine cube on the index set \(\KP_+(k)\) with legs \((c(R))_{R \in \KP_+(k)}\): by Newton's formula (10) 🔗,
Insert this into the definition of \(\Delta(f; x; c)\) and exchange sums: a family \(\KH \subseteq \KP_+(k)\) occurs in the term of \(T\) exactly when \(\bigcup \KH \subseteq T\), so its total coefficient is \(\sum_{\bigcup \KH \subseteq T \subseteq [k]} (-1)^{k - |T|} = [\bigcup \KH = [k]]\) by the Boolean sieve (5) 🔗. Only covers survive.
(18) Corollary (Discrete Faà di Bruno, [DFDB]). For arbitrary maps \(X \xrightarrow{\phi} Y \xrightarrow{f} G\) and \(v_1, \dots, v_k \in E\),
Proof. Let \(c = \Aff(v_1, \dots, v_k)\). Combine the adjunction (15) 🔗, the coordinate formula (12) 🔗, and affine reconstruction (17) 🔗: \(\Delta(f \circ \phi; x; c) = \Delta(f; y; \Delta_+(\phi; x)c) = \sum_{\KC} \Delta(f; y; (\Delta(\phi; x; v_A))_{A \in \KC})\).
(19) Remark. The coefficient systems and iterated covering formulas are developed in [DFDB]. The order-two formula is computed in An Order-Two Cube under Pushforward and Collapse 🔗.
Contravariant differences
The cubical pushforward induces a pullback on co-cubes. Applied to cubical jets, this gives the contravariant chain rule.
(20) Definition (Cubical jet). For \(f: X \to G\) define:
- \(\Delta(f; x) := \Delta(f; x; -)\) is the cubical jet of \(f\) at \(x\).
- \(\langle \Delta(f; x), c \rangle := \Delta(f; x; c)\) for \(c \in CT_k(X; x)\); on \(CT_0(X; x)\) its value is \(f(x)\).
(21) Remark (Prolongation). By Taylor duality (10) 🔗, the cubical jet and the vertex values of \(f\) determine each other by Möbius inversion.
(22) Proposition (Prolongation, not quotient). Let \(F(X) = \Map(X, \IR)\) and let \(I_x = \set{f : f(x) = 0}\) be the vanishing ideal of \(x\).
1) \(I_x\) is idempotent, \(I_x^2 = I_x\): the filtration by powers of \(I_x\) is constant, and \(F(X)/I_x^m = \IR\) for every \(m \geq 1\).
2) The cubical jet is injective: the values of \(\Delta(f;\, x)\) on cubes of order \(\leq 1\) already determine \(f\), via \(f(x + v) = f(x) + \Delta(f;\, x;\, v)\).
Proof. 1) For \(f \in I_x\) let \(g \in I_x\) take the value \(1\) away from \(x\); then \(f = fg \in I_x^2\). 2) The order-zero value is \(f(x)\), and the order-one values are the increments \(f(x + v) - f(x)\).
(23) Definition (Cubical pullback). For an arbitrary map \(\phi: X \to Y\), the cubical pullback is precomposition with the cubical pushforward,
By construction, pushforward and pullback are adjoint under the evaluation pairing,
(24) Theorem (Cubical jet pullback). For arbitrary maps \(\phi: X \to Y\) and observables \(f: Y \to G\),
Proof. Evaluate on \(c\): by the discrete adjunction (15) 🔗, \(\langle \Delta^+(\phi;\, x)\, \Delta(f;\, y),\; c \rangle = \Delta(f;\, y;\, \Delta_+(\phi;\, x)\, c) = \Delta(f \circ \phi;\, x;\, c)\).
(25) Theorem (Contravariant functoriality). For arbitrary maps \(\phi: X \to Y\) and \(\psi: Y \to Z\),
(26) Proposition (The pullback is multiplicative). \(\Delta^+(\phi;\, x)\) is an algebra morphism for the pointwise product of co-cubes.
Proof. Precomposition with any map is multiplicative for pointwise products: \((\omega \eta) \circ \Delta_+(\phi; x) = (\omega \circ \Delta_+(\phi; x))\,(\eta \circ \Delta_+(\phi; x))\).
From differences to differentials
Weighted collapse sends cubical probes to symmetric probes. It intertwines the cubical and differential pushforwards and identifies collapsing cube measures with point-supported distributions. We first introduce the smooth row's spaces. The product, coproduct, and their adjunction are developed where they are used, in Covariant differentials 🔗, Contravariant differentials 🔗, and the appendix Appendix: The symmetric bialgebra 🔗.
(27) Definition (Symmetric tangent and cotangent spaces). For \(r, k \geq 0\) define:
- \(\SYM^r(E)\) is the \(r\)-th algebraic symmetric power of \(E\), with \(\SYM^0(E) := \IR \cdot 1\).
- \(ST_k(X; x) := \SYM_{\leq k}(T_x X) = \Vsum_{r=0}^{k} \SYM^r(T_x X)\) is the symmetric tangent space of order \(k\).
- \(ST_*(X; x) := \SYM(T_x X)\) is the symmetric tangent space of all orders.
- On the cotangent side, \(\mathcal L_s^r(E;\IR)\) is the Banach space of continuous symmetric \(r\)-linear forms \(E^r\to\IR\). Equivalently, it is the continuous dual of \(\SYM^r(E)\) equipped with the symmetric projective tensor norm.
- \(ST^k(X; x) := \Vsum_{r=0}^k \mathcal L_s^r(T_xX;\IR)\) is the symmetric cotangent space of order \(k\). Symmetrized tensor product, truncated above degree \(k\), makes it an algebra.
- \(\STH^*(X; x) := \prod_{r \geq 0} \mathcal L_s^r(T_xX;\IR)\) is its degree completion. Its product is defined degreewise, so every coefficient is a finite sum.
- For \(a\in\mathcal L_s^r(E;\IR)\) and \(\xi\in\SYM^r(E)\), set \(\langle a,\xi\rangle:=r!\,a(\xi)\), and pair distinct degrees by zero. This is the permanent pairing between \(ST^k\) and \(ST_k\), and between \(\STH^*\) and \(ST_*\). On decomposables, \(\langle \ell_1 \cdots \ell_r, v_1 \cdots v_r \rangle = \sum_{\tau \in S_r} \prod_i \ell_i(v_{\tau(i)})\). We write \(\langle -, - \rangle_E\) when needed.
Elements of \(ST_k\) are called probes; monomials are written \(v_1 \cdots v_r\).
(28) Definition (Smooth functions and distributions). Let \(X\) be finite-dimensional. Define:
- \(\mathcal{E}(X) := C^\infty(X, \IR)\) with its standard Fréchet topology.
- \(\mathcal{E}'(X) := \mathcal{E}(X)'\) is the space of compactly supported distributions.
- \(\mathcal{E}'(X; x)\) is the space of distributions supported at \(x\).
- \(\mathcal{E}'(X; x)_{\leq k}\) is its subspace of distributions of order at most \(k\).
- \(\phi^* f := f \circ \phi\) is the pullback of smooth functions.
- \(\langle \phi_* u, f \rangle := \langle u, \phi^* f \rangle\) is the pushforward of distributions.
(29) Lemma (Finitely supported measures are distributions). Let \(X\) be finite-dimensional. Restriction along \(\mathcal{E}(X) \subseteq F(X)\) embeds the finitely supported measures into the distributions, \(F'(X) \mono \mathcal{E}'(X)\), as the distributions of order \(0\) with finite support. In particular every cube measure \(\delta(x;\, c)\) of (14) 🔗 is a distribution.
Proof. Evaluation at a point is continuous on \(\mathcal{E}(X)\) and extends to \(C^0(X, \IR)\), so \(\mu = \sum_i \lambda_i\, \delta(z_i)\) defines a distribution of order \(0\) supported in the finite set \(\set{z_i}\); the map is injective since the \(\delta(z_i)\) are linearly independent as functionals on \(\mathcal{E}(X)\).
Weighted collapse
The passage to differential calculus stays inside the cubical world: the cube is collapsed onto its basepoint with the weight \(t^{|A|}\) on the coordinate \(c(A)\). This section provides the analytic lemma governing the collapse and the resulting partition formula.
(30) Lemma (Weight bound). For \(k\geq1\), every \(\KC \in \Cov(k)\) satisfies \(\mathrm{wt}(\KC) \geq k\), with equality if and only if \(\KC \in \Part(k)\).
Proof. \(\sum_{A \in \KC} |A| \geq |\bigcup \KC| = k\), with equality iff the blocks are pairwise disjoint.
(31) Lemma (Iterated fundamental theorem). Let \(\phi\) be \(C^r\) on an open set \(U \subseteq X\), and let \(x \in U\) and \(u_1, \dots, u_r \in E\) be such that the parallelotope \(x + [0,1] u_1 + \dots + [0,1] u_r\) lies in \(U\). Then
Proof. Set \(g(\theta) := \phi(x + \sum_i \theta_i u_i)\) on \([0,1]^r\). Since the parametrization is affine, \(g\) is \(C^r\) with \(\del_{\theta_1} \cdots \del_{\theta_r} g(\theta) = D(\phi; x + \sum_i \theta_i u_i; u_1, \dots, u_r)\). For a continuous Banach-space-valued function, the fundamental theorem of calculus in one variable and Fubini give, one variable at a time, \(\int_{[0,1]^r} \del_1 \cdots \del_r g \, d\theta = \sum_{T \subseteq [r]} (-1)^{r - |T|} g(1_T)\), the alternating sum over the corners of the cube, which is \(\Delta(\phi; x; u_\bullet)\).
(32) Corollary (Difference asymptotics). Let \(\phi\) be \(C^k\) near \(x\) and \(R > 0\).
1) For \(1 \leq r \leq k\), \(t_1, \dots, t_r \in \IR\), and \(\|u_j\| \leq R\),
where \(\sup_{\|u_j\| \leq R} \|\eta(t, u)\| \to 0\) as \(\max_j |t_j| \to 0\).
2) For \(r > k\) there are \(\rho > 0\) and \(K < \infty\) such that for all \(w_1, \dots, w_r \in E\) with \(\|w_j\| \leq \rho\) and every \(k\)-element subset \(M \subseteq [r]\),
Proof. 1) By (31) 🔗 with directions \(t_j u_j\) and multilinearity of the \(r\)-th derivative, the difference equals \(t_1 \cdots t_r \int D(\phi; x + \sum_j \theta_j t_j u_j; u_\bullet)\, d\theta\), and \(\|\eta\| \leq R^r \sup_{\xi} \|D^r(\phi; \xi) - D^r(\phi; x)\|\), the supremum over \(\xi\) in a ball that shrinks to \(x\); conclude by continuity of \(D^r(\phi; \cdot)\) at \(x\).
2) Since \(\Delta\) is symmetric in its directions we may take \(M = [k]\). The commuting operator factorization \(\Delta_{w_1} \cdots \Delta_{w_r} = (\prod_{j > k} \Delta_{w_j}) (\prod_{j \leq k} \Delta_{w_j})\) expands into
For \(\rho\) small enough, all basepoints \(x + w_T\) and their attached parallelotopes lie in a fixed ball \(B\) on which \(\phi\) is \(C^k\) with \(\sup_B \|D^k\phi\| =: M_k < \infty\); by (31) 🔗, each summand is bounded by \(M_k \prod_{j \leq k} \|w_j\|\), and there are \(2^{r - k}\) summands. Take \(K = 2^{r-k} M_k\).
(33) Definition (Weighted rescaling). For \(t \in \IR\) and \(c \in CT_k(X; x)\), set \((\lambda_t c)(A) := t^{|A|} c(A)\). Then \(\lambda_t \Aff(v_1, \dots, v_k) = \Aff(t v_1, \dots, t v_k)\) and \(\del_T \lambda_t = \lambda_t \del_T\).
(34) Theorem (Collapse and partition formula). Let \(k\geq1\) and let \(\phi\) be \(C^k\) near \(x\). For every \(c\in CT_k(X;x)\), the limit
exists uniformly as \(c\) ranges over bounded subsets of \(CT_k(X;x)\), and
The right side is a continuous function of \(c\), and on affine cubes \(D^\square(\phi;x;\Aff(v_1,\dots,v_k))=D(\phi;x;v_1,\dots,v_k)\).
Proof. Fix \(R\) with \(\|c\| \leq R\). For small \(t\) all vertices of \(\lambda_t c\) lie in the domain of \(\phi\). By (12) 🔗 and affine reconstruction (17) 🔗,
Consider a cover \(\KC\) with \(r := |\KC|\) blocks. If \(r \leq k\), part 1 of (32) 🔗 with \(t_A := t^{|A|}\), \(u_A := c(A)\) gives
with \(\eta_\KC(t) \to 0\) uniformly for \(\|c\| \leq R\). If \(r > k\), then \(\KC\) contains a block of size \(\geq 2\): the blocks are distinct nonempty subsets of \([k]\) and there are only \(k<r\) singletons. Choose a \(k\)-element subset \(M \subseteq \KC\) containing such a block; then \(\sum_{A \in M} |A| \geq k + 1\), and part 2 of (32) 🔗 bounds the term by \(K R^k t^{k+1}\).
Divide by \(t^k\). Covers with \(r > k\) are \(O(t)\). Covers with \(r \leq k\) carry \(t^{\mathrm{wt}(\KC)-k}\) with \(\mathrm{wt}(\KC) \geq k\) (30) 🔗; those of weight \(>k\) vanish in the limit with bounded cofactor, and the partitions, exactly the covers of weight \(k\) (30) 🔗, converge to their leading term. All estimates are uniform for \(\|c\| \leq R\). Each summand is a bounded multilinear expression in \(c\), whence continuity. For affine cubes only the partition into singletons contributes.
(35) Definition (Cubical differential). For \(\phi \in C^k\) near \(x\), the cubical differential pushforward is the map \(D^\square_+(\phi;x):CT_k(X;x)\to CT_k(Y;y)\) defined by
Its top coordinate is the cubical differential of (34) 🔗. For observables, set \(D^\square(f;x;\del_\emptyset c):=f(x)\).
(36) Theorem (Differential functoriality). For \(\phi\) \(C^k\) near \(x\) and \(\psi\) \(C^k\) near \(y\),
Proof. It suffices to prove the top coordinate; applying it to each face gives the remaining coordinates. Fix \(c\in CT_k(X;x)\) and set \(c_t(A):=t^{-|A|}\Delta_+(\phi;x;\lambda_t c)(A)\). Then \(\Delta_+(\phi;x;\lambda_t c)=\lambda_t c_t\), and (34) 🔗 applied to each face gives \(c_t\to c':=D^\square_+(\phi;x)c\). In particular, \((c_t)\) is bounded. Exact functoriality (13) 🔗 gives
By the uniformity in (34) 🔗 applied to \(\psi\) at \(y\), the right side differs from \(\sum_{\pi \in \Part(k)} D(\psi;y;(c_t(A))_{A\in\pi})\) by a quantity tending to \(0\), and by continuity of the partition expression this sum converges to \(D^\square(\psi;y;c')\).
The symbol map
The cubical differential depends on the cube only through a symmetric-algebra element. The symbol map makes this exact and transfers the calculus to \(\SYM_{\leq k}\).
(37) Definition (Symbol map). The symbol map is
where the products are taken in the symmetric algebra. On affine cubes, \(\sigma_k(\Aff(v_1,\dots,v_k))=v_1\cdots v_k\).
(38) Lemma (Spanning). For every \(1 \leq r \leq k\) and \(v_1, \dots, v_r \in E\), the product \(v_1 \cdots v_r\) lies in the image of \(\sigma_k\). Consequently the image of \(\sigma_k\), together with \(1\), spans \(ST_k(X; x)\).
Proof. Choose a partition \([k] = B_1 \sqcup \dots \sqcup B_r\) into nonempty blocks and let \(c(B_j) := v_j\) and \(c(A) := 0\) for all other \(A\). A partition \(\pi \in \Part(k)\) contributes to \(\sigma_k(c)\) only if all its blocks lie in \(\set{B_1, \dots, B_r}\), which forces \(\pi = \set{B_1, \dots, B_r}\); hence \(\sigma_k(c) = v_1 \cdots v_r\).
(39) Proposition (Symbol factorization). For \(f \in C^k(X, G)\) near \(x\) and \(c \in CT_k(X; x)\),
Proof. The top-coordinate partition formula (34) 🔗 reads \(D^\square(f; x; c) = \sum_\pi D(f; x; (c(A))_{A \in \pi})\), which is the extended pairing evaluated at \(\sigma_k(c)\).
(40) Theorem (Symbol intertwining). For \(\phi\) \(C^k\) near \(x\),
Proof. By the partition formula (34) 🔗,
Expanding the product, the index data is a pair \((\pi, (\rho_A)_{A \in \pi})\) with \(\rho_A \in \Part(A)\). Such pairs correspond bijectively to pairs \((\rho, \KQ)\) of a partition \(\rho \in \Part(k)\) and a partition \(\KQ\) of the block set of \(\rho\): put \(\rho := \bigsqcup_{A \in \pi} \rho_A\) and \(\KQ := \set{\rho_A : A \in \pi}\); conversely \(\pi = \set{\bigcup Q : Q \in \KQ}\). Under this bijection the term of \((\pi, (\rho_A))\) equals \(\prod_{Q \in \KQ} D(\phi; x; (c(B))_{B \in Q})\). On the other side, by (52) 🔗 applied to the product of the \(|\rho|\) vectors \((c(B))_{B \in \rho}\),
The two sums agree term by term.
(41) Proposition (Contravariant collapse). Let \(f\) be \(C^k\) near \(x\) and \(c \in CT_k(X; x)\). Then
the top Möbius coordinate of the cubical jet converges to its differential pairing.
(42) Remark. In the language of the smooth jets of Contravariant differentials 🔗, the right side is the pairing \(\langle t(x;\, f),\, \sigma_k(c) \rangle\) (79) 🔗: the Taylor coefficients of \(f\) are the leading asymptotics of its cubical jet.
Higher symbols
For a fixed cube \(c\in CT_k(X;x)\) and observable \(f\), consider the collapse function
The symbol \(\sigma_k(c)\) gives its leading Taylor coefficient: by (34) 🔗 and (39) 🔗, the function vanishes to order \(k\) and begins with \(t^kD(f;x;\sigma_k(c))\). The same cover calculus computes every Taylor coefficient before pairing with \(f\).
(43) Definition (Higher symbols). For a cover \(\KC\) and multiplicities \(\nu:\KC\to\IN_{>0}\), set \(\mathrm{wt}(\nu):=\sum_{A\in\KC}|A|\nu_A\) and \(\nu!:=\prod_{A\in\KC}\nu_A!\). For \(m\geq0\), define
Thus \(\nu_A\) records how many times the direction \(c(A)\) occurs, and the weight \(\mathrm{wt}(\nu)\) records its power of \(t\). Minimal weight forces \(\sigma_{k,0}=\sigma_k\): only partitions with every multiplicity equal to one contribute.
(44) Theorem (Curved collapse). Let \(M\geq0\) and let \(f\) be \(C^{k+M}\) near \(x\). There is a residual \(R_{k,M}(f;x;c,t)\) such that
For every bounded \(B\subseteq CT_k(X;x)\), the residual satisfies \(\sup_{c\in B}\|R_{k,M}(f;x;c,t)\|\to0\) as \(t\to0\).
Proof. Affine reconstruction (17) 🔗 writes \(\Delta(f;x;\lambda_t c)\) exactly as the sum, over \(\KC\in\Cov(k)\), of the affine differences \(\Delta(f;x;(t^{|A|}c(A))_{A\in\KC})\). For a fixed cover \(\KC\), expand this difference as its alternating sum over \(J\subseteq\KC\) and apply Taylor's formula at \(x\), with Peano remainder, to \(f(x+\sum_{A\in J}t^{|A|}c(A))\).
In the Taylor polynomial, the multinomial indexed by \(\nu:\KC\to\IN_0\) has coefficient \(1/\nu!\) and contributes \(t^{\mathrm{wt}(\nu)}D(f;x;\prod_{A\in\KC}c(A)^{\nu_A})\). Its coefficient in the alternating sum is \(\sum_{\operatorname{supp}\nu\subseteq J\subseteq\KC} (-1)^{|\KC|-|J|}=[\operatorname{supp}\nu=\KC]\) by the Boolean sieve (5) 🔗. Hence precisely the multiplicities \(\nu_A>0\) survive.
Since \(\KC\) covers \([k]\), the weight \(\mathrm{wt}(\nu)\) is at least \(k\). Grouping the surviving terms of weights \(k,k+1,\dots,k+M\) gives \(\sigma_{k,0},\dots,\sigma_{k,M}\). The remaining terms in the finite Taylor polynomial are \(O(t^{k+M+1})\), and the Peano remainders are \(o(t^{k+M})\), uniformly for \(c\) in bounded sets. Dividing their sum by \(t^{k+M}\) defines \(R_{k,M}\) and proves the claim.
(45) Corollary (Affine collapse). Let \(n\geq k\), let \(f\) be \(C^n\) near \(x\), and \(v_1,\dots,v_k\in E\). With the multi-index notation \(v^\nu:=v_1^{\nu_1}\cdots v_k^{\nu_k}\), \(\nu!:=\nu_1!\cdots \nu_k!\), and \(\mathrm{wt}(\nu)=\nu_1+\dots+\nu_k\), the higher symbols of the affine cube specialize to
and the collapse expansion reads
For \(k=1\) this is the Taylor formula. In general, the constraint \(\nu_i>0\) keeps exactly the Taylor monomials that depend on all \(k\) directions: the iterated difference annihilates every other term.
Proof. All nonsingleton coordinates of \(\Aff(v_1,\dots,v_k)\) vanish, so in (43) 🔗 only the cover by singletons contributes, and the multiplicity function becomes a multi-index \(\nu\in\IN_0^k\) with \(\nu_i>0\) and weight \(\mathrm{wt}(\nu)=\nu_1+\dots+\nu_k\). Substitute into (44) 🔗 with \(M=n-k\) and group by \(\mathrm{wt}(\nu)=k+m\).
(46) Remark (Möbius inversion and the Taylor formula). The affine collapse is the Möbius transform of the multivariate Taylor formula: vertex values are the sums of the face differences (4) 🔗. Applying this once to (45) 🔗 merges a face \(T\) and a multi-index \(\nu\in\IN_{>0}^T\) into an arbitrary \(\nu\in\IN_0^S\) with \(T=\operatorname{supp}\nu\), and gives
the multivariate Taylor formula with Peano remainder.
Distributional collapse
Recall the distributions \(\mathcal{E}'(X)\) and the point-supported distributions \(\mathcal{E}'(X; x)\) with their order filtration from (28) 🔗. By (29) 🔗, every cube measure is a distribution.
(47) Theorem (Structure of point-supported distributions, [Hormander1983], Theorem 2.3.4). The map \(\xi \mapsto \delta(x; \xi)\) defined by
is a linear isomorphism \(ST_k(X; x) \xrightarrow{\;\sim\;} \mathcal{E}'(X; x)_{\leq k}\) between the symmetric tangent space and the distributions of order \(\leq k\) supported in \(\set{x}\). In particular \(\delta(x; 1) = \delta(x)\) is evaluation at \(x\).
The convergence in the collapse theorem can now be stated distributionally.
(48) Corollary (Distributional collapse). For \(c \in CT_k(X; x)\), the rescaled cube measures converge weakly in \(\mathcal{E}'(X)\), that is, pointwise on \(\mathcal{E}(X)\):
The left side is a finitely supported measure; the right side is a point distribution in \(\mathcal{E}'(X; x)\).
Proof. For every \(f \in \mathcal{E}(X)\), \(t^{-k} \langle \delta(x; \lambda_t c), f \rangle = t^{-k} \Delta(f; x; \lambda_t c) \to D(f; x; \sigma_k(c)) = \langle \delta(x; \sigma_k(c)), f \rangle\) by (34) 🔗 and (39) 🔗.
(49) Remark. Discretely, probes are the finitely supported measures \(\delta(x; c)\) and \(\Delta_+\) is their pushforward (15) 🔗; after collapse, probes are the point distributions \(\delta(x; \xi)\) and \(D_+\) is their pushforward (63) 🔗. The symbol map records the collapse of the former onto the latter. Pairing with observables gives the dual statement: Newton's exact finite expansion converges degreewise to the corresponding Taylor jet. For affine cubes, the higher symbols identify the repeated-direction Taylor coefficients (45) 🔗.
Covariant differentials
The differential \(D(\phi; x)\) extends uniquely to a coalgebra morphism \(D_+(\phi; x)\) on symmetric probes. We establish its functoriality, adjunction, product rule, and realization as pushforward of point-supported distributions.
Symmetric probes and pushforward
The symmetric tangent space \(ST_k(X; x)\) (27) 🔗 carries, besides the truncated symmetric product \(\xi \cdot \eta := \pi_{\leq k}(\xi \eta)\), a coproduct.
(50) Definition (Coproduct). The coproduct on \(ST_k(X; x)\) is the linear map \(\Delta^{\times}: ST_k(X; x) \to ST_k(X; x) \tensor ST_k(X; x)\) with \(\Delta^{\times}(1) = 1 \tensor 1\) and
where \(v_I := \prod_{i \in I} v_i\); the right side is symmetric and multilinear, so \(\Delta^{\times}\) is well defined.
The truncated product and the coproduct are paired structures on \(ST_k(X; x)\): truncation makes it an algebra quotient and the degree filtration makes it a subcoalgebra. We do not regard the truncated space as a bialgebra. The untruncated bialgebra and the coalgebra facts used below are collected in the appendix Appendix: The symmetric bialgebra 🔗.
(51) Definition (Differential). Let \(\phi: X \to Y\) be \(C^k\) near \(x\).
- The differential of \(\phi\) at \(x\) is the linear map \(D(\phi; x): ST_k(X; x) \to T_y Y\) defined on monomials by the Fréchet derivatives,
and \(D(\phi; x; 1) := 0\). It collects the derivatives of all orders into a single linear map, landing in degree one on the target. - For an observable \(f: X \to G\) the same formula defines the differential pairing \(D(f; x): ST_k(X; x) \to G\), with the degree-zero convention \(D(f; x; s \cdot 1) := s\, f(x)\).
(52) Definition (Symmetric pushforward). For \(\phi\) \(C^k\) near \(x\), the symmetric pushforward \(D_+(\phi; x): ST_k(X; x) \to ST_k(Y; y)\) is the linear map with \(D_+(\phi; x; 1) := 1\) and
The right side is symmetric and multilinear in \((v_1, \dots, v_r)\), so it defines a linear map on \(\SYM^r(E)\).
(53) Proposition (Coalgebra property). \(D_+(\phi; x)\) is a coalgebra morphism:
Proof. Evaluate on \(v_1\cdots v_r\) and set \(w_A:=D(\phi;x;v_A)\). Applying \(\Delta^{\times}\) to each partition term splits its block set:
The data \((\pi,\pi_1,\pi_2)\) correspond bijectively to \((I\sqcup J=[r],\pi_1\in\Part(I),\pi_2\in\Part(J))\), with \(I=\bigcup\pi_1\) and \(J=\bigcup\pi_2\). The resulting sum is \(\sum_{I\sqcup J}D_+(v_I)\tensor D_+(v_J) =(D_+\tensor D_+)\Delta^{\times}(v_1\cdots v_r)\).
(54) Corollary (Coalgebra lift). \(D_+(\phi;x)\) is the unique coalgebra morphism \(ST_k(X;x)\to ST_k(Y;y)\) whose degree-one component is the differential \(D(\phi;x)\).
Proof. The coalgebra property is (53) 🔗. Uniqueness follows from the truncated coalgebra-lift corollary (102) 🔗, applied to \(D(\phi;x)\); its partition expansion is (52) 🔗.
Functoriality, adjunction, and Faà di Bruno
(55) Corollary (Smooth functoriality). For \(\phi\) \(C^k\) near \(x\) and \(\psi\) \(C^k\) near \(y\),
Proof. The identity is transported across the bridge, where the cubical differential \(D^\square_+\) carries the functoriality. On \(\sigma_k(c)\), combine (40) 🔗 with (36) 🔗: \(D_+(\psi \circ \phi; x)\,\sigma_k(c) = \sigma_k(D^\square_+(\psi \circ \phi; x)c) = \sigma_k(D^\square_+(\psi; y)\, D^\square_+(\phi; x)c) = D_+(\psi; y)\, D_+(\phi; x)\, \sigma_k(c)\). Both sides are linear, fix \(1\), and the image of \(\sigma_k\) together with \(1\) spans (38) 🔗.
(56) Theorem (Smooth adjunction). For \(\phi\) \(C^k\) near \(x\), \(f \in C^k(Y, G)\) near \(y\), and \(\xi \in ST_k(X; x)\),
Proof. Both sides are linear in \(\xi\) and agree at \(\xi = 1\). For \(\xi = \sigma_k(c)\) the computation crosses back to the bridge, routing through the cubical differential \(D^\square\): apply symbol factorization to \(f \circ \phi\), differential functoriality with \(\psi = f\) (a \(C^k\) map into the Banach space \(G\)), symbol factorization for \(f\) at \(y\), and the intertwining:
Conclude by (38) 🔗.
(57) Corollary (Partition Faà di Bruno, [Fraenkel1978] [CS1996]). For \(\phi\), \(f\) as above and \(v_1, \dots, v_k \in E\),
Proof. Apply smooth functoriality (55) 🔗 to \(\psi = f\), a \(C^k\) map into the Banach space \(G\), evaluate at \(v_1 \cdots v_k\), and compare degree-one components. By (52) 🔗, the degree-one component of \(D_+(f;\, y;\, \eta)\) is \(D(f;\, y;\, \eta)\) for \(\eta\) of pure positive degree: the left side contributes \(D(f \circ \phi;\, x;\, v_1 \cdots v_k)\), the right side the partition sum.
(58) Remark (Proof dependencies). The proof of the partition formula uses the discrete Faà di Bruno identity (18) 🔗, the iterated fundamental theorem (31) 🔗, and the continuity statement in the collapse theorem (34) 🔗. It does not invoke a prior higher chain rule. Alternatively, the smooth pushforward and its functoriality follow from the coalgebra-lift uniqueness (102) 🔗 together with the classical Faà di Bruno formula.
Product rule
(59) Proposition (Algebra behavior of the pushforward). Let \(\phi\) be \(C^k\) near \(x\). \(D_+(\phi; x)\) is an algebra morphism for the truncated symmetric product if and only if \(D^r(\phi; x)=0\) for \(2\leq r\leq k\); in that case \(D_+(\phi; x)=\SYM_{\leq k}(D(\phi; x))\).
Proof. If \(D^r(\phi; x) = 0\) for \(2 \leq r \leq k\), only the partition into singletons survives in (52) 🔗, and \(D_+(v_1 \cdots v_r) = \prod_i D(\phi; x; v_i)\) is multiplicative. Conversely, suppose \(D_+\) is multiplicative and induct on \(r\): comparing \(D_+(v_1 v_2) = D(\phi; x; v_1) \cdot D(\phi; x; v_2) + D(\phi; x; v_1, v_2)\) with \(D_+(v_1)D_+(v_2)\) forces \(D^2(\phi; x) = 0\) by polarization; for the step, all partitions with a block of size \(2, \dots, r-1\) drop out by the induction hypothesis, and comparing with \(D_+(v_1) \cdots D_+(v_r)\) leaves \(D(\phi; x; v_1, \dots, v_r) = 0\).
(60) Theorem (Smooth product rule). For \(f, g \in C^k(X, A)\) near \(x\) and \(v_1, \dots, v_k \in E\), the higher Leibniz rule holds:
Proof. Apply the discrete product rule (16) 🔗 to the affine cube \(\Aff(t v_1, \dots, t v_k)\) and divide by \(t^k\). By the difference asymptotics (32) 🔗, \(\Delta(f;\, x;\, t v_I) = t^{|I|}(D(f;\, x;\, v_I) + o(1))\) and likewise for \(g\), including the cases \(I = \emptyset\) with value \(f(x)\); the left side is \(t^{-k}\, \Delta(fg;\, x;\, t v_1, \dots, t v_k) \to D(fg;\, x;\, v_1 \cdots v_k)\). A pair with \(I \cap J \neq \emptyset\) carries \(t^{|I| + |J|} = t^{k + |I \cap J|}\) and vanishes after division; the disjoint pairs converge to the product of the limits, multiplication being continuous on \(A\).
Distributions
(61) Definition (Point distributions). For \(\xi \in ST_k(X; x)\), the point distribution \(\delta(x; \xi)\) is the linear functional
so \(\delta(x; 1) = \delta(x)\) is evaluation at \(x\). For \(\phi\) \(C^k\) near \(x\), the pushforward of such a functional is defined by \(\langle \phi_* u, f \rangle := \langle u, \phi^* f \rangle\); this is well defined since \(\phi^* f\) is again \(C^k\) near \(x\) [FDB]. The same formulas apply to vector-valued test functions \(f \in C^k(X, G)\).
(62) Remark (Sign-free basis). The basis \(\delta(x; \xi)\) is sign-free (47) 🔗: it represents the operator datum \(\xi\) directly rather than through \(\del^\alpha \delta(x) = (-1)^{|\alpha|} \delta(x; e^\alpha)\)-type conventions; in one direction, \(\del_v \delta(x) = -\delta(x; v)\).
(63) Theorem (Distribution pushforward). For \(\phi\) \(C^k\) near \(x\) and \(\xi \in ST_k(X; x)\),
Proof. \(\langle \phi_* \delta(x; \xi), f \rangle = D(\phi^* f; x; \xi) = D(f; y; D_+(\phi; x)\xi) = \langle \delta(y; D_+(\phi; x)\xi), f \rangle\) by the smooth adjunction (56) 🔗.
(64) Proposition (Filtration and principal symbol). \(D_+(\phi; x)\) preserves the degree filtration \(ST_r(X; x) \subseteq ST_k(X; x)\), and on the associated graded it is the multiplicative extension of the first derivative:
Proof. A partition \(\pi \in \Part(r)\) contributes in degree \(|\pi| \leq r\), with equality only for the partition into singletons, whose term is \(\prod_i D(\phi; x; v_i)\).
Contravariant differentials
The differential pullback acts on completed symmetric cotangent spaces. We define it degreewise as the adjoint of \(D_+\), then identify its action on linear forms with Taylor expansion and prove the multiplicative extension.
The symmetric cotangent space and its pairing
The completed symmetric cotangent space \(\STH^*(X;x)=\prod_{r\geq0}\mathcal L_s^r(E;\IR)\) and the permanent pairing \(\langle a,\xi\rangle=r!\,a(\xi)\) are defined in (27) 🔗. Its symmetric product is given on homogeneous forms by
(65) Remark (Polynomial evaluation). The factorial normalization makes the permanent pairing agree with ordinary polynomial evaluation. If \(a=\sum_{r=0}^k a_r\) represents \(p_a(x+v):=\sum_{r=0}^k a_r(v,\dots,v)\), then, with \(\exp(v):=\sum_{r\geq0}v^r/r!\), \(p_a(x+v)=\langle a,\exp(v)\rangle\). Thus evaluation at \(x+v\) is represented on the symmetric tangent side by the exponential probe \(\exp(v)\).
Products of linear forms span \(\mathcal L_s^r(E;\IR)\) in finite dimensions. In infinite dimensions they form the finite-type subspace, which may be proper. The definitions below therefore use continuous symmetric forms directly.
(66) Lemma (Separation). The pairing separates forms: if \(\alpha\in\STH^*(X;x)\) pairs to zero with every \(\xi\in ST_*(X;x)\), then \(\alpha=0\).
Proof. Degreewise, \(\langle \alpha_r,\, v^r \rangle = r!\, \alpha_r(v, \dots, v)\) for \(v \in E\), and a symmetric multilinear form vanishing on the diagonal vanishes, by polarization.
(67) Lemma (Product and coproduct are adjoint). For \(\alpha,\beta\in\STH^*(X;x)\) and \(\xi\in ST_*(X;x)\),
Proof. Bilinearity reduces to \(\alpha \in \mathcal L_s^r(E;\IR)\), \(\beta \in \mathcal L_s^s(E;\IR)\), \(\xi = v_1 \cdots v_n\) with \(n = r + s\). In the defining sum of the symmetric product, the permutations \(\tau\) carrying \(\{\tau(1), \dots, \tau(r)\}\) to a fixed \(r\)-set \(I\) number \(r!\, s!\), so \(\langle \alpha \cdot \beta, v_1 \cdots v_n \rangle = n! \, (\alpha \cdot \beta)(v_\bullet) = r!\, s! \sum_{|I| = r} \alpha(v_I)\, \beta(v_J) = \sum_{I \sqcup J = [n]} \langle \alpha, v_I \rangle \langle \beta, v_J \rangle\), which is the right side by (50) 🔗.
Differential pullback
(68) Definition (Pullback). Let \(\phi:X\to Y\) be smooth near \(x\), with \(\phi(x)=y\), and let \(\alpha=(\alpha_q)_{q\geq0}\in\STH^*(Y;y)\). Define \(D^+(\phi;x)\alpha\in\STH^*(X;x)\) degreewise by
where the degree-zero monomial is \(1\). The right side is a continuous symmetric \(r\)-linear form. Moreover, it only uses \(\alpha_0,\dots,\alpha_r\), because \(D_+\) does not increase degree. Thus the formula defines a map on the degree completion and commutes with every truncation. Hence \(D^+(\phi;x):\STH^*(Y;y)\to\STH^*(X;x)\) is well defined.
(69) Proposition (Pushforward-pullback adjunction). For \(\alpha\in\STH^*(Y;y)\) and \(\xi\in ST_*(X;x)\),
Proof. For \(\xi=v_1\cdots v_r\), the permanent pairing and (68) 🔗 give \(\langle D^+(\phi;x)\alpha,v_1\cdots v_r\rangle_E =r!\,(D^+(\phi;x)\alpha)_r(v_1,\dots,v_r)\); the factor \(r!\) cancels the \(1/r!\) in the definition, leaving \(\langle\alpha,D_+(\phi;x;v_1\cdots v_r)\rangle_F\). The result follows for every \(\xi\in ST_*(X;x)\) by linearity.
(70) Proposition (The pullback is an algebra morphism). \(D^+(\phi;x)\) is a unital algebra morphism on the completed symmetric cotangent algebra.
Proof. Pair with \(\xi\in ST_*(X;x)\). By the product-coproduct adjunction (67) 🔗, the defining adjunction, and the coalgebra property of \(D_+\) (53) 🔗,
Separation (66) 🔗 gives multiplicativity. The same argument with \(1\) gives unitality.
(71) Corollary (Taylor pullback on generators). For \(\ell\in F^*\), let \(\ell_y(z):=\ell(z-y)\). Then
the full Taylor expansion at \(x\) of the centered observable \(\ell_y\circ\phi\). Consequently,
Proof. In degree \(r\), pairing with \(\ell\) selects the degree-one part of \(D_+(\phi;x;v_1\cdots v_r)\), namely the term indexed by the one-block partition. This gives the Taylor coefficient \(\frac1{r!}\ell\circ D^r(\phi;x)\). The product formula follows from (70) 🔗.
(72) Remark (Completion). In finite dimensions, \(\STH^*(Y;y)\) is the degree completion of the free symmetric algebra on \(F^*\). Each \(D^+(\phi;x;\ell)\) has zero constant term, so a product of \(q\) generators begins in degree \(q\). The generator formula therefore induces compatible maps on every degree truncation and extends uniquely to a degree-continuous algebra morphism on \(\STH^*(Y;y)\). In infinite dimensions, products of linear forms need not exhaust the continuous symmetric forms; the degreewise definition (68) 🔗 defines \(D^+\) on the full completion.
Functoriality
(73) Theorem (Contravariant functoriality).
Proof. For every \(\xi\), by the adjunction (69) 🔗 and smooth functoriality (55) 🔗,
conclude by separation (66) 🔗.
(74) Remark (Bialgebra structure). The pullback \(D^+\) is an algebra morphism and the pushforward \(D_+\) is a coalgebra morphism. The failure of the opposite compatibilities is measured by the higher derivatives of \(\phi\).
Jets and the Taylor realization
Jet pullback is defined formally on quotients by powers of the maximal ideal. The Taylor map realizes these quotients on symmetric cotangent spaces and identifies the formal pullback with \(D^+\). Throughout this subsection \(X\) is finite-dimensional.
(75) Definition (Maximal ideal, jets, completion).
- \(\mathfrak{m}_x := \set{f \in \mathcal{E}(X) : f(x) = 0}\) is the maximal ideal of \(x\).
- The \(k\)-jet space at \(x\) is the quotient algebra \(\mathcal{E}_k(X; x) := \mathcal{E}(X) / \mathfrak{m}_x^{k+1}\); the \(k\)-jet of \(f \in \mathcal{E}(X)\) is its class \(\hat{f} \in \mathcal{E}_k(X; x)\), classically written \(j^k_x f\). The stage is carried by the space and inferred from context.
- \(\hat{\mathcal{E}}(X; x) := \varprojlim_k\, \mathcal{E}_k(X; x)\) is the completion of the functions at \(x\), with the hat map \(f \mapsto \hat{f}\) and the projective limit topology.
- \(\langle \hat{f}, u \rangle := u(f)\), where \(f \in \mathcal{E}(X)\) is any lift of \(\hat{f}\), defines a bilinear pairing between \(\hat{\mathcal{E}}(X; x)\) and \(\mathcal{E}'(X; x)\) (28) 🔗.
(76) Proposition (Duality of completed functions and point distributions). The pairing \(\langle \hat{f}, u \rangle = u(f)\) is well-defined and induces an isomorphism \(\mathcal{E}'(X; x) \xrightarrow{\sim} \hat{\mathcal{E}}(X; x)'\).
Proof. Well-definedness: a point-supported distribution has finite order, and if \(u\) has order \(\leq k\) it vanishes on \(\mathfrak{m}_x^{k+1}\): such functions are flat to order \(k\) at \(x\) by the Leibniz rule, and flat functions are \(C^k\)-approximable by functions vanishing near \(x\) [Hormander1983], Theorem 2.3.3; so \(u\) factors through the quotient \(\mathcal{E}(X) / \mathfrak{m}_x^{k+1}\). Injectivity: if \(u \neq 0\) then \(u(f) \neq 0\) for some \(f\). Surjectivity: a continuous functional on \(\hat{\mathcal{E}}(X; x) = \varprojlim_k \mathcal{E}(X) / \mathfrak{m}_x^{k+1}\) factors through some finite stage; composing with the quotient map gives a distribution on \(\mathcal{E}(X)\) that vanishes on \(\mathfrak{m}_x^{k+1}\), hence is supported in \(\set{x}\), since a function vanishing near \(x\) lies in every power of \(\mathfrak{m}_x\), and has order \(\leq k\).
(77) Proposition (Formal functoriality). For every smooth \(\phi: X \to Y\) with \(\phi(x) = y\), the pullback \(\phi^*\) satisfies \(\phi^*\, \mathfrak{m}_y \subseteq \mathfrak{m}_x\) and hence \(\phi^*\, \mathfrak{m}_y^{k+1} \subseteq \mathfrak{m}_x^{k+1}\): it descends to algebra morphisms \(\mathcal{E}_k(Y; y) \to \mathcal{E}_k(X; x)\) and \(\hat{\phi}^*: \hat{\mathcal{E}}(Y; y) \to \hat{\mathcal{E}}(X; x)\) with
Proof. \(f(y) = 0\) implies \(f(\phi(x)) = 0\), and \(\phi^*\) is multiplicative, so the ideal containments hold and the quotient maps exist; functoriality is that of \(\phi^*\).
(78) Definition (Taylor map). For \(f\in C^\infty(X,\IR)\), its Taylor expansion at \(x\) is
Its order-\(k\) Taylor polynomial is the truncation \(\pi_{\leq k}t(x;f)\).
(79) Lemma (Jet pairing). \(\langle t(x;\, f),\, \xi \rangle = D(f;\, x;\, \xi)\) for \(f \in C^k(X, \IR)\) and \(\xi \in ST_k(X; x)\).
Proof. Degreewise, \(r! \cdot \frac{1}{r!} D(f; x; \xi_r) = D(f; x; \xi_r)\).
(80) Lemma (Jet realization). For \(\alpha = \sum_r \alpha_r \in ST^k(X; x)\), the polynomial \(p_\alpha(x+v):=\sum_{r=0}^k\alpha_r(v,\dots,v)\) satisfies \(t(x;\, p_\alpha) = \alpha\). In particular the Taylor map \(t(x;\, -)\) is surjective.
Proof. The \(r\)-th derivative at \(v = 0\) of the \(s\)-homogeneous term vanishes for \(r \neq s\) and equals \(r!\, \alpha_r\) for \(r = s\), by polarization.
(81) Proposition (Jets are multiplicative). \(t(x;\, fg) = \pi_{\leq k}(t(x;\, f) \cdot t(x;\, g))\) for \(f, g \in C^k(X, \IR)\).
Proof. In degree \(n \leq k\), as in the proof of (67) 🔗,
which equals \(\frac{1}{n!} D(fg; x; v_1, \dots, v_n)\) by the Leibniz rule (60) 🔗.
(82) Proposition (Taylor realization). On \(\mathcal{E}(X)\) the Taylor map \(t(x;\, -)\) is an algebra morphism onto \(ST^k(X; x)\) with kernel \(\mathfrak{m}_x^{k+1}\); it descends to an isomorphism of algebras
Proof. Multiplicativity is (81) 🔗, and surjectivity is the jet realization (80) 🔗 on the smooth polynomials \(p_\alpha\). The kernel consists of the functions flat to order \(k\) at \(x\): all derivatives of order \(\leq k\) vanish. Flatness is a differential condition, membership in \(\mathfrak{m}_x^{k+1}\) an algebraic one; the two inclusions need separate arguments. \(\mathfrak{m}_x^{k+1} \subseteq \ker t(x;\, -)\): a derivative of order \(\leq k\) of a product of \(k + 1\) functions vanishing at \(x\) leaves, in each Leibniz term, at least one factor undifferentiated. \(\ker t(x;\, -) \subseteq \mathfrak{m}_x^{k+1}\) is Hadamard's lemma iterated: writing \(f\) by Taylor's formula with integral remainder, a smooth function flat to order \(k\) at \(x\) is a sum of products of \(k + 1\) coordinate functions centered at \(x\) with smooth coefficients, hence lies in \(\mathfrak{m}_x^{k+1}\) [Nestruev2020]. This step uses smoothness and the finite dimension of \(X\).
(83) Corollary (Realization of the completion and its dual). The stagewise isomorphisms assemble to an isomorphism of algebras
the interface of the smooth ladder in the introduction. Under it and the structure isomorphism \(ST_k(X; x) \isom \mathcal{E}'(X; x)_{\leq k}\) (47) 🔗, the formal duality of (76) 🔗 becomes the permanent pairing:
Proof. The isomorphisms (82) 🔗 commute with the quotient maps \(\mathcal{E}_{k+1}(X; x) \to \mathcal{E}_k(X; x)\) and the truncations \(\pi_{\leq k}\), so they induce an isomorphism of the limits. The pairing identity is the definition of the point distribution (61) 🔗 together with the jet pairing (79) 🔗.
(84) Theorem (Jet pullback). For \(\phi\) \(C^k\) near \(x\) and \(f \in C^k(Y, \IR)\),
and \(D^+(\phi; x)\) is the unique linear map with this property: under the Taylor realization (82) 🔗, the formal pullback \(\mathcal{E}_k(Y; y) \to \mathcal{E}_k(X; x)\) of (77) 🔗 becomes the operator \(D^+(\phi;\, x)\).
Proof. Pair with \(\xi \in ST_k(X; x)\): by the jet pairing (79) 🔗, the smooth adjunction (56) 🔗, and the pushforward-pullback adjunction (69) 🔗, \(\langle t(x;\, f \circ \phi),\, \xi \rangle = D(\phi^* f;\, x;\, \xi) = D(f;\, y;\, D_+(\phi;\, x)\, \xi) = \langle t(y;\, f),\, D_+(\phi;\, x)\, \xi \rangle = \langle D^+(\phi;\, x)\, t(y;\, f),\, \xi \rangle\); conclude by separation (66) 🔗. Uniqueness follows from surjectivity of the Taylor map at \(y\) (80) 🔗.
(85) Remark (Formal and explicit pullback). Formal jet pullback is ideal-theoretic and classical. The Taylor realization identifies it with the explicit operator \(D^+\) on the symmetric cotangent algebra.
Exhibits
Gallery
Throughout, \(X, Y, Z\) are affine spaces over real Banach spaces \(E, F, G\), with maps and basepoints
observables \(f, g\) take values in a Banach space, or in a Banach algebra where they are multiplied, and the order \(k \geq 1\) is fixed. Smooth formulas assume maps and observables \(C^k\) near the basepoints at which they are evaluated; discrete formulas assume no regularity. For a tuple \(v_1, \dots, v_k\) and \(A \subseteq [k]\) we write \(v_A := \prod_{i \in A} v_i\); differentials and differences are overloaded by the type of their argument, so no order superscripts appear. The spaces and the two ladder diagrams are introduced in the introduction; the interface maps and operators are constructed in the body. Here we collect the formulas and adjunctions proved there.
The four sectors, with their model spaces, operators, and pairings, and the formal and total spaces they realize:
| Discrete covariant \(\Delta_+\) | Discrete contravariant \(\Delta^+\) | Differential covariant \(D_+\) | Differential contravariant \(D^+\) | |
|---|---|---|---|---|
| Model spaces | \(CT_*(X; x)\) | \(\CTH^*(X; x)\) | \(ST_*(X; x)\) | \(\STH^*(X; x)\) |
| Operators | \(\Delta_+(\phi; x)\) | \(\Delta^+(\phi; x)\) | \(D_+(\phi; x)\) | \(D^+(\phi; x)\) |
| Pairing | \(\langle \omega, c \rangle = \omega(c)\) | \(\langle a, \xi \rangle = r!\, a(\xi)\) | ||
| Formal spaces | \(\mathcal{E}'(X; x)\) | \(\hat{\mathcal{E}}(X; x)\) | ||
| Operators | \(\phi_*\) | \(\hat{\phi}^*\) | ||
| Pairing | \(\langle \hat{f}, u \rangle = u(f)\) | |||
| Total spaces | \(F'(X)\) | \(F(X)\) | \(\mathcal{E}'(X)\) | \(\mathcal{E}(X)\) |
| Operators | \(\phi_*\) | \(\phi^*\) | \(\phi_*\) | \(\phi^*\) |
| Pairing | \(\langle \mu, f \rangle = \sum \lambda_i f(z_i)\) | \(\langle u, f \rangle\) |
The four operators. The cubical pushforward is given coordinatewise by face differences, the cubical pullback by precomposition:
The smooth operators are determined by their values on monomials:
Structure morphisms. The pullbacks preserve products, while the smooth pushforward preserves the coproduct:
Functoriality. Pushforwards preserve composition and pullbacks reverse it:
Adjunctions. Pullback is adjoint to pushforward in both sectors:
Realizations. Cube measures and cubical jets realize the discrete operators; point-supported distributions and Taylor series realize the smooth operators:
Product rules. On affine probes:
Faà di Bruno formulas. On affine probes:
Collapse junction. Let \(\phi\) and \(f\) be \(C^k\) near \(x\). The symbol map is the junction between the discrete and smooth calculi: paired against an observable, the rescaled difference converges to the corresponding differential (34) 🔗, (39) 🔗,
An Order-Two Cube under Pushforward and Collapse
Fix an arbitrary map \(\phi:X\to Y\) with \(y=\phi(x)\) and directions \(v_1,v_2\in E\). At order two, the pushforward, affine reconstruction, weighted collapse, and symbol map can all be written explicitly.
The image of a parallelogram. The affine probe \(\Aff(v_1, v_2)\) at \(x\) has vertices \(x\), \(x + v_1\), \(x + v_2\), \(x + v_1 + v_2\) and vanishing defect. Its pushforward \(c := \Delta_+(\phi;\, x)\, \Aff(v_1, v_2)\) (11) 🔗 is the vertexwise image, re-read in Möbius coordinates (12) 🔗:
The legs are the image increments, and the defect \(c_{12} = \phi(x + v_1 + v_2) - \phi(x + v_1) - \phi(x + v_2) + \phi(x)\) measures the failure of the four image points to close a parallelogram: an arbitrary map does not preserve affinity, and the pushforward records this failure without any regularity assumption.
Affine reconstruction. Pair an observable \(f: Y \to \IR\) against the curved image. Affine reconstruction (17) 🔗 expands the curved pairing over the five covers of \([2]\):
Together with the adjunction \(\Delta(f \circ \phi;\, x;\, v_1, v_2) = \Delta(f;\, y;\, c)\) (15) 🔗, this is the covering Faà di Bruno formula (18) 🔗 at order two. It has five cover terms at finite scale; after collapse, the smooth formula has two partition terms.
Collapse. Now let \(\phi\) be \(C^2\) near \(x\) and \(f\) be \(C^2\) near \(y\), and rescale the source probe, \(v_i \mapsto t v_i\). The image coordinates have the scales given by (32) 🔗: the legs are first differences and the defect is a second difference.
so after weighted rescaling the image cube converges to the cube \(c'\) with legs \(D(\phi;\, x;\, v_i)\) and defect \(D(\phi;\, x;\, v_1, v_2)\). Each cover term of the reconstruction scales with its weight:
| cover | weight | order | role |
|---|---|---|---|
| \(\set{\set{1}, \set{2}}\) | 2 | \(t^2\) | principal partition term |
| \(\set{\set{1,2}}\) | 2 | \(t^2\) | principal partition term |
| \(\set{\set{1}, \set{1,2}}\) | 3 | \(t^3\) | first correction |
| \(\set{\set{2}, \set{1,2}}\) | 3 | \(t^3\) | first correction |
| \(\set{\set{1}, \set{2}, \set{1,2}}\) | 4 | \(t^4\) | higher error |
By (30) 🔗, the covers of weight two are exactly the partitions. After division by \(t^2\), these terms remain and the three terms of higher weight converge to zero. The weight-three pair gives the first correction (44) 🔗.
The symbol. The two survivors assemble into the symmetric algebra: by the collapse theorem (34) 🔗,
with \(\sigma_2(c') = c'_1 c'_2 + c'_{12}\) (37) 🔗. Substituting the limits,
(52) 🔗. This is the intertwining (40) 🔗 at order two: the collapsed cubical pushforward is the symmetric pushforward. The limit identity is the partition Faà di Bruno formula (57) 🔗,
The lower-degree correction. The Möbius defect measures the failure of the image vertices to form an affine parallelogram. Under collapse it converges to \(D(\phi;x;v_1,v_2)\), the degree-one summand of \(D_+(\phi;x;v_1v_2)\). Pairing this summand with \(D(f;y)\) gives the lower-degree term \(D(f;y;D(\phi;x;v_1,v_2))\) in the second-order chain rule. The Laplace and stencil exhibits below show how inverse pushforward turns these lower-degree terms into the Christoffel correction in a coordinate formula.
Inverse Differences and Differentiation
Pushing point operators through coordinate changes, in the examples below, requires the pushforward along a local inverse. Functoriality reduces this to inverting \(D_+(\phi; x)\), and splitting off the diagonal makes the inversion a finite Neumann sum. This separates the inverse of the first derivative from the higher-order correction terms.
(86) Proposition (Inverse pushforward). Let \(\phi\) be \(C^k\) near \(x\) with invertible differential \(D(\phi; x) \in L(E, F)\). Split the pushforward by partition type,
where \(S\) collects the singleton partitions and \(N\) the partitions containing a block of size at least two. Then \(S^{-1} N\) is nilpotent, \(D_+(\phi; x)\) is invertible, and
If \(\phi\) has a \(C^k\) local inverse at \(x\), then \(D_+(\phi^{-1};\, y) = D_+(\phi;\, x)^{-1}\).
Proof. A partition of \([r]\) with a block of size at least two has at most \(r - 1\) blocks, so its term in (52) 🔗 lands in degree \(\leq r - 1\): \(N\) strictly lowers the degree and vanishes in degrees \(\leq 1\), while \(S\) is degreewise invertible with the stated inverse. Hence \(S^{-1} N\) strictly lowers the degree, \((S^{-1} N)^k = 0\) on \(ST_k(X; x)\), and the finite geometric sum inverts \(1 + S^{-1} N\); then \(D_+(\phi; x) = S\,(1 + S^{-1} N)\) is invertible with the stated sum. No smallness enters. For the last claim, apply functoriality (55) 🔗 to \(\phi^{-1} \circ \phi = \id\) and \(\phi \circ \phi^{-1} = \id\).
In degree two, the degree-one component of the inverse formula gives \(D(\phi^{-1};y;w_1,w_2)=-A\,D(\phi;x;Aw_1,Aw_2)\), where \(A:=D(\phi;x)^{-1}\).
(87) Remark (Inverse differencing). For comparison, let \(k\geq1\). The cubical pushforward \(\Delta_+(\phi;x):CT_k(X;x)\to CT_k(Y;y)\) is bijective if and only if \(\phi\) is bijective. In that case
This follows directly from the conjugation in (11) 🔗. If \(d=\Delta_+(\phi;x)c\), the source vertices are recovered by \(\zeta_x(c)(T)=\phi^{-1}(\zeta_y(d)(T))\).
(88) Remark. Expanded, the Neumann sum is the tree expansion of the classical inverse Faà di Bruno formula; in the Hopf-algebraic setting this unipotent inversion is performed by the antipode [FM2014].
(89) Remark (Comparison under collapse). The stencil example The Exact Polar Five-Point Stencil 🔗 rescales the vertexwise inversions of a collapsing cube family. Its collapse gives the inverse of the linearization and the finite correction terms of (86) 🔗. In numerical practice the vertexwise equations \(\phi(q) = v\) are solved by Newton iteration, whose linearized first step is the diagonal term of the Neumann sum.
Laplace Operator in an Arbitrary Chart
The polar-coordinate Laplacian is a classical textbook exercise whose direct chain- and product-rule derivation typically takes about two pages. The formula in an arbitrary chart is its classical Christoffel-symbol generalization [KobayashiNomizu1963]. Inverse differential pushforward separates the calculation into a quadratic term and one linear correction.
Let \(\phi:U\to\IR^n\) be a \(C^2\) chart near \(x\) with invertible differential \(D^1(\phi;x)\). Set \(y:=\phi(x)\) and write \(D:=D^1(\phi;x)\), \(E:=D^{-1}\), \(D^*:=\SYM(D)\), and \(E^*:=\SYM(E)=(D^*)^{-1}\). Indices \(i,j,k\) refer to the chart basis and \(a\) to the Cartesian basis. The Cartesian Laplacian is represented by the probe \(L:=\sum_a e_a e_a\in ST_2(\IR^n;y)\). We compute its inverse pushforward \(\hat L:=D_+(\phi;x)^{-1}L\).
Invert the pushforward. Write \(D_+(\phi;x)=D^*+N\), where \(N\) lowers degree. The finite inversion formula (86) 🔗 stops after one correction on \(L\):
Compute the quadratic term. Since \(E^*\) is the algebra map induced by \(E\), it maps the Cartesian sum of squares to
These coefficients form the inverse metric.
Compute the linear correction. On a quadratic monomial, \(N\) retains the one-block partition, so \(N(e_i e_j)=D(\phi;x;e_i,e_j)\). Define \(\Gamma^k_{ij}\) by the second identity below. Then
Read off the operator. Substitution in the inverse formula gives the probe and its associated point operator. Thus we have established the following proposition.
(90) Proposition (Laplace pushforward). Let \(\phi:U\to\IR^n\) be a \(C^2\) chart near \(x\) with invertible differential \(D^1(\phi;x)\). With \(g^{ij}\) and \(\Gamma^k_{ij}\) defined above, let \(F\) be of class \(C^2\) near \(y=\phi(x)\) and set \(G:=F\circ\phi\). Then
(91) Remark (Classical identification). With the notation above, \((g^{ij})=EE^{\mathsf T} =(D^{\mathsf T}D)^{-1}\). Thus \(g^{ij}\) is the inverse of the pullback metric. The coefficients \(\Gamma^k_{ij}\) are its Christoffel symbols, and the proposition is the coordinate expression of the Laplace--Beltrami operator [KobayashiNomizu1963].
(92) Corollary (Polar coordinates). Let \(I\subset\IR\) be an open interval of length less than \(2\pi\) and set \(U:=(0,\infty)\times I\). For \(\phi(r,\theta)=(r\cos\theta,r\sin\theta)\),
Proof. Direct differentiation gives
The two terms required by the general calculation are
The mixed entry \(D(\phi;x;e_r,e_\theta)=(-\sin\theta,\cos\theta)\) is nonzero, but \(g^{r\theta}=0\), so it does not enter the contraction. The remaining diagonal term \(D(\phi;x;e_r,e_r)\) vanishes. Hence \(g^{rr}=1\), \(g^{\theta\theta}=r^{-2}\), and \(\Gamma^r_{\theta\theta}=-r\). The inverse-pushforward formula gives the stated probe.
Higher Laplace Operators
The first higher power of the Laplacian is \(\nabla^4=(\del_1^2+\del_2^2)^2\), called the biharmonic operator. Powers of constant-coefficient operators are represented by powers of their probes, so the biharmonic operator is represented by \(L^2\in ST_4(\IR^2;y)\). In polar coordinates, direct expansion squares the polar Laplacian and differentiates its variable coefficients by the product rule. Instead, apply the inverse-pushforward formula (86) 🔗 in degree four. For the polar chart data of (92) 🔗, set \(\widehat{L^2}:=D_+(\phi;x)^{-1}L^2\) and write \(\widehat{L^2}=\sum_{q=1}^4\widehat L_q\) by symmetric degree. With \(D_+(\phi;x)=D^*+N\) as above, solve \((D^*+N)\widehat{L^2}=L^2\) from the top degree downward. If \(\pi_q\) denotes projection onto symmetric degree \(q\), then
Substituting the derivatives of the polar chart and collecting by degree gives
Their sum is the polar biharmonic probe. The degree-four line is the inverse linear image of \(L^2\); each lower line is the correction forced by the higher derivatives of the chart. Putting these components together gives:
(93) Proposition (Polar biharmonic operator). Let \(I\subset\IR\) be an open interval of length less than \(2\pi\), set \(U:=(0,\infty)\times I\), and let \(\phi(r,\theta)=(r\cos\theta,r\sin\theta)\). For \(x=(r,\theta)\in U\), set \(y:=\phi(x)\). If \(F\) is of class \(C^4\) near \(y\) and \(G:=F\circ\phi\), then
More generally, the same downward recursion applied to \(D_+(\phi;x)^{-1}L^m\) gives the coordinate expression of \(\nabla^{2m}\).
The Exact Polar Five-Point Stencil
The discrete adjunction gives the corresponding identity at finite mesh width, before taking a collapse limit. The five-point Laplacian at \(y \in \IR^2\) with mesh width \(t\),
is the pairing of \(f\) with the cube measures of two affine cubes at \(y\). With the polar chart \(\phi\), \(x\), and \(y\) of (92) 🔗, set \(V:=\phi(U)\) and take \(t\) small enough that \(y\pm te_i\in V\). Vertexwise inversion, the argument of (87) 🔗 restricted to these cubes, transports the stencil to polar coordinates: the cubes
have legs \(c^i_1 = \phi^{-1}(y + t e_i) - x\) and \(c^i_2 = \phi^{-1}(y - t e_i) - x\), the polar coordinates of the stencil points, and Möbius defect \(c^i(\set{1,2}) = -(c^i_1 + c^i_2)\), since the top vertex returns to \(y\). The curved polar stencil is exact at every mesh width: by the discrete adjunction (15) 🔗,
Under collapse the rescaled legs converge, \(t^{-1} c^i_j \to \pm A e_i\), and the rescaled defects converge to second derivatives of the inverse, \(t^{-2}\, c^i(\set{1,2}) \to -\, D(\phi^{-1};\, y;\, e_i, e_i)\), supplied from forward data by inverse differentiation (86) 🔗. By the uniformity and continuity in (34) 🔗, as in the proof of (36) 🔗,
the polar Laplacian probe of (92) 🔗: the legs contribute the diagonal terms, while the Möbius defects converge to the linear correction \(\tfrac{1}{r}\,e_r\).
Polynomials
On polynomials the collapse requires no limit: the expansion of (44) 🔗 terminates and holds with vanishing residual. A map \(p:X\to\IR\) is a continuous polynomial of degree \(\leq d\) if \(p(x+w)=\sum_{s=0}^d A_s(w,\dots,w)\) for continuous symmetric \(s\)-linear maps \(A_s:E^s\to\IR\).
(94) Proposition (Polynomial collapse). Let \(p:X\to\IR\) be a continuous polynomial of degree \(\leq d\).
1) If \(d\leq k\), the discrete pairing on affine cubes equals the differential pairing:
2) For \(c\in CT_k(X;x)\) and \(t\in\IR\), the collapse expansion is finite and exact:
In particular \(\Delta(p;x;\lambda_t c)\) is a polynomial in \(t\) that vanishes below degree \(k\), with \(t^k\)-coefficient \(D(p;x;\sigma_k(c))\).
Proof. 1) Write \(p(x+w)=\sum_{s\leq k}\frac1{s!}D(p;x;w,\dots,w)\). For the \(s\)-homogeneous part, expand \(D(p;x;(\sum_{i\in T}v_i)^{\times s})\) multilinearly and sum with signs. The Boolean sieve kills every term that misses an index \(i\in[k]\). For \(s<k\) no term remains; for \(s=k\) the \(k!\) orderings of \((v_1,\dots,v_k)\) cancel the factor \(1/k!\).
2) Run the proof of (44) 🔗 with the Taylor expansion of \(p\) at \(x\), which represents \(p\) without remainder: every Peano term vanishes, and a multinomial of tensor degree \(\sum_A\nu_A>d\) pairs to zero against the derivatives of \(p\). The surviving weights satisfy \(k\leq\mathrm{wt}(\nu)\leq k\sum_A\nu_A\leq kd\), so the expansion terminates at \(m=k(d-1)\).
Globalization: Bundles over Manifolds
Smooth functoriality gives the cocycle identity required to glue the local model spaces over a manifold.
Let \(M\) be a smooth \(n\)-manifold with atlas \(\kappa_i: U_i \to \IR^n\) and chart transitions \(\varphi_{ji} := \kappa_j \circ \kappa_i^{-1}\), and fix \(k \geq 1\). Over each chart domain take the constant bundle with fiber \(ST_k(\IR^n)\), and glue over an overlap by the pushforward of the transition,
The entries of these matrices are polynomials in the derivatives of \(\varphi_{ji}\), so they depend smoothly on \(x\). The cocycle condition asks that gluing from chart \(i\) to chart \(l\) through an intermediate chart \(j\) agrees with gluing directly, and this is smooth functoriality (55) 🔗 applied to \(\varphi_{li} = \varphi_{lj} \circ \varphi_{ji}\). The result is a vector bundle \(ST_k(M)\), and a smooth map \(\phi: M \to N\) induces a bundle map \(D_+(\phi): ST_k(M) \to \phi^*\, ST_k(N)\), given in charts by \(D_+(\kappa' \circ \phi \circ \kappa^{-1})\) and well defined across charts by functoriality. The same construction with \(D^+\) glues the contravariant bundle \(ST^k(M)\) with its pullback \(D^+(\phi)\), and the permanent pairing and the adjunction glue along.
The transitions preserve the degree filtration, whose graded pieces are the symmetric powers \(\SYM^r(TM)\) (64) 🔗. The constants split off canonically, since the transitions are counital, but the rest of the filtration does not: a grading of \(ST_2(M)\), equivalently a splitting of
is the same datum as a torsion-free connection, and the Christoffel transformation law is the change of splitting. On affine spaces the translations provide the splitting, which is why the affine theory acts on the direct sum.
These bundles are classical. \(ST_k(M)\) is Pohl's higher-order tangent bundle [Pohl1962], with the Weil-functor account in [KMS1993], §35. Its sections are the differential operators of order \(\leq k\) on \(M\), acting through the pairing, so the cocycle above is the transformation law of the coefficient tensors of a differential operator. Contravariantly, \(ST^k(M)\) is the bundle of \(k\)-jets of functions [Ehresmann1951], and its sheaf of sections is the \(k\)-jet truncation of the structure sheaf; the covariant sections form the dual sheaf of point-supported distribution fields.
\(D_+\) as \(L_\infty\)-Morphism
The calculus of this article is the most elementary case of the \(L_\infty\) machinery of deformation theory, in the coalgebra conventions of [KDQ03], §4.1--4.3, and [LV2012], §10.2: the deformation problem is the deformation of a point inside an affine space, and the Lie algebra controlling it is abelian. Throughout, \(\phi: X \to Y\) is \(C^\infty\) near \(x\) and all orders are taken.
The dictionary. Write \(E=T_xX\) and \(F=T_yY\) for the translation spaces and set \(\mathfrak g := E[-1]\) and \(\mathfrak h := F[-1]\), regarded as abelian \(L_\infty\)-algebras: all higher brackets vanish, \(\ell_k = 0\) for \(k \geq 1\). The Chevalley--Eilenberg coalgebra of \(\mathfrak g\) is the cofree conilpotent cocommutative coalgebra on \(\mathfrak g[1] = E\), which is the covariant model space with its coproduct (50) 🔗 and zero coderivation:
Geometrically, deformations of the point \(x\) inside \(X\) are translations by \(E\); the vanishing differential and brackets record that these deformations are unobstructed and unconstrained.
Taylor coefficients. In the coalgebra formulation, an \(L_\infty\)-morphism \(\mathcal F: \mathfrak g \rightsquigarrow \mathfrak h\) is a morphism of Chevalley--Eilenberg coalgebras intertwining the coderivations. For abelian source and target the intertwining condition is empty: an \(L_\infty\)-morphism \(\mathfrak g \rightsquigarrow \mathfrak h\) is exactly a counital coalgebra morphism \(\SYM(E) \to \SYM(F)\), and by the cofree property (101) 🔗 it is freely determined by its Taylor coefficients ([KDQ03], §4.2), the components \(\mathcal F_r: \SYM^r(\mathfrak g[1]) \to \mathfrak h[1]\). For the coalgebra morphism \(D_+(\phi;\, x)\) (53) 🔗 the name is literal: the Taylor coefficients are the higher derivatives,
The lift. Conversely, an \(L_\infty\)-morphism is assembled from its Taylor coefficients by cofreeness: \(CE_\bullet(\mathfrak h)\) is the cofree conilpotent coalgebra on \(\mathfrak h[1] = F\), so a linear map \(\SYM(E) \to F\) extends uniquely to a counital coalgebra morphism. The appendix proves this couniversal property for the symmetric coalgebra (101) 🔗: the extension is the convolution exponential of the coefficient map, and its expansion on monomials is the partition sum. The Taylor coefficients above assemble into the differential \(D(\phi;\, x)\) of (51) 🔗, and the exponential is the pushforward:
Composition. Smooth functoriality (55) 🔗 is composition of \(L_\infty\)-morphisms, and the partition Faà di Bruno formula (57) 🔗 is the resulting composition rule for Taylor coefficients. Kontsevich's gloss for the coalgebra morphism of a map of formal pointed manifolds, "the pushforward on distributions supported at zero" ([KDQ03], §4.1), is realized analytically by (63) 🔗.
Smooth maps between affine spaces thus form a geometric family of \(L_\infty\)-morphisms in which the Taylor coefficients are the classical higher derivatives, every intertwining condition is empty, and composition is the Faà di Bruno formula. Despite its striking simplicity, we have not found this example stated in the \(L_\infty\) literature.
Further connections
(95) Remark (Operator propagation through composites). By (63) 🔗 and functoriality, a point operator on a composite is evaluated by applying the pushforwards successively: \((\psi_m \circ \cdots \circ \psi_1)_*\delta(x;\xi)= \delta(z;D_+(\psi_m;x_{m-1})\cdots D_+(\psi_1;x_0)\xi)\). The test function is applied after the final pushforward. For \(\deg \xi = 2\) this is the forward-Laplacian scheme of [ForwardLaplacian2023]; forward- and Taylor-mode automatic differentiation [GriewankWalther2008] [Bettencourt2019] [Betancourt2018] [Sangha2025] are the jet reading of the same functoriality. Cube probes compose through arbitrary maps by (13) 🔗. A single collapse after composition extracts the derivative; the resulting discrete jet calculus is developed in [DFDB].
(96) Remark (Moments and cumulants). The symbol formula \(\sigma_k(c) = \sum_\pi \prod_{A \in \pi} c(A)\) is the multivariate cumulant-to-moment transform, with cube coordinates in the role of joint cumulants and symbol components in that of moment tensors; the pushforward \(D_+\) is the transformation law of moment tensors under nonlinear maps. See [McCullagh1987], Chapters 2 and 3, where these transformation laws are developed. The cubical formula gives the corresponding finite-increment identity with covering-indexed correction terms.
(97) Remark (Interaction effects). For functions on the Boolean cube, the Möbius coordinates of a cube are the main effects and interactions of a \(2^k\) factorial design: Yates's algorithm [Yates1937] is the fast Möbius transform, and the interaction indices of set functions are the same transform [Grabisch2016]. The pushforward \(\Delta_+\) is the exact transformation law of interaction effects under a nonlinear response map, with the covering formula (18) 🔗 as its coordinate expansion.
Appendix: The symmetric bialgebra
This appendix collects the coalgebra background behind the coalgebra lift (54) 🔗. Throughout, \(V\) is a real vector space. The permanent pairing lives in the Setting (27) 🔗, the product-coproduct adjunction in the body (67) 🔗.
(98) Definition (Symmetric bialgebra). Let \(\SYM(V) = \Vsum_{r \geq 0} \SYM^r(V)\) be the symmetric algebra with its degree filtration \(F_k := \Vsum_{r \leq k} \SYM^r(V)\).
- The coproduct \(\Delta^{\times}: \SYM(V) \to \SYM(V) \tensor \SYM(V)\) is the unique algebra morphism with \(\Delta^{\times}(v) = v \tensor 1 + 1 \tensor v\) for \(v \in V\); the counit \(\eps\) is the degree-zero projection. On monomials, \(\Delta^{\times}(v_1 \cdots v_r) = \sum_{I \sqcup J = [r]} v_I \tensor v_J\) with \(v_\emptyset := 1\), matching (50) 🔗.
- The reduced coproduct of an element \(a\) of positive degree is \(\uDelta(a) := \Delta^{\times}(a) - a \tensor 1 - 1 \tensor a\).
- A counital coalgebra \(C\) with coaugmentation \(1_C\) is conilpotent if for every \(c \in \ker \eps_C\) there is an \(m\) with \(\uDelta^{(m)}(c) = 0\).
(99) Lemma (Conilpotence). The coproduct preserves the degree filtration, so each \(\SYM_{\leq k}(V)\) is a subcoalgebra; the reduced coproduct is strictly degree-decreasing, \(\uDelta(F_k) \subseteq F_{k-1} \tensor F_{k-1}\), so \(\SYM(V)\) and every \(\SYM_{\leq k}(V)\) are conilpotent.
Proof. In the monomial formula both tensor factors have degree \(\leq r\), and in the reduced part both are proper sub-monomials, of degree \(\leq r - 1\).
(100) Definition (Convolution). For a counital cocommutative coalgebra \(C\) and a commutative unital algebra \(A\), the convolution product on \(\Hom(C, A)\) is
associative and commutative with unit \(1_\star := \eta_A \circ \eps_C\).
(101) Proposition (Cofree property; unique coalgebra lift). Let \(C\) be a conilpotent cocommutative coalgebra and \(f: C \to V\) linear with \(f(1_C) = 0\), viewed as a map into \(\SYM(V)\) of pure degree one. Then the convolution exponential
is a finite sum on every element of \(C\), and it is the unique counital coalgebra morphism \(C \to \SYM(V)\) with degree-one component \(f\).
Proof. Finiteness. For \(c \in \ker \eps_C\) and \(m\) large, conilpotence makes every term of \(\Delta_C^{(m-1)}(c)\) carry a tensor factor \(1_C\), which \(f\) kills; hence \(f^{\star m}(c) = 0\), and \(f^{\star m}(1_C) = f(1_C)^m = 0\) for \(m \geq 1\).
Morphism. The value \(f(c) \in V\) is primitive, so \(\Delta^{\times} \circ f = (L + R)(f)\) with \(L(f)(c) := f(c) \tensor 1\) and \(R(f)(c) := 1 \tensor f(c)\), and \(L(f)\), \(R(f)\) commute under the convolution of \(\Hom(C, \SYM(V) \tensor \SYM(V))\). Exponentiating, \(\Delta^{\times} \circ \exp_\star(f) = \exp_\star(L(f)) \star \exp_\star(R(f)) = (\exp_\star(f) \tensor \exp_\star(f)) \circ \Delta_C\); counitality holds since all positive-degree terms have \(\eps = 0\).
Uniqueness. Let \(g, h\) be counital coalgebra morphisms with the same degree-one component. Counitality gives \(\pi_0 g = \eps_C = \pi_0 h\). For \(r \geq 2\), taking the components of \(\Delta^{\times} \circ g = (g \tensor g) \circ \Delta_C\) in \(\SYM^i \tensor \SYM^j\) with \(i + j = r\), \(i, j \geq 1\),
depends only on the components of degree \(< r\), and \(\uDelta\) is injective on \(\SYM^r(V)\) for \(r \geq 2\) since \(\mu \circ \uDelta = (2^r - 2)\, \id\). Induct on \(r\).
(102) Corollary (Coalgebra lift, truncated form). Let \(f: ST_k(X; x) \to F\) be linear with \(f(1) = 0\). There is a unique counital coalgebra morphism \(f^+: ST_k(X; x) \to \SYM(F)\) with degree-one component \(f\); it preserves the degree filtration, hence takes values in \(ST_k(Y; y)\), and on monomials it is the partition sum
In particular the symmetric pushforward (52) 🔗 is the unique coalgebra morphism with degree-one component the differential \(D(\phi; x)\).
Proof. \(ST_k(X; x) = \SYM_{\leq k}(T_x X)\) is a conilpotent subcoalgebra (99) 🔗, so (101) 🔗 applies with \(f^+ = \exp_\star(f)\). On a monomial, the iterated coproduct is the sum over ordered decompositions \([r] = I_1 \sqcup \dots \sqcup I_m\) into possibly empty blocks; \(f\) kills the empty blocks, and \(\frac{1}{m!}\) converts ordered partitions into unordered ones, giving the partition sum. Each term has degree \(|\pi| \leq r\), whence the filtration bound.
Comments