Chain level Koszul duality between the Gravity and Hypercommutative operads

Paolo Salvatore; Tommaso Rossi

REVIEW 2 major objections 5 minor 1 cited by

Chain level Koszul duality between the Gravity and Hypercommutative operads

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Pith's one-line read This paper constructs a finite-dimensional chain model for the hypercommutative operad and proves it is, up to an operadic suspension, the Koszul dual of the gravity operad at chain level.

desk verdict Real construction, real gap: the operad-level Koszul duality is asserted but not proven; the arity-wise identification goes through. read the letter →

arxiv 2412.03474 v1 pith:Z5X43UBK submitted 2024-12-04 math.AT math.AG

Tommaso Rossi , Paolo Salvatore This is my paper · ORCID

classification math.ATmath.AG MSC 18M8555P4814H10

keywords hypercommutativeoperadgravityKoszuldualityDeligne-Mumfordmodulispacenestedcacticellularchainmodeloperadicbarconstructionsuspension

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper constructs a small, finite-dimensional chain model for the hypercommutative operad, the operad whose $n$-ary operations are the homology of the moduli space of stable genus-zero curves with $n+1$ marked points. The model is the cellular chain complex of a regular CW-decomposition of $\overline{\mathcal{M}}_{0,n+1}$ built from nested cacti, so each arity is finite-dimensional in each degree. The paper proves this chain model is weakly equivalent to the operad of singular chains on the Deligne-Mumford moduli spaces. It then shows that, up to an operadic suspension, the model is exactly the linear dual of the operadic bar construction of a chain model for the gravity operad. This refines the known homology-level Koszul duality between the two operads to the chain level and makes it computable.

What carries the argument

The central object is the space of nested cacti $N^\sigma_n(C/S^1)$, a regular CW-complex whose cells are indexed by nested trees whose vertices are decorated by cells of the space of unbased cacti. Theorem 5.6 gives a homeomorphism $\phi_{a_1,\ldots,a_n}\colon \overline{\mathcal{M}}_{0,n+1}\to N^\sigma_n(C/S^1)$ for any choice of positive weights, producing a regular CW-decomposition of the moduli space. The dual cell decomposition of this CW-complex defines $C_*^{\mathrm{dual}}(\overline{\mathcal{M}})$. Proposition 6.2 identifies the cellular chains on the original decomposition with the operadic bar construction $B(\mathrm{grav})$, up to a degree-two shift; the W-construction (the standard resolution of an operad by trees with edge lengths) then yields the zig-zag of quasi-isomorphisms that connects the dual-cell complex to singular chains.

What would settle it

Compute the radial and angular parameters of the nested cactus assigned to a simultaneous three-charge collision, such as $[z_0+\epsilon w_1, z_0+\epsilon w_2, z_0+\epsilon w_3, z_4]$ as $\epsilon\to 0$, and check that the limiting nested cactus is exactly the stratum predicted by the stable dual graph; a mismatch would disprove the homeomorphism of Theorem 5.6 and with it the chain-level duality theorem.

Watch

Extended reading notes

Core claim

The central claim is Theorem 8.1: for $\mathrm{grav}$, the chain model of the gravity operad built from cellular chains on unbased cacti, the Koszul dual $D(\mathrm{grav}) = B(\mathrm{grav})^*$ is isomorphic, after the operadic suspension $\Lambda^{-2}$, to $C_*^{\mathrm{dual}}(\overline{\mathcal{M}})$, the dual-cell chain complex of the Deligne-Mumford operad. The companion statement, Theorem 7.6, says that $C_*^{\mathrm{dual}}(\overline{\mathcal{M}})$ is weakly equivalent to the singular chain operad $C_*(\overline{\mathcal{M}})$, so the dual-cell complex is a genuine chain model for the hypercommutative operad. The identification runs through Proposition 6.2, which identifies the bar construction $B(\mathrm{grav})(n)$ with a degree-two shift of the cellular chains of $\overline{\mathcal{M}}_{0,n+1}$, and through Poincaré duality on the regular CW-decomposition.

Load-bearing premise

The proof that the map $\phi_{a_1,\ldots,a_n}\colon \overline{\mathcal{M}}_{0,n+1}\to N^\sigma_n(C/S^1)$ is a homeomorphism verifies continuity only for one explicit two-charge collision, stating that the general case follows by a similar computation; if some collision pattern breaks continuity, the CW-decomposition and the chain model built on it collapse.

Editorial extensions

If this is right

The hypercommutative operad acquires a finite-dimensional chain model in each arity, so homology computations of hypercommutative algebras become explicit rather than singular-chain computations.
The chain-level identification $D(\mathrm{grav}) = \Lambda^{-2}C_*^{\mathrm{dual}}(\overline{\mathcal{M}})$ means algebras over the gravity operad and algebras over the hypercommutative operad are related by operadic bar and cobar duality.
Although $\overline{\mathcal{M}}$ is not shown to be a cellular operad, its W-construction is cellular, giving a cellular model for the homotopy-coherent version of the operad.
The construction gives a topological, Poincaré-duality explanation of the Koszul dual operations, rather than only an algebraic presentation.
The chain-level result verifies the chain-complex analogue of the spectrum-level Koszul duality conjecture discussed in [29].

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper leaves open whether the CW-decompositions can be made independent of the weights; if not, the dual-cell model may be the best available replacement for a cellular operad structure on $\overline{\mathcal{M}}$ itself.
A natural test extension is to compute the induced hypercommutative algebra structure on the homology of a Calabi-Yau manifold using the dual-cell model, checking that it matches the Gromov-Witten construction.
The same nested-cactus cell structure might yield an explicit cellular model for the framed little disks operad and its $S^1$-homotopy fixed points, connecting the duality to string topology operations.
One could probe the spectrum-level conjecture by comparing the cell counts and degrees of $B(\mathrm{grav})$ with the stable cells of $(\Sigma^\infty_+ D_2)^{hS^1}$ as predicted in [29].

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Briefly: this paper does something genuinely new — it builds a regular CW-decomposition of M_{0,n+1} from nested cacti, forms the dual cell complex, and defines a cellular operad structure on it by grafting. That part (Sections 5–6) is substantial and mostly convincing. Theorem 7.6, that C_*^{dual}(M) is a chain model for Hycom, also seems to hang together, modulo the same kind of "similar computation" gaps.

The soft spot is exactly where the stress-test note points. Theorem 8.1 claims D(grav) = Λ^{-2} C_*^{dual}(M) as operads. The proof identifies the underlying n-ary chain complexes: Proposition 6.2 gives B(grav)(n) ≅ s^2 C_*^{cell}(M_{0,n+1}) as chain complexes, and dualizing plus Poincaré duality gives the arity-wise identification. But nothing in the proof compares the two operad structures. The left side gets its composition from the cooperad structure of B(grav), which is transferred through that chain isomorphism and ultimately comes from the operad structure on grav defined via the transfer τ. The right side is the grafting operad on dual cells from Definition 6.5 and Theorem 6.9. These are two independently defined structures, and no lemma says they agree. You need a naturality check: the identification should intertwine the cooperadic decomposition with the cellular chain map induced by the topological grafting. That check is absent. This is not a technicality — it is the difference between Koszul duality of operads and Koszul duality of their underlying collections. So the advertised refinement of Getzler's theorem is not established as written.

The continuity gap in Theorem 5.6 is real but smaller. The proof runs one explicit three-charge collision and then hand-waves the general case. For a paper whose whole edifice is the homeomorphism M_{0,n+1} ≅ N^σ_n(C/S^1), that needs a complete proof, but it is the kind of thing a referee can plausibly ask the authors to fill in without changing the architecture.

The citation pattern is fine; the reliance on [26] and Ward's model is legitimate, and the paper is honest about the open cellular-operad conjecture.

For whom: operad theorists and anyone computing in string topology or quantum cohomology. The construction of C_*^{dual}(M) is a useful finite-dimensional chain model even if the Koszul-duality statement needs repair.

Recommendation: send to peer review, but ask for a major revision. The authors should either prove the operadic compatibility in Theorem 8.1 or weaken the statement to an isomorphism of underlying chain complexes and state the operad-level claim as a conjecture. The paper deserves referee time; it just needs to be finished.

Referee Report

2 major / 5 minor

Summary. The paper constructs, for each n, a regular CW decomposition of the Deligne-Mumford compactification M_{0,n+1} using spaces of nested cacti (Theorem 5.6), then considers the associated dual-cell chain complexes C_*^{dual}(M)(n). It defines an operad structure on the collection C_*^{dual}(M) via grafting maps on nested cacti (Definition 6.5 and Theorem 6.9) and proves, by a zig-zag through the Boardman–Vogt construction, that C_*^{dual}(M) is weakly equivalent as an operad to the singular chain operad C_*(M) (Theorem 7.6). The final theorem (Theorem 8.1) asserts that the linear dual of the bar construction B(grav), where grav is the cacti-based chain model for the gravity operad, is isomorphic as an operad to Λ^{-2} C_*^{dual}(M), thereby refining Getzler's homology-level Koszul duality between the hypercommutative and gravity operads.

Significance. If the operad-level statements are fully justified, this is a valuable result: it provides an explicit finite-dimensional chain model for the hypercommutative operad, relates it to the bar construction of a chain-level gravity operad, and gives a topological construction based on a regular CW decomposition of M_{0,n+1}. The paper is honest about its open problems, including the fact that M itself is not shown to be a cellular operad (Section 1.1 and Paragraph 7.3). The combinatorial machinery is substantial and largely self-contained, building on Salvatore's earlier cell decomposition [26] and Ward's chain model [30]. However, as detailed below, the proof of the central Koszul-duality theorem currently lacks the required compatibility check between operad structures, and the foundational homeomorphism of Theorem 5.6 is proved only for a representative collision. These issues are local and fixable, but they are load-bearing.

major comments (2)

[§8, Theorem 8.1] The proof of Theorem 8.1 identifies only the underlying n-ary chain complexes, not the operad structures. After using Proposition 6.2 and Poincaré duality to obtain an arity-wise isomorphism B(grav)^*(n) ≅ s^{2-2n} C_*^{dual}(M)(n), the proof stops; it never verifies that this isomorphism is compatible with the operadic compositions. The operad structure on D(grav) is induced by the cooperad structure of B(grav), while the operad structure on Λ^{-2} C_*^{dual}(M) is defined independently in Section 6.3 by grafting maps on nested cacti (Definition 6.5, Theorem 6.9). No lemma compares these two structures. In particular, Proposition 6.2 itself proves only an isomorphism of chain complexes B(grav)(n) ≅ s^2 C_*^{cell}(M_{0,n+1}); it does not identify the cooperad structure on B(grav) with a cooperad structure on the cellular chains. Therefore Theorem 8.1, as written, establishes at most an isomorphism of the underlying arity-wise chain complexes, not the advertised operad-level Koszul duality. A compatibility lemma comparing the bar/cobar decompositions with the dual-cell grafting compositions is needed.
[§5.4, Theorem 5.6] The proof of continuity of φ_{a_1,...,a_n}: M_{0,n+1} → N^σ_n(C/S^1) is incomplete. The proof establishes bijectivity and uses compactness/Hausdorff to reduce to continuity, but then verifies continuity only for the explicit three-charge collision [z_0 − εw, z_0 + εw, z_3] in M_{0,4}. The general case is dispatched with the sentence 'In the general case we can do a similar computation.' This homeomorphism is load-bearing: it is used to transfer the regular CW structure of nested cacti to M_{0,n+1} (Corollary 5.8), and all subsequent constructions — dual cells, the operad C_*^{dual}(M), and the proofs of Theorems 7.3 and 7.6 — depend on it. The manuscript should either provide a complete argument for arbitrary collisions and weights or give a precise reference containing the full proof.

minor comments (5)

[§6.1, Proposition 6.2] The statement would be clearer if it explicitly said 'isomorphism of chain complexes' rather than just '=', since the proof does not address the cooperad structure; this is related to the major comment on Theorem 8.1.
[§6.3, Definition 6.5] The text asserts that the grafting maps define an operad structure on N(C/S^1), but associativity and equivariance are not verified. A short argument or an explicit statement that these are immediate from the nested-tree formalism would be helpful.
[§7.2, Theorem 7.6] The zig-zag uses the fact that the retraction r: W N(C/S^1) → N(C/S^1) is a cellular homotopy equivalence and that the map from cellular chains to singular chains is a quasi-isomorphism; these facts are standard, but they are invoked rather than stated precisely.
[§5.3, Lemma 5.5] The inverse map f^{-1} is described as continuous, but the proof of continuity is quite compressed given the complexity of the construction. Expanding this point would improve readability.
[Throughout] There are several typographical issues: 'Haursdorff' (page 35), 'stucture' (page 37), 'costruction' (page 3), 'Poincarè' (throughout), 'baricenter' (page 46), and a missing parenthesis in 'M_{0,n+1}) is 2n − 4' in the proof of Theorem 8.1. These do not affect the mathematics.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: Theorem 8.1 is derived from a genuine homeomorphism/CW-decomposition and a checked bar-construction identification, not from a definitional or fitted input; the significant flagged concerns are omitted operad-compatibility and continuity verifications, which are rigor gaps rather than circular reductions.

full rationale

I find no circularity. The derivation chain is genuine: Theorem 5.6 proves a homeomorphism Mbar_{0,n+1} -> N^sigma_n(C/S^1) (topological input fixing a regular CW-structure); Proposition 6.2 identifies B(grav)(n) with s^2 C^cell_*(Mbar_{0,n+1}) by matching the cell combinatorics (nested trees decorated by unbased cacti) with the free-cooperad indexing and then verifying that the cellular boundary (Move 1 = cacti boundary, Move 2 = edge contraction) equals the bar differential d1 + d2 — a check that could in principle fail, not a definitional fiat; and Definition 6.5/Theorem 6.9 equip the dual cells with an operad structure by topological grafting, independently of the bar construction. Theorem 8.1 then compares the two sides, so the equality is not written into either definition. Although the paper leans heavily on the second author's [26] (Theorems 3.4, 4.4, 4.5; Proposition 5.4), that is a peer-reviewed published paper whose assumptions do not include the present target result, so it counts as real evidence and does not raise the circularity score. Per protocol I explicitly flag, and weigh as correctness risks, the following omitted proofs/limitations: (1) Theorem 5.6 defers the general continuity case: 'In the general case we can do a similar computation to show the continuity of phi_{a_1,...,a_n}'; (2) Theorem 7.3 asserts continuity in the weights — 'the homeomorphisms phi_{aS} ... depend continuously on aS' — without a proof; (3) Theorem 8.1's proof computes only arity-wise: 'B(grav)^*(n) = s^{-2}C^*_cell(Mbar_{0,n+1}) = s^{2-2n}C^dual_*(Mbar_{0,n+1}) = (Lambda^{-2}C^dual_*(M))(n)', and never verifies compatibility with the operadic compositions of D(grav) and Lambda^{-2}C^dual_*(M), so the operad-level statement is not established as written; (4) the paper concedes the CW-decompositions are not operad-compatible (Remark 5.10) and that proving M is a cellular operad remains open (Sections 1.1 and 7.3). These are gaps in rigor and completeness, not circular reductions to the paper's own inputs, so the circularity score stays at 0.

Assumptions & free parameters 2 free parameters · 4 assumptions · 1 invented entities

The central claim rests on the second author's prior cell-decomposition work [26], on Ward's chain model for the gravity operad [30], and on standard tools (bar construction, Poincare duality). No numerical data are fitted; the only auxiliary choices are weights and sections, and the construction is claimed to be independent of them.

free parameters (2)

weights a_1,...,a_n = arbitrary positive reals summing to 1
Each choice defines the homeomorphism to nested cacti (Theorem 5.6) and thus a CW decomposition of M_{0,n+1}; the paper asserts the resulting chain complex is independent of this choice but does not prove it in detail.
sections sigma_k
The space of nested cacti N^sigma depends on chosen sections of C_k to C_k/S^1; any choice yields a homeomorphic CW complex (Remark 5.7).

assumptions (4)

domain assumption The cacti operad Cact is a chain model for the little disks operad D_2 (Theorem 4.5, attributed to [26]).
Used to construct grav and to identify the cellular chains of nested cacti with the bar construction; external published result.
domain assumption grav := sC_cell(C_n/S^1) is a chain model for the Gravity operad (Paragraph 4.3, based on Ward [30]).
Central input for Theorem 8.1; external result.
standard math Standard operadic bar construction and Poincare duality for regular CW complexes.
Used in Propositions 6.2 and Theorem 8.1 to identify B(grav) with shifted cellular chains and to pass to dual cells.
domain assumption The second author's cell decomposition of the Fulton-MacPherson operad and the homeomorphism T r_n to M_{0,n+1} (Theorem 3.4, [26]).
The paper extends this to M_{0,n+1}; external published work by one of the authors.

invented entities (1)

Space of nested cacti N^sigma_n(C/S^1)
purpose: Provides a regular CW decomposition of M_{0,n+1} and a topological model supporting the cellular chain operad.
New combinatorial model constructed in Section 5; its validity is shown internally via Theorem 5.6, not by external data.

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Pith. "Pith review of Chain level Koszul duality between the Gravity and Hypercommutative operads." pith.science (2026). https://pith.science/paper/Z5X43UBK

@misc{pith2026241203474,
  author       = {Pith},
  title        = {Pith review of: Chain level Koszul duality between the Gravity and Hypercommutative operads},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z5X43UBK}},
  note         = {Machine review of arXiv:2412.03474}
}

abstract

Let $\overline{\mathcal{M}}_{0,n+1}$ be the moduli space of genus zero stable curves with $(n+1)$-marked points. The collection $\overline{\mathcal{M}}=\{\overline{\mathcal{M}}_{0,n+1}\}_{n\geq 2}$ forms an operad in the category of complex projective varieties; its homology $Hycom= H_*(\overline{\mathcal{M}})$ is called the Hypercommutative operad. In this paper we construct a chain model for the hypercommutative operad, i.e. an operad of chain complexes $C_*^{dual}(\overline{\mathcal{M}})$ which is weakly equivalent to the operad of singular chains $C_*(\overline{\mathcal{M}})$. We prove that $C_*^{dual}(\overline{\mathcal{M}})$ is the linear dual of the bar construction $B(grav)$, where $grav$ is a chain model of the gravity operad based on cacti without basepoint. This shows that the Gravity and Hypercommutative operad are Koszul dual also at the chain level, refining a previous result of Getzler. The construction is topological, since $C_*^{dual}(\overline{\mathcal{M}})(n)$ is the cellular complex associated to a regular CW-decomposition of $\overline{\mathcal{M}}_{0,n+1}$.

Figures

Figures reproduced from arXiv: 2412.03474 by the authors.

**Figure 1.** This picture shows how to associate a labelled tree with black and white vertices to a point (z1, z2, z3) ∈ F3(C). On the left we see the flow lines of the electric field E(z): some of them are of finite length (the black ones) and connect points of the configuration (white vertices) to zeroes of E(z) (black vertices). by the second author: given a configuration of n points (z1, . . . , zn) ∈ Fn(C), think of each zi… view at source ↗

**Figure 2.** In this picture we see three nodal curves with marked points, and below their dual graphs: the curve on the left is stable, while the others are not. The middle one does not satisfy the stability condition, while the curve on the right is not stable since its dual graph is not a tree. 2.1.1. Pointed stable curves. Let M0,n+1 be the moduli space of genus zero Riemann surfaces with n + 1 marked points. Grothendieck a… view at source ↗

**Figure 3.** This picture shows how to associate an admissible tree to a configuration of points (z1, . . . , zn) ∈ M0,n+1. The orientation of the grey flow lines is omitted in order to have a clearer picture: their source is the point ∞ ∈ C ∪ {∞}, their endpoint is one of the black/white vertices. (2) For each white vertex w, a function g : Ew → [0, 1] such that P e∈Ew g(e) = 1. We interpret 2πg(e) as the angle between the edge… view at source ↗

Figures from the paper (21 more)

**Figure 4.** Figure 4: This picture shows all admissible trees with 3 leaves. Definition 3.3. For each tree T ∈ Tn with set of black vertices B, let σB be the topological space of functions f : B → (0, 1] such that: • max(f) = 1 • For any edge e = (b, b′ ) of T we have f(b) ≥ f(b ′ ) Definit…

**Figure 5.** Figure 5: This picture shows some identifications that we need to do in the definition of T rn [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]

**Figure 6.** Figure 6: The (open) cell decomposition of T r3. There are two 0-cells (black vertices), six 1-dimensional open cells (black edges) and three 2-dimensional open cells [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]

**Figure 7.** Figure 7: On the left there is an element x ∈ C4, on the right its associated cactus c(x). The base point of the circle S 1 is depicted in red and corresponds to a base point on the cactus c(x) (which we depict as a red spine). 4.2. Cell decomposition. To any point x ∈ Cn we can…

**Figure 8.** Figure 8: On the left there are some cacti (without base point), on the right the corresponding dual graphs. 1 2 3 3 1 2 1 2 1 2 1 2 1 2 C2 = 1 2 3 1 2 3 ∂ [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]

**Figure 9.** Figure 9: On top there is a full description of C2 ∼= S 1 : there are two zero cells (2, 1) (on the left) and (1, 2) (on the right). The 1-cells are (2, 1, 2) (on the top) and (1, 2, 1) (on the bottom). Below we see the cell (2, 3, 2, 1, 2) ∼= ∆0 × ∆2 × ∆0 of C3 and the codimens…

**Figure 10.** Figure 10: On the left we see a cactus x, whose associated sequence is (2, 1, 2). On the right we see the image of the corresponding cactus map cx : [0, 2] → [0, 1]2 Theorem 4.4 ( [26]). The space of cacti Cn is (S 1 × Σn)-equivariantly homotopy equivalent to Fn(C). 4.3. Opera…

**Figure 11.** Figure 11: On the left we see three cacti with two lobes. The red bullet on the leftmost cactus denotes the local base point of lobe 1. On the right we see a picture of the composite θ2,2,2(x, x1, x2). consider the product of dilations D : [0, 1]k → Y k j=1 [0, nj ] (t1, . . . ,…

**Figure 12.** Figure 12: An example of how the transfer τ works (with F2 coefficients). (1) One can do the same construction as before but with the additional data of a base point (also called a spine) for each lobe. We call the resulting space fCn the space of cacti with spines (or framed ca…

**Figure 13.** Figure 13: An example of how the partial composition of grav works (with F2 coefficients). 1 3 2 3 1 2 1 2 3 1 3 2 2 1 3 [PITH_FULL_IMAGE:figures/full_fig_p023_13.png]

**Figure 14.** Figure 14: This picture shows the CW-complex C3/S1 ≃ M0,4. There are two zero cells and three edges. We summarize these observations in the next table: Space Fn(C)/S1 Fn(C) fFn(C) Chain model Cn/S1 Cn fCn Operad in Top D2 fD2 Operad in Ch(Z) grav Cact fCact 5. CW-decompositions …

**Figure 15.** Figure 15: In this picture we see two graphical representations of the nested tree S = {{1, 2, 3, 4}, {1, 2, 3}, {1, 2}}. In this example there are two internal edges, three vertices and each vertex has valence two. S3 ∈ S such that S1 ⊆ S3 ⊆ S2. We say that the internal edge (S…

**Figure 16.** Figure 16: In this picture the element (xR, αS, xS, αT , xT ) is depicted on the left: xR is the bigger cactus, xS is the middle one and xT is the smaller. They are nested one into the other according to the nested tree S = (1(23(45))). The base points depicted on the cacti are…

**Figure 17.** Figure 17: In this picture we see two nested cactus that are identified in Nσ 5 (C/S1 ): the nested tree underlying the leftmost (resp. rightmost) nested cactus is S = (1(23(45))) (resp. T = (123(45))). t1, t2 ∈ (0, 1] are radial parameters, α, α1, α2 ∈ S 1 are angular paramet…

**Figure 18.** Figure 18: The cell in this picture has dimension three (it is a cylinder). If we compute the boundary of the cactus with three lobes (move 1) we obtain the cells in the red rectangle, corresponding to the base and the top of the cylinder. If we contract the internal edge (move…

**Figure 19.** Figure 19: An explicit example of the procedure that assigns a nested tree to a labelled tree. cactus associated to S, then consider yS := σ|S|(xS) ∈ C|S| . This is a pointed cactus, so each lobe of yS has a canonical base point. Let b be a black vertex associated to the interse…

**Figure 20.** Figure 20: In this picture we see the nested cactus associated to the labelled tree on the left. We assume that the label of the black vertices are the same as in [PITH_FULL_IMAGE:figures/full_fig_p033_20.png]

**Figure 21.** Figure 21: This picture represent the collision of two charges. Proof. ϕa1...,an is bijective, the domain is compact and the target is Hausdorff. So it is enough to prove the continuity of ϕa1...,an . By definition ϕa1...,an restricted to M0,n+1 ⊆ M0,n+1 is the map ϕa1,...,an of…

**Figure 22.** Figure 22: In this picture we see the cell decomposition of M0,4. On the left there are the cells. On the right it is depicted the cell A1, together with its boundary [PITH_FULL_IMAGE:figures/full_fig_p037_22.png]

**Figure 23.** Figure 23: In blue we see the CW-decomposition of M0,4 given by nested cacti. In red we see the dual cells. C1, C2, three 1-cells B1, B2, B3 and three 2-cells A1, A2, A3 (see [PITH_FULL_IMAGE:figures/full_fig_p042_23.png]

**Figure 24.** Figure 24: In this picture we see the grafting S ◦3 T , where S = ((12)(345)) and T = ((12)3). • t ′ T and α ′ T if U = T +i for some T ∈ T . In the case U = R+i (where R is the root of T ) we set the radial parameter to be 0 (and therefore we do not have to specify an angular p…

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