Operadic structure on the Gerstenhaber-Schack complex for prestacks

We introduce an operad which acts on the Gerstenhaber-Schack complex of a prestack as defined by Dinh Van and Lowen, and which in particular allows us to endow this complex with an underlying $L_{\infty}$-structure. We make use of the operad $\operatorname{Quilt}$ which was used by Hawkins in order to solve the presheaf case. Due to the additional difficulty posed by the presence of twists, we have to use $\operatorname{Quilt}$ in a fundamentally different way (even for presheaves) in order to allow for an extension to prestacks. The resulting $L_{\infty}$-algebra governs the deformation theory of the prestack.


Introduction
The deformation theory of algebras due to Gerstenhaber furnishes the guiding example for algebraic deformation theory. For an algebra A, the Hochschild complex C(A) is a dg Lie algebra governing the deformation theory of A through the Maurer-Cartan formalism. This dg Lie structure is the shadow of a richer operadic structure, which can be expressed by saying that C(A) is a homotopy G-algebra [6]. This structure, which captures both the brace operations and the cup product, is a special case of a B ∞ -structure [7]. Importantly, this purely algebraic structure constitutes a stepping stone in the proof of the Deligne conjecture, proving C(A) to be an algebra over the chain little disk operad [13] [10].
The deformation theory of algebras was later extended to presheaves of algebras by Gerstenhaber and Schack, who in particular introduced a bicomplex computing the natural bimodule Ext groups [4], [5]. However, this GS-complex C(A) of a presheaf A does not control deformations of A as a presheaf, but rather as a twisted presheaf, see for instance [11], [2]. From this point of view, it is more natural to develop deformation theory at once on the level of twisted presheaves or, more generally prestacks, that is, pseudofunctors taking values in the 2-category of linear categories (over some fixed commutative ground ring). In [2], Dinh Van and Lowen established a Gerstenhaber-Schack complex for prestacks, involving a differential which features an infinite sequence of higher components in addition to the classical simplicial and Hochshild differentials. Further, for a prestack A, they construct a homotopy equivalence C GS (A) ∼ = CC(A!) between the Gerstenhaber-Schack complex C GS (A) and the Hochschild complex CC(A!) of the Grothendieck construction A! of A. Through homotopy transfer, this endows the GS-complex with an L ∞ -structure. This result improves upon the existence of a quasi-isomorphism, which is a consequence of the Cohomology Comparison Theorem due to Gerstenhaber and Schack for presheaves [5] and to Lowen and Van den Bergh for prestacks [12].
Although the GS-complex does not possess a B ∞ -structure, its elements -linear maps involving different levels of the prestack -can be composed in an operadic fashion. As such, it makes sense to investigate this higher structure in its own right, and use it directly in order to establish an underlying L ∞ -structure. For particular types of presheaves, explicit L ∞ -structures on the GS-complex have been established by Frégier, Markl and Yau in [3] and by Barmeier and Frégier in [1].
Let Brace be the brace operad and F 2 S the homotopy G-operad. In the case of a presheaf (A, m, f ), in [8], Hawkins introduces an operad Quilt ⊆ F 2 S ⊗ H Brace which he later extends to an operad mQuilt acting on the GS-complex. These operads are naturally endowed with L ∞ -operations as desired. The action of Quilt on the GS-complex considered by Hawkins only involves the restriction functors f of the presheaf, the multiplication m being incorporated later on in mQuilt. Unfortunately, the way in which functoriality of f is built into these actions, does not allow for an extension to twisted presheaves or prestacks.
The goal of this paper is to solve the problem of establishing a natural operadic structure with underlying L ∞ -structure on C GS (A) in the case of a general prestack (A, m, f, c) with twists c. As part of our solution, we use Quilt in a fundamentally different way in relation to the GS-complex, but still allowing us to make use of the naturally associated L ∞ -structure from [8]. In section §3, we capture the higher structure of C GS (A) by introducing the new operad Patch ⊆ mNSOp ⊗ H NSOp over which C GS (A) is shown to be an algebra (see Theorem 3.24). Here, (m)NSOp is the operad of nonsymmetric operads (with multiplication).
In [6], Gerstenhaber and Voronov obtain a brace algebra structure on an operad and a homotopy G-algebra structure on an operad with multiplication. Based upon the expression of these results in terms of the underlying operads NSOp and mNSOp in §2, we construct a morphism Quilt −→ Patch s (see Proposition 3.27) as a restriction of where the operads with subscript denote the (uncolored) graded operads associated to the unsubscripted colored operads. This gives rise to the composition R : Quilt −→ Patch s −→ End(sC GS (A)) which incorporates the multiplication m and the restrictions f of A.
In §4, we extend the action R to ] extending those on Quilt from [8] (see Theorem 4.10) by adding an infinite series of higher components containing twists. Under the action of Quilt this neatly corresponds to and extends the differential on C GS (A) obtained in [2]. In the final section 4.5 we briefly discuss the relation of this L ∞ -structure with the deformation theory of the prestack A. The present work naturally grew out of [2], and at the time when [8] appeared large parts of an operadic approach to the GS complex of a prestack had already been developed independently by us. Given the efficient way in which Hawkins' description of Quilt gives rise to an L ∞ -structure, we decided it was worthwhile to build on this approach to the presheaf case, albeit in a way which "flips and refines" the action of Quilt in order to make it useful for general prestacks. As a consequence, when we follow through Hawkins' approach, in comparison we manage to incorporate not only the restrictions f , but also the multiplications m in an initial action of Quilt on the GS complex. In analogy with the way in which Hawkins extends his action from Quilt to mQuilt in order to incorporate the multiplications m, we establish an extension from Quilt to Quilt b [[c]] in order to incorporate the twists c.
The current paper is part of a larger project in which it is our goal to understand the homotopy equivalence C GS (A) ∼ = CC(A!) from [2] operadically, showing in particular that the L ∞ -structure from [2] and the one established in the present paper actually coincide.
Acknowledgement. The authors are grateful to an anonymous referee for their meticulous reading of an earlier version of the manuscript, and in particular for several corrections and valuable suggestions which greatly helped improve the paper. The third named author thanks Severin Barmeier and Eli Hawkins for bringing the preprints [1], respectively [8] to her attention.

Gerstenhaber-Voronov operadically
In the seminal paper [6], Gerstenhaber and Voronov define a brace-algebra structure on the totalisation of a non-symmetric operad O. Moreover, in presence of a multiplication, they define a homotopy Galgebra structure on O incorporating both the cup product and the Gerstenhaber-bracket.
In this section we describe the morphisms of operads underlying these results. To this end, in §2.3, we recall the colored operad NSOp encoding non-symmetric operads, and we describe the natural extension mNSOp which adds a multiplication. Let NSOp s and mNSOp st be their totalised graded (uncolored) operads with suspended, respectively standard degree ( see §2.5 and §2.6). Let Brace be the brace operad (see §2.1) and F 2 S the Gerstenhaber-Voronov operad encoding homotopy G-algebras (see §2.2). The main goal of this section is the definition of morphisms of dg-operads φ : Brace −→ NSOp s andφ : F 2 S −→ mNSOp st (see Theorems 2.16 and 2.34 respectively). In these definitions, we have to pay particular attention to the choice of signs. For this, we will make use of morphisms of operads (m)NSOp −→ Multi∆ landing in the multicategory associated to the simplex category ∆ (see Proposition 2.11). For both uncolored as colored operads, we use the term morphism of operads. In case confusion may arise, we add a subscript to differentiate the uncolored operads from their colored counterparts.
2.1. The operad Brace. Throughout, we work over a fixed commutative ground ring k.
The operad Brace encoding brace algebras is defined using trees, that is, planar rooted trees. Following the presentation from [8, §2.2], for a tree T we denote the set of vertices by V T , the set of edges by E T , the "vertical" partial order on V T generated by E T by ≤ T , and the "horizontal" partial order on V T by T . For (u, v) ∈ E T we call u the parent of v and v a child of u. For n ∈ N, put [n] := {0, . . . , n} and n := {1, . . . , n}. Let Tree(n) denote the set of trees with vertex set n and let Brace(n) be the free k-module on Tree(n) endowed with the S n -action given by permuting the vertices, i.e., T σ is the tree defined by replacing vertex i in T by σ −1 (i). The operadic composition on Brace is based upon substitution of trees, as follows. For trees T ∈ Tree(m), T ′ ∈ Tree(n) and 1 ≤ i ≤ m, we denote by Ext(T, T ′ , i) ⊆ Tree(m + n − 1) the set of trees extending T by T ′ at i (that is, U ∈ Ext(T, T ′ , i) has T ′ as a subtree which upon removal reduces to the vertex i of T ). We then define Underlying every such extension lie two maps n α ֒→ n + m − 1 β ։ m acting on the vertices, where α embeds n vertices consecutively and β contracts the image of α to the vertex i. We call the pair (α, β) the extension of m by n at i. We refer to [8, §2.2] for more details.
2.2. The operad F 2 S. The operad F 2 S encodes homotopy G-algebras [6]. Again, we largely follow the exposition from [8, §2.3]. Given a set A, a word over A is an element of the free monoid on A. For a word W = a 1 a 2 . . . a k , correspondiong to the function W : k −→ A : i −→ a i , the i-th letter of W is the couple (i, a i ). We will often identify a word with its graph For a ∈ A, a letter (i, a) ∈ W is called an occurrence of a in W . The letter (i, a) is a caesura if there is a later occurrence of a in W , that is, a letter (j, a) with i < j. We say that By definition, F 2 S(n) is the free k-module generated by the words W over n such that: For a word W ∈ F 2 S(n) and u ∈ n , let (i u , u) be the first occurrence of u in W . Then we obtain a total order u ↓ v ⇐⇒ i u ≤ i v on n .
The operadic composition on F 2 S is based upon merging of words, as follows. For words W ∈ F 2 S(m), W ′ ∈ F 2 S(n) and 1 ≤ i ≤ m, we denote by Ext(W, W ′ , i) ⊆ F 2 S(m + n − 1) the set of extensions of W by W ′ at i (that is, X ∈ Ext(W, W ′ , i) if up to relabelling and deleting repetitions, W ′ is a subword of X and upon collapsing the letters from W to i, relabelling and deleting repetitions, we recover W ).
In order to define the composition, we need the sign of an extension.
Sign of Extension. Let W ∈ F 2 S(m) and let int(W ) be the set of interposed elements of m ordered by their first occurrence in W . For X ∈ Ext(W, W ′ , i) the relabelling gives rise to two maps α : Moreover, it is possible to talk about the boundary of a word, inducing a differential.
Boundary. Given a word W ∈ F 2 S(n) and a letter (i, a) of W for which a is repeated in W , then define ∂ i W ∈ F 2 S(n) as the word obtained by deleting the letter (i, a) from W (and relabelling). If a is not repeated, then set ∂ i W = 0.
Sign of Deletion. Given a word W ∈ F 2 S(n) of length k, then we define sgn W : k −→ {−1, 1} by setting sgn W (i) = (−1) k if (i, a i ) is the k-th caesura of W , and otherwise sgn W (i) = (−1) k+1 if it is the last occurrence, but the previous occurrence is the k-th caesura of W .
The S-module F 2 S defines a dg-operad with operadic composition given by and boundary given by The following lemma, which we include for the convenience of the reader, shows how F 2 S encodes the algebraic operations of a homotopy G-algebra.
Notations. To avoid too large expressions, we leave out certain bracketings by setting as default the bracketing Lemma 2.1. Let M 2 := 12, M 1,0 = 1 and M 1,k := 121 . . . 1(k + 1)1 for k ≥ 1, then F 2 S is generated by these elements and the following holds It is a straightforward computation to determine that M 2 and M 1,k satisfy these relations.
Let W ∈ F 2 S(n), we then show that it lies in the suboperad generated by M 2 , (M 1,k ) k≥1 using only the above relations. We prove this by induction on n. If n = 1, then W = 1. So assume n > 1 and apply a permutation such that the first letter of W is 1, then W is of the form where W i is the image of a non-empty word W ′ i ∈ F 2 S(n i ) under the map γ i : n i −→ n , except W k+1 which is possibly empty. Due to no interlacing we also know that the images Im(γ i ) are pair-wise disjoint. Hence, we can apply a permutation to assume that max Im(γ i ) < min Im(γ i ′ ) holds for every i < i ′ . In this case, we have that By induction, this shows that W is generated by M 2 and (M 1,k ) l≥0 .
2.3. The operads NSOp and mNSOp. It is well-known that non-symmetric operads can be encoded using a colored operad NSOp which can be defined using indexed trees, that is, for q 1 , . . . , q n ∈ N and q ′ = 1 + n i=1 (q i − 1), NSOp(q 1 , . . . , q n ; q ′ ) is the set of pairs (T, I) where T ∈ Tree(n) and I : We will often write I to denote the indexed tree (T, I). Moreover, NSOp is generated by those trees with a single edge, that is, for every q 1 , q 2 and 1 ≤ i ≤ q 1 , with the following pair of relations Note that these are the well-known associativity relations for non-symmetric operads.
Definition 2.2. Let mNSOp be the N-colored operad generated by NSOp and an element m ∈ mNSOp(; 2) satisfying the relation More explicitly, every representative of an element X ∈ mNSOp(q 1 , . . . , q n ; q) is of the form I • i1 m • i2 . . . • i k m for I ∈ NSOp and appropriate i 1 , . . . , i k ∈ N. Due to equivariance, we can always consider a representative of X of the form for I ∈ NSOp(q 1 , . . . , q n , 2, . . . , 2; q).
. . , q n ; q), the partial orders < I and ⊳ I on n are independent of the representative of X. We denote them by < X and ⊳ X .
Proof. We proceed by induction on k the number of m's in X. For k = 0 or k = 1, there is nothing to show, so assume k > 1. It is clear that if the lemma holds for X, then the relations that hold for < and ⊳ for trees, also hold for < X and ⊳ X . In particular, if the lemma holds for X and X ′ and (α, β) is the extension of n by m at i, then for a, b / ∈ Im(α) we have by induction that the lemma holds for X 0 . Moreover, we have for a, b ∈ n that which proves the lemma for X.

2.4.
The morphisms (m)NSOp −→ Multi∆. Let C be a small category. We denote by MultiC the Ob(C)-colored operad for which MultiC(c 1 , . . . , c n ; c) is freely generated as a k-module by n-tuples (ζ 1 , . . . , ζ n ) of C-morphisms with ζ i : c i −→ c, S n acts by permutating labels, and composition is defined in the obvious way. Let ∆ be the simplex category. Next, we construct a morphism of operads NSOp −→ Multi∆ by associating to every indexed tree I in NSOp(q 1 , . . . , q n ; q) a n-tuple ζ I in Multi∆(q 1 , . . . , q n ; q) which assigns to each vertex a, considered as an q a -corolla, a numbering denoting where its inputs are amongst the inputs of the indexed tree as a whole. It suffices to define the morphism on the generators E i ∈ NSOp and show that it respects the relations.
We will employ it as in the following example.
Example 2.6. Let A be a k-linear category, then its Hochschild complex is defined as For a Hochschild cochain φ ∈ C n (A) and a n-simplex A 0 an ← A n in A, we have that φ A0,...,An (a 1 , . . . , a n ) ∈ A(A n , A 0 ).
Let φ 1 ∈ C q1 (A) and φ 2 ∈ C q2 (A), then each E i ∈ NSOp(q 1 , q 2 ; q 1 + q 2 − 1) determines a cochain φ 1 • i φ 2 ∈ C q1+q2−1 (A) as follows which we can visualize using n-corollas On the other hand, we compute the right-hand side ζ ′ := ζ Ei−1+j • 1 ζ Ei and obtain Remark 2.8. In appendix A we have added a generator-free description of this morphism and an alternative proof of lemma 2.7, which we consider insightful and valuable, especially for concrete computations of signs in later sections. It is also possible to associate to an element of mNSOp an element of Multi∆.
for t ∈ n is independent of the representative I of X.
In this case, we write ζ X .
Proof. We prove the lemma by induction on k the number of occurrences of m. The cases k = 0 and k = 1 are trivial, so assume k > 1. Let X 0,i : then by induction and lemma 2.7 we have for t ∈ n that ζ X0,1,t = ζ X0,t = ζ X0,2,t which proves the lemma. Moreover, this last morphism is surjective, but not an isomorphism. This is due to the existence of vertices with zero inputs which collapse information. We consider a simple example. Hence, we can consider mNSOp as a finer operad than Multi∆ and thus encoding more information. where an element x ∈ NSOp(p 1 , . . . , p n ; p) is graded as |x| = r i=1 (p i − 1) − (p − 1) = 0 (this is the suspended grading, whence the subscript) and NSOp s (n) is the subspace generated by sequences of elements with constant grading. The composition on NSOp s is derived from the composition of NSOp where it is set to 0 when the colors do not match. Note in particular that the S n -action on NSOp s (n) is affected by this grading: permuting two vertices i and j introduces the signs (−1) (pi−1)(pj −1) . Definition 2.13. Let (T, I) ∈ NSOp(p 1 , . . . , p n ; p), then (T, I) is a coloring of T and we write Clr(T, p 1 , . . . , p n ) as the set of all such colorings of T .
In order to define the sign sgn T (I) for T ∈ Brace(n), we use the morphism of operads  Proof. Per definition of sgn T (I) we see that φ is equivariant. Hence, we only need to verify that sgn T •1T ′ (I • 1 I ′ ) = sgn T (I) sgn T ′ (I ′ ) for T ∈ Brace(n), T ′ ∈ Brace(m) and I ∈ Clr(T, p 1 , . . . , p n ) and I ′ ∈ Clr(T ′ , p ′ 1 , . . . , p ′ m ). This equation holds as we can decompose the shuffle χ ′′ : where χ and χ ′ are the corresponding shuffles determining sgn T (I) and sgn T ′ (I ′ ).
2.6. The morphism F 2 S −→ mNSOp st . In order to define the morphismφ : F 2 S −→ mNSOp st properly, we again need to compile the colored operad mNSOp to obtain a graded non-colored operad mNSOp st (n) ⊆ q1,...,qn,q mNSOp(q 1 , . . . , q n ; q) where an element x ∈ mNSOp(q 1 , . . . , q n ; q) is graded as deg(x) = r i=1 q i − q (standard grading) and mNSOp st (n) is generated by the sequences of constant grading. The composition on mNSOp st is derived from the composition of mNSOp where it is set to 0 when the colors do not match. Note in particular that the S n -action on mNSOp st (n) is affected by this grading: permuting two vertices i and j introduces the sign (−1) qiqj .
• each vertex n + 1, . . . , n + k has exactly two children in We write Clr(W, q 1 , . . . , q n ) for the set of all such colorings for W .
Note however that not all elements of mNSOp color a word of F 2 S: the following set of elements m 1 r ∈ mNSOp(q 1 ; q 1 + 1) for r ∈ {1, 2}, colors no word in F 2 S because the vertex plugged by m does not have two children.
Lemma 2.20. Definition 2.17 is well-defined, that is, it is independent of the chosen representative I of X.
Proof. Due to lemma 2.4, both < X and ⊳ X are well-defined. We show that the condition stipulating that all vertices of I that are plugged by m's have exactly two children, is independent of the representative of We construct a word for every element of mNSOp satisfying the above criteria.
. . , q n ; q) such that each vertex a > n in I has exactly two children, then we construct a word W X ∈ F 2 S(n) such that X ∈ Clr(W X , q 1 , . . . , q n , q).
• To every tree T we can associate a word W T ∈ F 2 S(n + k) (see [8, §2.3]).
• Suppose for X 0 ∈ mNSOp such that X 0 • n+1 m = X we have an associated word W X0 ∈ F 2 S(n+1), then let W X be the word given by deleting all occurrences of n+1. Then W X ∈ F 2 S(n) because n + 1 had two children, so no degeneracy can occur.
We consider an example of this procedure.
for some 1 ≤ i ≤ q 1 from example 2.19 and show how construction 2.21 assigns a word. First, we associate to the indexed tree the word 1424341 and then delete all occurrences of 4 as it is plugged by an instance of m. As a result, we obtain the word 1231.
Proof. This clearly holds for X = I ∈ NSOp(q 1 , . . . , q n ; q). Assume the lemma holds for X 0 ∈ mNSOp and X = X 0 • n+1 m, then W X0 = W 0 (n + 1)W 1 (n + 1)W 2 (n + 1)W 3 for W 0 and W 3 possibly empty. In this case, W X = W 0 W 1 W 2 W 3 and it is easy to see that X ∈ Clr(W X , q 1 , . . . , q n ). Now reversely, if X ∈ Clr(W, q 1 , . . . , q n ) and a ⊳ b are the two children of n + 1 in X 0 , then W = W 0 W 1 W 2 W 3 where W 1 = a . . . a, W 2 = b . . . b and W 0 and W 3 are possibly empty. In that case, X 0 ∈ Clr(W 0 (n + 1)W 1 (n + 1)W 2 (n + 1)W 3 , q 1 , . . . , q n , 2) and thus by induction W X0 = W 0 (n + 1)W 1 (n + 1)W 2 (n + 1)W 3 . Hence, Lemma 2.24. Let X ∈ Clr(V, q 1 , . . . , q n , q) and Y ∈ Clr(W, q ′ 1 , . . . , q ′ m , q i ), then there exists a unique U ∈ Ext(V, W, i) such that X • i Y ∈ Clr(U, q 1 , . . . , q ′ 1 , . . . , q ′ m , . . . , q n , q) Proof. By construction 2.21 we obtain a word U ∈ F 2 S(n + m − 1) such that Z := X • i Y ∈ Clr(U, . . .). We show that U ∈ Ext(V, W, i): let U α be the word obtained from deleting from U occurrences of vertices not in the image of α and eliminating consecutive repetitions (uu → u). It is easy to check that . . α(u) . . . α(v) . . . α(u) . . ., and that all occurrences of u are left to those of v in W iff the same holds for α(u) and α(v) in U α . Let U β be the word obtained from U by relabelling by β and eliminating consecutive repetitions. To verify that U β = V is straight forward, except in the following case: As α(v) and α(v ′ ) are part of the same subtree α(Y ), i.e. the image of Y under α in Z, there must be some vertex a in the tree underlying α(Y ) (possibly plugged by m) such that a lies underneath u. As Y is a coloring of a word, the conditions imply that a is not plugged by an instance of m (otherwise it would not have two children in Y ). As a result, there is some vertex .. This clearly also holds reversibly.
where two subwords W j and W j ′ do not share any occurrence of the same number, and W j is of the form a j . . . a j . As Z is a coloring of U , we have that α(a 1 ) ⊳ Z . . . ⊳ Z α(a t ) and no vertex of Im(α) lies under any a j . In this case, there exists some vertex a ∈ {n + m, . . . , n + m + k + l − 1} such that a ≤ I ′′ α(a j ) which is ≤ I ′′ -maximal for these conditions (otherwise when applying β to U we will not obtain V ). Let I ′ be the minimal subtree of I ′′ on the root a containing Im(α). By contracting this subtree to a point we obtain a tree I such that, after permutation of some vertices, we obtain I It now suffices to show that X : , which is a straight forward computation using the facts X • i Y = Z, Z ∈ Clr(U, . . .) and U ∈ Ext(V, W, i).

2.6.2.
Signs. In order to define a sign sgn W (X) for W ∈ F 2 S(n) and X ∈ Clr(W, q 1 , . . . , q n ) we use the morphism of operads Lemma 2.26. Let X ∈ Clr(W, q 1 , . . . , q n ; q) for W ∈ F 2 S(n) and q i > 0, then ζ X is a coloring of W in the sense of [8,Def. 4.13], that is, We first show it holds for X = (T, I) ∈ NSOp by induction on the number of vertices: let be its decomposition into its root u with maximal subtrees I i . In this case, we have W = uW 1 u . . . uW k u where the subwords W i represent the subtrees I i . Let γ i : k i ֒→ n be the maps embedding the tree I i 0 onto I i in I and I i 0 ∈ Clr(W i 0 , . . .), then they extend to a map γ i : By induction, the lemma holds for I i 0 ∈ Clr(W i 0 , . . .). Let (p 1 , u), . . . , (p k+1 , u) be all the occurrences of u in W , then we define then it is easy to verify that these satisfy the above conditions. Now assume X = [X 0 • n+1 m] such that the lemma holds for X 0 ∈ Clr(W X0 , . . .), then W X is obtained from W X0 by deleting all occurrences of (n + 1). As the vertex n + 1 has exactly two children in X 0 and Let us define analogously the sign corresponding to the horizontal part of sgn Q (ζ X , I).
Construction 2.27. We work with the following alphabet . . , n and define the word The second word J W (X) is the concatenation of two words J 0,W (X) and J 1,W (X) defined as follows Note that we start from position 0.
Definition 2.28. For X ∈ mNSOp(q 1 , . . . , q n ; q) where we replace those q i = 0 by 2, we define sgn W (X) as the sign of the shuffle transforming J(q 1 , . . . , q n ) to J W (X). for which we calculate the words J W (X) and J W ′ (X ′ ) and their corresponding signs. In the first case, we have For the second case, we calculate is not taken to the front of the word as 3 is not interposed in W ′ .
Proof. We can assume that all q i and q ′ j are not zero. We can decompose sgn W (X) in three components • sign of the shuffle σ shuffling J 0,W (X) to 0 v1 . . . 0 v k for v 1 < . . . < v k the interposed vertices of X, • sign of the shuffle τ shuffling J 1,W (X) to concatenation of 1 i . . . (q i − 1) i for i interposed and 0 i . . . (q i − 1) i for i not interposed. We call this latter sequence J int W (q 1 , . . . , q n ). • sign of the shuffle ρ shuffling We add ′ and ′′ to denote the correspondings shuffles for X ′ and X ′′ : . Further we clearly have (−1) τ ′′ = (−1) τ +τ ′ by simply applying them one after the other and renaming using α and Proposition 2.31. We have a morphism of graded operads Proof. By definition of sgn W (X) the above linear maps are equivariant. By lemma 2.24, 2.25 and 2.30 they define a morphism of graded operads.
We make mNSOp st into a dg-operad with the hochschild differential, then φ will be a morphism of dg-operads.
We consider the associated derivation Proposition 2.33. ∂ D defines a differential making mNSOp into a dg-operad, for which holds Proof. The first part follows directly if D • 1 D = 0 which is an easy computation (see [6,Prop. 2]). In order to prove the second part we only need to show this for the generators of W , i.e. 12 and 121 . . . 1k1 for k ≥ 1.
Theorem 2.34. We have a morphism of dg-operads Proof. This is the direct consequence of propositions 2.31 and 2.33.

The Gerstenhaber-Schack Complex For Prestacks
Let (A, m, f, c) be a prestack over a small category U and let C GS (A) be the associated Gerstenhaber-Schack complex as defined in [2] (see §3.1). In loc. cit., a homotopy equivalence C GS (A) ∼ = CC(A!) is constructed with the Hochschild complex CC(A!) of the Grothendieck contruction A! of A. Through homotopy transfer, this allows to endow the GS-complex with an L ∞ -structure. However, it is desirable to have a direct description available of this structure, without reference to transfer.
In the case of a presheaf, originally considered by Gerstenhaber and Schack, in [8], Hawkins introduces an operad Quilt ⊆ F 2 S ⊗ H Brace which he later extends to an operad mQuilt acting on the GS-complex. These operads are naturally endowed with L ∞ -operations as desired. The action of Quilt on the GScomplex considered by Hawkins only involves the restriction functors f of the presheaf, the multiplication m being incorporated later on in mQuilt. Unfortunately, the way in which functoriality of f is built into these actions, does not allow for an extension to twisted presheaves or prestacks.
In our solution for the prestack case, we propose to use Quilt in a fundamentally different way in relation to the GS-complex, but still allowing us to make use of the naturally associated L ∞ -structure. In this section we capture the higher structure of C GS (A) by introducing the operad Patch ⊆ mNSOp ⊗ H NSOp (see §3.3) over which the bicomplex C •,• (A), of which C GS (A) is the totalisation, is shown to be an algebra (see Theorem 3.24). Next, we construct a morphism Quilt −→ Patch s (see Proposition 3.27) as a restriction ofφ This morphism is such that the resulting composition incorporates the multiplications m and the restrictions f . Note that in Hawkins' approach to the presheaf case, the initial action of Quilt on End(sC GS (A)) only incorporates the restrictions. As far as the structure of both approaches goes, the auxiliary operad Patch we use is the counterpart of the operad ColorQuilt from [8,Def. 4.6]. In §4, we will further extend the action R in order to incorporate the twists.
3.1. The GS complex. In this section, we recall the notions of prestack and its associated Gerstenhaber-Schack complex, thus fixing terminology and notations. We use the same terminology as in [2], [12]. A prestack is a pseudofunctor taking values in k-linear categories. Let U be a small category.
For u = 1 or v = 1, we require that c u,v = 1. Moreover, the natural isomorphisms have to satisfy the following coherence condition for every triple w : Given such a prestack A, we have an associated Gerstenhaber-Schack complex C GS (A). In [2] this is defined as the totalisation of a bicomplex C •,• (A). We first review some notations. For each 1 ≤ k ≤ p − 1, denote by L k (σ) and R k (σ) the following simplices and we consider the following natural isomorphisms We write c σ,k,A = c σ,k (A) and ǫ σ,k,A = ǫ σ,k (A) for A ∈ A(U p ). We also define a set P (σ) of formal paths from σ # to σ * inductively. A formal path is finite sequence of couples (τ, i) consisting of a simplex σ and a natural number i. We set where ∂ i denotes the ith face-operator of the nerve N p (U). Given such a formal path r = (r 1 , . . . , r p−1 ) we define its sign By interpreting the data (σ, i) as the natural isomorphism ǫ σ,i , every formal path r ∈ P (σ) induces a sequence of natural isomorphisms r ∈ N p−1 (Fun(A(U p ), A(U 0 ))). Note that ǫ (u1,u2),1 = c u1,u2 and its associated sign is −1.
Let S t,p−1 denote the set of (t, p − 1)-shuffles, then given a formal path r ∈ P (σ), a shuffle β ∈ S t,p−1 and a tuple a = (A 0 Here, we give a more explicit definition of β(a, r): first we construct inductively a sequence (b 1 , . . . , b t+p−1 ) which formally represents a sequence of morphisms in A(U 0 ). Every b i is either of the form (τ, a i , A i−1 ) or (r i , A j ) for τ a simplex, a i and A j respectively a morphism and an object occurring in a, and r i an element of the formal path r. Define The differential d on the GS-complex is defined for θ ∈ C p,q (A) as a i+1 ), . . . , a 1 ) for σ = (u 1 , . . . , u p+j ) ∈ N p+j (U)(U 0 , U p+j ), a = (a 1 , . . . , a q−j+1 ) where a i ∈ A(U p+j )(A i , A i−1 ) and such that B is the sequence of objects underlying β(a, r).
We will also be interested in the subcomplex C GS (A) ⊆ C GS (A) of normalized and reduced cochains which is shown to be quasi-isomorphic to the GS complex (see [2,Prop. 3.16 A cochain θ is normalized if θ σ (A)(a) = 0 for every normal simplex a in A(U p ). We come back to this in section §4.
Elements of the GS complex have a neat geometric interpretation as rectangles: for θ ∈ C p,q (A) and the data (σ, A, a) from above, we can represent θ σ (A)(a) as the rectangle of data Similarly, we can draw different components of the differential d using rectangles, providing more insight in its rather technical definition. For the hochschild component d 0 we have The first component d 1 can similarly be drawn as Finally, we will draw d 2 as an example from which it is easy to deduce the higher components d j for j > 2. Namely, we have for shuffle β(q) = i, β(s) = s for s < i and β(s) = s + 1 for s ≥ i, and formal path r = ((u p , u p+1 ), 1). Note in particular that we can draw β(a, r) as follows We will use this rectangular interpretation as a guide in the next sections.
3.2. Endomorphism operad of a prestack. Although the GS-complex does not have partial compositions • i , its elements θ = (θ σ (A)) (σ,A) consist of parts that lie in the endomorphism operad End(A).
with partial compositions defined by composition of linear maps.

The Operad
Patch. In this section we define an N × N-colored operad Patch ⊆ mNSOp × NSOp. Its elements encode concrete (planar) patchworks of rectangles of size (p i , q i ) to form a rectangle of size (p, q).
Definition 3.5. Let Patch((q 1 , p 1 ), . . . , (q n , p n ); (q, p)) consists of the elements (X, J) ∈ mNSOp(q 1 , . . . , q n ; q)× NSOp(p 1 , . . . , p n ; p) such that Remark 3.6. Note that in order for Patch not to be empty, we need to allow a multiplication in one of its coordinates which is not present in the other coordinate.
This has a neat geometric interpretation as well: a (p, q)-rectangle has p inputs on the right-hand side, q inputs on top and a single output on respectively the bottom and the left-hand side 1 . . . q 1 . . .

p
We then interpret a patchwork (X, J) as an ordering of these rectangles: the first coordinate X represent the vertical ordering (from top to bottom) and the second coordinate J the horizontal ordering (from right to left). The multiplications m form a single exception: they appear only vertically, thus we draw them as flat rectangles, that is, having no horizontal input and output. From this perspective, the conditions impose planarity on the patchwork such that we have below < X above above ⊳ J below left ⊳ X right left < J right Note that when we write down a patchwork using rectangles, possible 'open spaces' can appear. Moreover, it is possible that multiple rectangles are vertically the 'lowest' elements due to the insertion of multiplication elements m. However, horizontally there can only appear a single most left rectangle which is (horizontally) connected to all other rectangles. We give an example.  Proof. Let (X, J) ∈ Patch((q 1 , p 1 ), . . . , (q n , p n ); (q, p)) and (X ′ , J ′ ) ∈ Patch((q ′ 1 , p ′ 1 ), . . . , (q ′ m , p ′ m ); (q i , p i )) and we set X ′′ := X • i X ′ and J ′′ := J • i J ′ . Let (α, β) be the extension of n by m at i, then for a, b ∈ m we compute and for c, d / ∈ Im(α) we compute For c / ∈ Im(α) and b ∈ m , we have Completely symmetrically, this also shows that c < J ′′ d =⇒ c ⊳ X ′′ d for c, d ∈ n + m − 1 .
We again compile the colored operad Patch to obtain a graded non-colored operad Patch s (n) ⊆ q1,...,qn,q p1,...,pn Patch((q 1 , p 1 ), . . . , (q n , p n ); (q, p)) where an element x ∈ Patch((q 1 , p 1 ), . . . , (q n , p n ); (q, p)) is graded as and Patch s (n) is generated as a k-module by the sequences of constant degree. Its composition is derived from Patch where it is set to 0 when the colors do not match. Note in particular that the S n -action on Patch(n) is affected by this grading: permuting two vertices i and j introduces a sign (−1) (qi+pi−1)(qj +pj −1) . Lemma 3.9. Patch s is a dg-suboperad of (mNSOp st ⊗ H NSOp s , (∂ D , Id)).
3.4. The morphism Patch s −→ End(sC GS (A)). In this section we make the GS-complex C GS (A) of a prestack A into a Patch s -algebra. We do so by making its underlying bicomplex C •,• (A) into a Patch-algebra. We first fix some notations. Remark 3.11. Note that we apply the reflection as we count the horizontal inputs of a patchwork from top to bottom (see example 3.7) instead of bottom to top (see further, example 3.22).
Given a patchwork (X, J) ∈ Patch, we now determine which simplices we need to fill in the 'open spaces' in between the rectangles. We first sketch the idea.
Given a simplex σ in U and a vertex a, we want to determine two sorts of simplices: for every vertical input i = I(a, b) for some vertex b, we want to determine a simplex σ(a, b) that we place between them. For the other vertical inputs 1 ≤ i ≤ q a , we determine a simplex σ a (i) to place on top of a at input i. To do so, we determine the set of left-most vertices which do not "surpass" the ith input and that lie higher than vertex a. In the drawing below, this set consists of the vertices e 1 , e 2 and e 3 . To calculate σ(a, b), we restrict this set to those vertices that still lie below vertex b, in this case, the vertices e 2 and e 3 .
We observe that each element of the GS complex composes in U the subsimplex corresponding to its horizontal inputs. Hence, using our auxiliary set, we contract the corresponding subsimplices and obtain σ(a, b) and σ a (i). Note that we have not yet treated the multiplications m. In order to do so, we have to add the following complexity. Let X = [I • n+1 m • n+1 . . . • n+1 m] where I is an indexed tree with n + k vertices, then we call a vertex a of I non-plugged in X if in X it is not inserted by a multiplication element m. We continue with the above chosen representation of X where a non-plugged is equivalent to stating a ≤ n. Remark 3.13. This is clearly independent of the representative I of X. Moreover, ↓ is for the given representative I the identity on n .
Next, we determine the auxiliary set.
Let σ be a p-simplex in U and A = (A 0 , . . . , A q ) (q + 1)-tuple of objects in A (U p ), then we define for every vertex a in I Θ a := θ ζJ,a(σ) a σ a (0) # A ζI,a(0) , . . . , σ a (q a ) # A ζI,a(qa) and for every i ∈ q a we make the compositions All these compositions together define Lemma 3.21. Construction 3.20 is independent of the representative I of X.
Proof. It suffices to verify the relation on the formal multiplication elements m in mNSOp. This follows directly from the associativity of the local composition m U of the category A(U ) for every U ∈ U.
Let us work out an example.
We compute (2) L(X, J)(θ 1 , . . . , θ n−1 , L(X ′ , J ′ )(θ n , . . . , θ n+m−1 )) σ (A) and show that it equals for σ ∈ N p (U) and A = (A 0 , . . . , A q ) objects in A(U p ). It is clear that per construction the blocks involved are composed according to X ′′ = X • n X ′ . Hence it suffices to verify that they correspond to the blocks Θ ′′ x in L(X ′′ , J ′′ ) and that the functors used to fill in the open spaces, agree.
First, for x a non plugged vertex of I ′′ in X ′′ , it is clear that Θ ′′ x is either Θ β(x) , or Θ ′ α −1 (x) evaluated at σ ′ = ζ J,n (σ). Next, we verify the simplices σ ′′ i (x). For its ith input, we have the following two cases: • if x does not lie in the image of (X ′ , J ′ ), then σ βx (i) = σ ′′ x (i) because if n ∈ min L βx (i) then it is replaced by α(r ′ ) for r ′ the root of J ′ for which holds ζ J ′′ ,α(r ′ ) = ζ J,n ζ J ′ ,r ′ .
Hence, the corresponding term in both calculations agrees. Next, we calculate σ ′′ (x, b) for b a child of x in (X ′′ , J ′′ ) that is not plugged. We again have three cases • if both x and b lie either outside or inside the image of (X ′ , J ′ ), then clearly σ ′′ (x, b) = σ(βx, βb) or σ ′ (α −1 x, α −1 b) for σ ′ = ζ J ′ ,n (σ) due to the previous reasoning and thus the terms agree. • if b lies in the image of (X ′ , J ′ ), i.e. b = α(b ′ ), and x does not, then b ′ is clearly the root of X ′ .
• if x lies in the image of (X ′ , J ′ ), i.e. x = α(x ′ ), and b does not, then min L ′′ α(x ′ ) (i) is the union of min L ′ x ′ (i) and min L n (i ′ ) for some i ′ .
Hence, we obtain in (2) the concatenation of σ(n, βb) and σ ′ x ′ (i) for σ ′ = ζ J,n (σ), which corresponds exactly to σ ′′ (x, b). In case either x or b is plugged, we possibly have to apply the functorial property of the restrictions, i.e.

Specifically, in the following cases
• let βx lie on top of n in (X, J) and ↓ x = α(y) for some vertex y of (X ′ , J ′ ).
In this case, Θ βx = m U ζ J,n (0) occurs in (2) and Θ ′′ x = m U ζ J,n ζ J ′ ,y (0) occurs in (3). Using functo- for an appropriate simplex τ . As a result they agree. Next, it is clear from the drawing that σ ′′ x (j) is the concatenation of σ βx (j) and τ . Moreover, for some vertex b, we have σ ′′ (x, b) as the concatenation of σ(βx, βb) and τ , except in the case that b is plugged as well. In the latter case, we can also pull Θ βb in (2) down to Θ βx and obtain m U ζ J,n ζ J ′ ,y (0) = Θ ′′ x = Θ ′′ b as in (3).
• The case where x lies in the image of (X ′ , J ′ ) such that ↓ x / ∈ Im(α), is analogous to the previous one. This finishes the proof. Insightfully, elements of Quilt can also be drawn as a stacking of rectangles in the plane, as extensively explained in [8, §3.2]. We will use Quilt in a fundamentally different way by switching the roles of its first and second component, and thus flipping the rectangles on their side. As such, we also draw the elements of Quilt on their side. We give an example. Note the double line above rectangle 4: this reflects the fact that 3 is not interposed, otherwise the corresponding word would be 142434.
Proof. Let u, v ∈ n , if u < I v, then u < T v and thus W = . . . v . . . u . . . and thus every occurrence of u in W is left of every occurrence of v in W . Hence, u ⊳ X v.
The other way around, if u < X v, then W = . . . u . . . v . . . u . . . and thus v ⊳ T u which is equivalent to v ⊳ I u.
As a direct consequence of Lemma 3.9 we have the following. Proof. Immediate from Theorem 3.24 and Proposition 3.27.
Our action of Quilt on the GS-complex of a prestack is orthogonal to the action constructed in [8,Thm. 4.26] in the case of presheaves, and thus also new for the latter case. This can be interpreted in a geometric sense: our action encodes a quilt Q = (W, T ) as a vertical patchwork according to W and a horizontal patchwork according to T . In Hawkins' action their roles are reversed, where the role of the multiplication is filled in by the identity 1 u,v : v * u * = (uv) * . This does not translate to the case of prestacks due to the occurring twists c u,v : v * u * −→ (uv) * .

Incorporating Twists
The morphism R : Quilt −→ End(sC GS (A)) from Corollary 3.28 only involves the multiplication m and the functors f of the data of a prestack (A, m, f, c). In this section, we will incorporate the twists c by adding a formal element with certain relations, resulting in the bounded powerseries operad Putting Theorems 4.10 and 4.17 together, we have thus endowed sC GS (A) with an L ∞ -structure. In the case of presheaves, this coincides on reduced and normalised cochains with the L ∞ -structure from [8,Thm. 7.13].
In the final section 4.5 we briefly discuss the relation of this structure with the deformation theory of the prestack A.

Powerseries operads.
In order to obtain an L ∞ -structure incorporating twists, we will make use of operads of formal power series.
be the S-submodule of bounded series which is graded by the series with coefficients of constant degree. ( Proof. These are straight-forward computations.   In particular, this means that for every n ≥ 2 the equation holds.
An important feature which we will need, is that we can write L 0 n as the antisymmetrization of elements P 0 n . Namely, we set P 0 n := In fact, the L ∞ -relations translate to the following where − denotes 'free k-module generated by−'.

Similar to how we drew elements from mNSOp as trees with vertices plugged by m, elements of Quilt[c]
can be drawn as quilts with rectangles plugged by c. For example, we have Note that depending on where we plug the elements c a sign is added.
This enables us to define the following. Set (a derivation by an element) which will be the new differential. The main theorem of this section is the following.  Remark 4.12. Note that we have used 2 out of the three conditions on c to prove this lemma.
The following lemma extends the L ∞ -equation of (L 0 n ) n for higher L r n .
Lemma 4.13. For n ≥ 1 and r ≥ 0, we have Proof. By applying ∂ and using equation (5) and ∂(c) = 0, we deduce Given 1 ≤ y 1 < . . . < y r ≤ n + r and 1 ≤ x ≤ p, we have a subdivision into two groups where • ys−s+1 c is inserted into either P 0 p or P 0 q . Hence, if we are also given a permutation σ ∈ S n , then we show that there exists unique integers i + j = r, k + l = n + 1, indices 1 ≤ z 1 < . . . < z i ≤ k + i and 1 ≤ z ′ 1 < . . . < z ′ j ≤ j + l, permutations τ ∈ S k+i , τ ′ ∈ S j+l and a shuffle χ ∈ Sh k−1,l such that In this case, we obtain We also compute the sign of y 1 , . . . , y r : let θ and θ ′ be the shuffles such that then we obtain that On the other hand, and thus we have We then define the (k − 1, l)-shuffle χ = (a 1 < . . . < a k−1 , b 1 < . . . < b l ) and It is then easy to see that Note that τ and z 1 , . . . , z i determine x and i 1 uniquely. Hence, given z 1 , . . . , z i , z ′ 1 , . . . , z ′ j , k, l we can uniquely determine y 1 , . . . , y r , p, q, n, r in the above manner. In order to show that equation (6)  As such, we can compute which completes the proof.
Lemma 4.14. L n are skew symmetric and ∂ ′ is a differential making Quilt b [[c]] into a dg-operad.
Proof. It is clear from the definition of L r n that they are skew symmetric and thus also L n . Per definition ∂ L1 is a derivation by construction and so is ∂, and thus so is ∂ ′ . It is clear from the definition of c that ∂ ′ (c) = 0. Hence, we only need to show that ∂ ′ ∂ ′ (Q) = 0 for every Q ∈ Quilt. Using lemma 4.11 and 4.13 we first prove that −∂L 1 = L 1 • 1 L 1 . Namely, we compute adding them gives Next, we compute ∂ L1 ∂ L1 Q. As L 1 has only a single input and has degree −1, we have for i = j that Hence, by also using using equation (9), we obtain As ∂∂Q = 0 we thus obtain ∂ ′ ∂ ′ Q = 0.
Proof of Theorem 4.10. We need to show for every n ≥ 2 that the equation holds, which is equivalent to This is equivalent to showing for every r ≥ 0 that the equation k,l≥1 (l,j) =(1,0) =(k,i) (l,j) =(0,1) =(k,i) holds, which follows from lemma 4.13 and L i 0 = 0 for i ≥ 0 (lemma 4.11).
Proof. This follows from C GS (A) having only non-negative bidegree. Namely, given We verify that R c ∂ ′ Q = ∂ d R c Q for Q ∈ Quilt. It suffices to verify that R c L j 1 = d j as ∂ corresponds to ∂ d0 . Let θ ∈ C p,q (A) and σ ∈ N p+j (U) and note that we write |θ| for the degree p + q of θ in C GS (A).
There exists p colorings (X i , J i ) p i=1 of P 0 2 occuring in the second term of (11), given by the patchworks for i = 1, . . . , p. They correspond to d p+1−i 1 and we verify their signs: we have sgn Q (X i , J i ) = sgn W (X i ) sgn T (J i ) determined by the two shuffles Hence, we obtain the sign Step 2: R c L j 1 = d j First, we name the terms of d j and write Next, we note that the only non-vanishing term of L j 1 is given by (−1) for t 1 < . . . < t j−1 in q and J an indexed tree coloring T . As a result, we have that θ, c, . . . , c) which sums over the terms sgn QT (X, J ′ ) (−1) We finish the proof by showing that a formal path r and a shuffle β ∈ S q+1−j,j−1 correspond uniquely to such a binary tree T and a coloring (X, J ′ ) of Q T such that Given a formal path r = (r 1 , . . . , r j−1 ) = ((τ 1 , i 1 ) , . . . , (τ j−1 , i j−1 )) we first define T and its coloring J as trees with vertex set {3, . . . , j + 1} inductively: • In the degenerate case j = 2, r is uniquely determined and we set T to be the one vertex tree and J the empty function. • For j > 2, let (T 0 , J 0 ) be the indexed tree corresponding to ((τ 1 , i 1 ), . . . , (τ j−2 , i j−2 )), in order to add vertex j + 1 we have three cases (1) if i j−1 < i j−2 , then set j ⊳ J (j + 1) and start over with (i j−1 , i j+1 ) (2) if i j−1 = i j−2 or i j−1 = i j−2 + 1, then let (j, j + 1) ∈ E T and set resp. J(j, j + 1) = 2 or = 1. (3) if i j−1 > i j−2 + 1, then set (j + 1) ⊳ J j and start over with (i j−1 , i j+1 − 1) which we can draw as follows Clearly, this process is reversible: given (T, J) we obtain a unique sequence r ∈ P (R p (σ)). By identifying the shuffle β ∈ Sh q+1−j,j−1 and X via t l = β(q − j + 1 + l), we clearly obtain that L(X, The remaining work is to verify the signs: sgn QT (X, J ′ ) consists of three components where (−1) σ = (−1) q as it corresponds to the shuffle 0, q, 0, . . . , 0, 1, p − 1, 1, . . . , 1 0, 1, q, p − 1, 0, 1, . . . , 0, 1.
, for t = q − j + 1, into the word We observe that shuffling the second part (2) almost corresponds to the shuffle β. However, there is in every interval β(s) 2 , . . . , (β(s + 1) − 1) 2 exactly one element too many. We remedy this by moving 1 (j−2) one place to the right, then 1 (j−3) two place, and so on. As such, its corresponding sign is (−1) whose sign is (−1) (j−1)(q+2) . Hence, we obtain that Next, we determined sgn T ′ (J ′ ) as the sign of the shuffle where [C J ] denotes the word obtained from the indexed tree J. We will show that the sign corresponding to the shuffle χ : As a consequence, we obtain that and thus that sgn(X, J ′ ) = (−1) β+r+q+j−1 (−1) (j−1)j 2 +(j−1)(p+q−1) .
Hence, we have which completes the proof.
We compute χ inductively: we have that [C J ] = A1 (j+1) B for certain words A and B and let χ 0 denote the shuffle AB 1 3 . . . 1 j . By induction we know that (−1) χ0 = (−1) r0+j−2 for the formal path r 0 = (r 1 , . . . , r j−2 ) and (−1) r = (−1) r0+ij−1 where r j−1 = (σ, i j−1 ). Moreover, we have (−1) χ = (−1) χ0+|B| where |B| denotes the length of B. We determine |B|. First, observe that the sequence associated to two indexed trees 3 4 1 3 4 2 and is respectively 1 h4 1 h3 and 1 h3 1 h4 . Thus, let 3 = v 1 < T . . . < T v t = j + 1 be the unique chain of vertices from the root of T to j + 1, then we can define the numbers • l as the the number of vertices to the right of vertex j + 1 in T , • k as the number of vertices in the above chain such that J(v g , v g+1 ) = 1. We then easily compute that • the height i j−1 of r j−1 = (∂ ij−1 σ, i j−1 ) is exactly (1 + l) + k, • length of B is l + k and thus by induction we obtain (−1) χ = (−1) χ0+|B| = (−1) χ0+ij−1+1 = (−1) r+(j−1) where the last equality follows from induction. However, the resulting L ∞ -algebra structure, in the case of presheaves, does correspond to the one obtained from [8,Thm. 5.6,Thm. 7.13]. This is essentially due to the multiplication of the prestack being unital. A mQuilt-algebra structure also induces a Gerstenhaber-algebra structure on cohomology [8,Thm. 6.11]. Writing down the relevant quilts, it is also easy to see that both mQuilt-algebra structures (ours and [8, Thm. 5.6]) induce the same Gerstenhaber-algebra structure on cohomology. We consider the set of (degree 1) Maurer-Cartan elements M C(C) = {θ ∈ C 1 | M C(θ) = 0} and for the appropriate notion of gauge equivalence (see [9]), we consider Putting Theorems 4.10 and 4.17 together, sC GS (A) is endowed with an L ∞ -structure which can be used to obtain higher order versions of these results.
For normalised, reduced cochains, it will be convenient to express the MC-equation in terms of the unsymmetrised components (P n ) n≥1 from Definition 4.5. The following characteristic-free expression of the MC-equation should be seen as the counterpart of the equation M C(θ) = d(θ) + θ{θ} for the first brace operation (or "dot product") on the Hochschild complex of an algebra. Note that we omit writing R c and consider everything as elements of End(sC GS (A)).
In particular, the MC-equation is quartic.
In this case, only the elements c ′ or c, and m ′ can be inserted in Q respectively in 1 and 2, with 2 a child of 1 in T . As c ′ is reduced, this means that Q(θ, . . . , θ) = 0 for n ≥ 4.
As c is not necessarily reduced, more quilts are possible. Note that we write θ[i] to refer to the element θ inserted in vertex i. The above reasoning still applies and we can once more apply this reasoning to obtain for n ≥ 6 that W = 12324546 . . . 42 and 1 and 3 have two children in T .  This means that 2 and 4 need be inserted by m ′ and hence 3 and 1 by either c ′ or c. In case either one is c ′ , we already know it is zero as c ′ is reduced. Thus, consider vertex 1 and c inserted by c, then, as m ′ [4] is a child of c [3] in T , c [3] is the unit of the corresponding category A(U ) and it is plugged into m ′ [2] which is normalised, whence we obtain that Q(θ 1 , . . . , θ n ) = 0 for n ≥ 6. Hence, P 0 n (θ 1 , . . . , θ n ) = 0 for n ≥ 6.
Combining the above reasonings, we have that P n (θ, . . . , θ) = 0 for n > 5. In the case n = 5 there has to be at least one c present and thus the non-zero terms are contained in P 4 . Proof. Using the fact that θ = (m ′ , f ′ , c ′ ) ∈ C 2 GS (A) is reduced and normalised, we compute M C(θ) and look at each component M C(θ) [p,q] ∈ C p,q (A) for p + q = 3. We will use that c 1,− = 1 = c −,1 , m is unital and that (m, f, c) satisfy the axioms of a prestack. Note that for a cochain θ = (θ σ (A)) σ,A , we omit writing the set of objects A explicitly where possible in order to lighten the equations below.
For (p, q) = (1, 2), let u : U 0 → U 1 in U and A = (A 0 , A 1 , A 2 ) objects in A(U 1 ), then we compute For (p, q) = (2, 1), let σ = (U 0 u1 → U 1 u2 → U 2 ) be a 2-simplex in U and A = (A 0 , A 1 ) objects in A(U 2 ), then we compute dθ σ [2,1] In this case, ζ I ′ ,j−i+1 determines where the leaves of j are placed in I ′ and ζ I,i determines where the leaves of i in I are placed. As a result, we see that ζ I ′′ ,j = ζ I,i • ζ I ′ ,j−i+1 which determines where the leaves of j are put in I ′′ . (3) If j ≥ i and j does not lie in the image of I ′ in I ′′ , then j − m + 1 lies in I above i. In this case, the subtrees I j−m+1 and I ′′ j are equal. If the parent a of j in I ′′ does not lie in Im(I ′ ), then we clearly have ζ I ′′ ,j = ζ I,j−m+1 . If a lies in Im(I ′ ), then ζ I ′′ ,a = ζ I,i • ζ I ′ ,a−i+1 due to the previous case. a i j I ′ Moreover, the index I ′′ (a, j) then equals I(i, j) − ζ I ′ ,a−i+1 (0). Hence, we have that ζ I ′′ ,j = ζ I,j−m+1