Abstract
We introduce an operad \({{\,\mathrm{Patch}\,}}\) 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 \({{\,\mathrm{Quilt}\,}}\) 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 \({{\,\mathrm{Quilt}\,}}\) in a fundamentally different way (even for presheaves) in order to allow for an extension to prestacks.
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1 Introduction
The deformation theory of algebras due to Gerstenhaber furnishes the guiding example for algebraic deformation theory. For an algebra A, the Hochschild complex \({\mathbf {C}}(A)\) 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 \({\mathbf {C}}(A)\) is a homotopy Galgebra [6]. This structure, which captures both the brace operations and the cup product, is a special case of a \(B_{\infty }\)structure [7]. Importantly, this purely algebraic structure constitutes a stepping stone in the proof of the Deligne conjecture, proving \({\mathbf {C}}(A)\) to be an algebra over the chain little disk operad [10, 13].
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 GScomplex \({\mathbf {C}}({\mathcal {A}})\) of a presheaf \({\mathcal {A}}\) does not control deformations of \({\mathcal {A}}\) as a presheaf, but rather as a twisted presheaf, see for instance [2, 11]. 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 2category 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 \({\mathcal {A}}\), they construct a homotopy equivalence \(\mathbf{C} _{GS}({\mathcal {A}}) \cong \mathbf {CC}({\mathcal {A}}!)\) between the Gerstenhaber–Schack complex \(\mathbf{C} _{GS}({\mathcal {A}})\) and the Hochschild complex \(\mathbf {CC}({\mathcal {A}}!)\) of the Grothendieck construction \({\mathcal {A}}!\) of \({\mathcal {A}}\). Through homotopy transfer, this endows the GScomplex with an \(L_{\infty }\)structure. This result improves upon the existence of a quasiisomorphism, 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 GScomplex does not possess a \(B_{\infty }\)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 \(L_{\infty }\)structure. For particular types of presheaves, explicit \(L_{\infty }\)structures on the GScomplex have been established by Frégier et al. [3] and by Barmeier and Frégier [1].
Let \({{\,\mathrm{Brace}\,}}\) be the brace operad and \({{\,\mathrm{F_{2}S}\,}}\) the homotopy Goperad. In the case of a presheaf \(({\mathcal {A}}, m, f)\), in [8], Hawkins introduces an operad \({{\,\mathrm{Quilt}\,}}\subseteq {{\,\mathrm{F_{2}S}\,}}\otimes _H {{\,\mathrm{Brace}\,}}\) which he later extends to an operad \({{\,\mathrm{mQuilt}\,}}\) acting on the GScomplex. These operads are naturally endowed with \(L_{\infty }\)operations as desired. The action of \({{\,\mathrm{Quilt}\,}}\) on the GScomplex considered by Hawkins only involves the restriction functors f of the presheaf, the multiplication m being incorporated later on in \({{\,\mathrm{mQuilt}\,}}\). 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 \(L_{\infty }\)structure on \(\mathbf{C} _{GS}({\mathcal {A}})\) in the case of a general prestack \(({\mathcal {A}}, m, f, c)\) with twists c. As part of our solution, we use \({{\,\mathrm{Quilt}\,}}\) in a fundamentally different way in relation to the GScomplex, but still allowing us to make use of the naturally associated \(L_{\infty }\)structure from [8]. In Sect. 3, we capture the higher structure of \(\mathbf{C} _{GS}({\mathcal {A}})\) by introducing the new operad \({{\,\mathrm{Patch}\,}}\subseteq {{\,\mathrm{mNSOp}\,}}\otimes _H {{\,\mathrm{NSOp}\,}}\) over which \(\mathbf{C} _{GS}({\mathcal {A}})\) is shown to be an algebra (see Theorem 3.24). Here, \({{\,\mathrm{(m)NSOp}\,}}\) is the operad of nonsymmetric operads (with multiplication).
In [6], Gerstenhaber and Voronov obtain a brace algebra structure on an operad and a homotopy Galgebra structure on an operad with multiplication. Based upon the expression of these results in terms of the underlying operads \({{\,\mathrm{NSOp}\,}}\) and \({{\,\mathrm{mNSOp}\,}}\) in Sect. 2, we construct a morphism \({{\,\mathrm{Quilt}\,}}\longrightarrow {{\,\mathrm{Patch}\,}}_s\) (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
which incorporates the multiplication m and the restrictions f of \({\mathcal {A}}\).
In Sect. 4, we extend the action R to
in order to incorporate the twists (see Theorem 4.17). Here, \({{\,\mathrm{Quilt_{b}}\,}}[[c]]\) is obtained from an operad of formal power series. Further, we establish \(L_{\infty }\)operations on \({{\,\mathrm{Quilt_{b}}\,}}[[c]]\) extending those on \({{\,\mathrm{Quilt}\,}}\) from [8] (see Theorem 4.10) by adding an infinite series of higher components containing twists. Under the action of \({{\,\mathrm{Quilt}\,}}\) this neatly corresponds to and extends the differential on \(\mathbf{C} _{GS}({\mathcal {A}})\) obtained in [2]. In the final Sect. 4.5 we briefly discuss the relation of this \(L_{\infty }\)structure with the deformation theory of the prestack \({\mathcal {A}}\).
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 \({{\,\mathrm{Quilt}\,}}\) gives rise to an \(L_{\infty }\)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 \({{\,\mathrm{Quilt}\,}}\) 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 \({{\,\mathrm{Quilt}\,}}\) on the GS complex. In analogy with the way in which Hawkins extends his action from \({{\,\mathrm{Quilt}\,}}\) to \(\mathrm {mQuilt}\) in order to incorporate the multiplications m, we establish an extension from \({{\,\mathrm{Quilt}\,}}\) to \({{\,\mathrm{Quilt_{b}}\,}}[[c]]\) 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 \(\mathbf{C} _{GS}({\mathcal {A}}) \cong \mathbf {CC}({\mathcal {A}}!)\) from [2] operadically, showing in particular that the \(L_{\infty }\)structure from [2] and the one established in the present paper actually coincide.
2 Gerstenhaber–Voronov operadically
In the seminal paper [6], Gerstenhaber and Voronov define a bracealgebra structure on the totalisation of a nonsymmetric operad \({\mathcal {O}}\). Moreover, in presence of a multiplication, they define a homotopy Galgebra structure on \({\mathcal {O}}\) incorporating both the cup product and the Gerstenhaberbracket.
In this section we describe the morphisms of operads underlying these results. To this end, in Sect. 2.3, we recall the colored operad \({{\,\mathrm{NSOp}\,}}\) encoding nonsymmetric operads, and we describe the natural extension \({{\,\mathrm{mNSOp}\,}}\) which adds a multiplication. Let \({{\,\mathrm{NSOp}\,}}_s\) and \({{\,\mathrm{mNSOp}\,}}_{st}\) be their totalised graded (uncolored) operads with suspended, respectively standard degree ( see Sects. 2.5 and 2.6). Let \({{\,\mathrm{Brace}\,}}\) be the brace operad (see Sect. 2.1) and \({{\,\mathrm{F_{2}S}\,}}\) the Gerstenhaber–Voronov operad encoding homotopy Galgebras (see Sect. 2.2). The main goal of this section is the definition of morphisms of dgoperads
and
(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 \({{\,\mathrm{(m)NSOp}\,}}\longrightarrow \text {Multi}\Delta \) landing in the multicategory associated to the simplex category \(\Delta \) (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 \({{\,\mathrm{Brace}\,}}\)
Throughout, we work over a fixed commutative ground ring k.
The operad \({{\,\mathrm{Brace}\,}}\) 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 \(\le _T\), and the “horizontal” partial order on \(V_T\) by \(\unlhd _{T}\). For \((u,v) \in E_{T}\) we call u the parent of v and v a child of u.
For \(n\in {\mathbb {N}}\), put \([n] := \{ 0,\ldots , n\}\) and \(\langle n \rangle := \{ 1,\ldots ,n \}\).
Let \({{\,\mathrm{Tree}\,}}(n)\) denote the set of trees with vertex set \(\langle n \rangle \) and let \({{\,\mathrm{Brace}\,}}(n)\) be the free kmodule on \({{\,\mathrm{Tree}\,}}(n)\) endowed with the \({\mathbb {S}}_{n}\)action given by permuting the vertices, i.e., \(T^{\sigma }\) is the tree defined by replacing vertex i in T by \(\sigma ^{1}(i)\). The operadic composition on \({{\,\mathrm{Brace}\,}}\) is based upon substitution of trees, as follows. For trees \(T \in {{\,\mathrm{Tree}\,}}(m)\), \(T' \in {{\,\mathrm{Tree}\,}}(n)\) and \(1 \le i \le m\), we denote by \(Ext(T,T',i) \subseteq {{\,\mathrm{Tree}\,}}(m + n 1)\) the set of trees extending T by \(T'\) at i (that is, \(U \in Ext(T,T',i)\) 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 \(\langle n \rangle \overset{\alpha }{\hookrightarrow } \langle n+m1\rangle \overset{\beta }{\twoheadrightarrow } \langle m \rangle \) acting on the vertices, where \(\alpha \) embeds n vertices consecutively and \(\beta \) contracts the image of \(\alpha \) to the vertex i. We call the pair \((\alpha ,\beta )\) the extension of m by n at i. We refer to [8, §2.2] for more details.
2.2 The operad \({{\,\mathrm{F_{2}S}\,}}\)
The operad \({{\,\mathrm{F_{2}S}\,}}\) encodes homotopy Galgebras [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 \(W = a_1a_2 \ldots a_k\), correspondiong to the function \(W: \langle k \rangle \longrightarrow A: i \longmapsto a_i\), the ith letter of W is the couple \((i, a_i)\). We will often identify a word with its graph \(W = \{ (i, a_i) \,\, \,\, i \in \langle k \rangle \} \subseteq \langle k \rangle \times A\), writing \((i, a_i) \in W\).
For \(a \in A\), a letter \((i,a) \in W\) 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 \(a \in A\) is interposed in W if \(W = \cdots ba \ldots b \ldots \). The length of \(W: \langle k \rangle \longrightarrow A\) is \(W = k\).
By definition, \({{\,\mathrm{F_{2}S}\,}}(n)\) is the free kmodule generated by the words W over \(\langle n \rangle \) such that:

(1)
\(W: \langle k \rangle \longrightarrow \langle n \rangle \) is surjective,

(2)
\(W \ne \cdots {u}{u} \ldots \) (nondegeneracy), and

(3)
For any \(u\ne v \in \langle n \rangle \), \(W \ne \cdots {u} \ldots {v} \ldots {u} \ldots {v} \ldots \) (no interlacing).
The set \({{\,\mathrm{F_{2}S}\,}}(n)\) is graded by setting \(deg(W) := W  n\) and naturally carries a \({\mathbb {S}}_{n}\)action by permuting letters, i.e. \(W^{\sigma } = \sigma ^{1}W\).
For a word \(W\in {{\,\mathrm{F_{2}S}\,}}(n)\) and \(u \in \langle n \rangle \), let \((i_u, u)\) be the first occurrence of u in W. Then we obtain a total order \(u \downarrow v \iff i_u \le i_v\) on \(\langle n \rangle \).
The operadic composition on \({{\,\mathrm{F_{2}S}\,}}\) is based upon merging of words, as follows. For words \(W\in {{\,\mathrm{F_{2}S}\,}}(m), W' \in {{\,\mathrm{F_{2}S}\,}}(n)\) and \(1 \le i \le m\), we denote by \(Ext(W,W',i) \subseteq {{\,\mathrm{F_{2}S}\,}}(m+n1)\) the set of extensions of W by \(W'\) at i (that is, \(X \in Ext(W,W',i)\) 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 \(W\in {{\,\mathrm{F_{2}S}\,}}(m)\) and let int(W) be the set of interposed elements of \(\langle m \rangle \) ordered by their first occurrence in W. For \(X \in Ext(W,W',i)\) the relabelling gives rise to two maps \(\alpha :int(W')\longrightarrow int(X)\) and \(\gamma : int(W) \longrightarrow int(X)\) where \(\gamma := \beta ^{1}\) except if i is interposed in W, then \(\gamma (i) := \alpha (a)\) for (1, a) the first letter of \(W'\).
As \(int(W) = deg(W)\), an extension X defines a unique \((deg(W),deg(W'))\)shuffle \(\chi \) and we define
Moreover, it is possible to talk about the boundary of a word, inducing a differential.
Boundary. Given a word \(W \in {{\,\mathrm{F_{2}S}\,}}(n)\) and a letter (i, a) of W for which a is repeated in W, then define \(\partial _{i}W\in {{\,\mathrm{F_{2}S}\,}}(n)\) as the word obtained by deleting the letter (i, a) from W (and relabelling). If a is not repeated, then set \(\partial _{i}W= 0\).
Sign of Deletion. Given a word \(W \in {{\,\mathrm{F_{2}S}\,}}(n)\) of length k, then we define \({{\,\mathrm{sgn}\,}}_{W}:\langle k \rangle \longrightarrow \{1,1\}\) by setting \({{\,\mathrm{sgn}\,}}_{W}(i)= (1)^{k}\) if \((i,a_i)\) is the kth caesura of W, and otherwise \({{\,\mathrm{sgn}\,}}_{W}(i)= (1)^{k+1}\) if it is the last occurrence, but the previous occurrence is the kth caesura of W.
The \({\mathbb {S}}\)module \({{\,\mathrm{F_{2}S}\,}}\) defines a dgoperad with operadic composition given by
and boundary given by
The following lemma, which we include for the convenience of the reader, shows how \({{\,\mathrm{F_{2}S}\,}}\) encodes the algebraic operations of a homotopy Galgebra.
Notations. To avoid too large expressions, we leave out certain bracketings by setting as default the bracketing
Moreover, we compress the following
Lemma 2.1
Let \(M_{2}:= 12, M_{1,0} = 1\) and \(M_{1,k} := 121\ldots 1(k+1) 1\) for \(k\ge 1\), then \({{\,\mathrm{F_{2}S}\,}}\) is generated by these elements and the following holds

(1)
\(\partial (M_{2}) = 0\)

(2)
\(\partial (M_{1,k}) = (M_{2}^{(12)}\circ _{2}M_{1,k1}) + \sum _{i=2}^{k} (1)^{i} M_{1,k1} \circ _{i} M_{2} + (1)^{k+1} M_{2} \circ _{1} M_{1,k1}\)
Proof
It is a straightforward computation to determine that \(M_{2}\) and \(M_{1,k}\) satisfy these relations.
Let \(W\in {{\,\mathrm{F_{2}S}\,}}(n)\), we then show that it lies in the suboperad generated by \(M_{2},(M_{1,k})_{k\ge 1}\) 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 nonempty word \(W_{i}'\in {{\,\mathrm{F_{2}S}\,}}(n_{i})\) under the map \(\gamma _{i} : \langle n_{i} \rangle \longrightarrow \langle n \rangle \), except \(W_{k+1}\) which is possibly empty. Due to no interlacing we also know that the images \({{\,\mathrm{Im}\,}}(\gamma _{i})\) are pairwise disjoint. Hence, we can apply a permutation to assume that \(\max {{\,\mathrm{Im}\,}}(\gamma _{i}) < \min {{\,\mathrm{Im}\,}}(\gamma _{i'})\) holds for every \(i < i'\). In this case, we have that
if \(W_{k+1} = \emptyset \), and
otherwise. By induction, this shows that W is generated by \(M_{2}\) and \((M_{1,k})_{l\ge 0}\). \(\square \)
2.3 The operads \({{\,\mathrm{NSOp}\,}}\) and \({{\,\mathrm{mNSOp}\,}}\)
It is wellknown that nonsymmetric operads can be encoded using a colored operad \({{\,\mathrm{NSOp}\,}}\) which can be defined using indexed trees, that is, for \(q_{1},\ldots ,q_{n} \in {\mathbb {N}}\) and \(q'= 1+ \sum _{i=1}^{n} (q_{i}1)\), \({{\,\mathrm{NSOp}\,}}(q_{1},\ldots ,q_{n};q')\) is the set of pairs (T, I) where \(T \in {{\,\mathrm{Tree}\,}}(n)\) and \(I:E_{T}\longrightarrow {\mathbb {N}}\) a function such that

For \((u,v)\in E_{T}\), \(1\le I(u,v) \le q_{u}\)

\((t,u),(t,v)\in E_{T}\) and \(u \vartriangleleft _{T}v \Longrightarrow I(t,u) < I(t,v)\)
We will often write I to denote the indexed tree (T, I). Moreover, \({{\,\mathrm{NSOp}\,}}\) is generated by those trees with a single edge, that is,
for every \(q_1,q_2\) and \(1 \le i \le q_1\), with the following pair of relations

(I)
for \(1 \le j \le q_2\) and \(1 \le i \le q_1\),

(II)
for \(1\le i<k \le q_1\).
Note that these are the wellknown associativity relations for nonsymmetric operads.
Definition 2.2
Let \({{\,\mathrm{mNSOp}\,}}\) be the \({\mathbb {N}}\)colored operad generated by \({{\,\mathrm{NSOp}\,}}\) and an element \(m\in {{\,\mathrm{mNSOp}\,}}(;2)\) satisfying the relation
Remark 2.3
We often write where we have already filled in the plugged in m’s.
More explicitly, every representative of an element \(X\in {{\,\mathrm{mNSOp}\,}}(q_{1},\ldots ,q_{n};q)\) is of the form \(I \circ _{i_{1}} m \circ _{i_{2}} \ldots \circ _{i_{k}} m\) for \(I \in {{\,\mathrm{NSOp}\,}}\) and appropriate \(i_{1},\ldots ,i_{k} \in {\mathbb {N}}\). Due to equivariance, we can always consider a representative of X of the form
for \(I \in {{\,\mathrm{NSOp}\,}}(q_{1},\ldots ,q_{n},2,\ldots ,2;q)\).
Lemma 2.4
Let \(X = [ I \circ _{n+1} m \circ _{n+1} \cdots \circ _{n+1} m] \in {{\,\mathrm{mNSOp}\,}}(q_{1},\ldots ,q_{n};q)\), the partial orders \(<_{I}\) and \(\vartriangleleft _{I}\) on \(\langle n \rangle \) are independent of the representative of X. We denote them by \(<_{X}\) and \(\vartriangleleft _{X}\).
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 \(\vartriangleleft \) for trees, also hold for \(<_{X}\) and \(\vartriangleleft _{X}\). In particular, if the lemma holds for X and \(X'\) and \((\alpha ,\beta )\) is the extension of n by m at i, then for \(a,b \notin {{\,\mathrm{Im}\,}}(\alpha )\) we have
Now, let such that \(X = X_{0,1} \circ _{n+1} m \circ _{n+1} m = X_{0,2} \circ _{n+1} m \circ _{n+1} m\), then we have by induction that the lemma holds for \(X_{0}\). Moreover, we have for \(a,b\in \langle n \rangle \) that
which proves the lemma for X. \(\square \)
2.4 The morphisms \({{\,\mathrm{(m)NSOp}\,}}\longrightarrow \text {Multi}\Delta \)
Let \({\mathcal {C}}\) be a small category. We denote by \(\mathrm {Multi} {\mathcal {C}}\) the \(\mathrm {Ob}({\mathcal {C}})\)colored operad for which \(\mathrm {Multi} {\mathcal {C}}(c_1, \ldots , c_n; c)\) is freely generated as a kmodule by ntuples \((\zeta _{1},\ldots ,\zeta _{n})\) of \({\mathcal {C}}\)morphisms with \(\zeta _i: c_i \longrightarrow c\), \({\mathbb {S}}_{n}\) acts by permutating labels, and composition is defined in the obvious way.
Let \(\Delta \) be the simplex category. Next, we construct a morphism of operads
by associating to every indexed tree I in \({{\,\mathrm{NSOp}\,}}(q_{1},\ldots ,q_{n};q)\) a ntuple \(\zeta _I\) in \(\text {Multi}\Delta (q_{1},\ldots ,q_{n};q)\) 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 \(E_i\in {{\,\mathrm{NSOp}\,}}\) and show that it respects the relations.
Construction 2.5
Let \(E_i\in {{\,\mathrm{NSOp}\,}}(q_1,q_2;q_1 +q_2 1)\) for \( 1 \le i \le q_1\), then we define
Then, \(\zeta _{E_i}\in \text {Multi}\Delta (q_1,q_2;q_1+q_21)\), that is, it is a tuple of nondecreasing maps. Moreover, if \(q_2 > 0\), then these are strictly increasing.
We will employ it as in the following example.
Example 2.6
Let \({\mathcal {A}}\) be a klinear category, then its Hochschild complex is defined as
For a Hochschild cochain \(\phi \in \mathbf{C} ^n({\mathcal {A}})\) and a nsimplex \(A_0 \overset{a_1}{\leftarrow } A_1 \overset{a_2}{\leftarrow } \cdots \overset{a_{n1}}{\leftarrow }A_{n1} \overset{a_n}{\leftarrow } A_n\) in \({\mathcal {A}}\), we have that \(\phi ^{A_0,\ldots ,A_n}(a_{1},\ldots ,a_{n}) \in {\mathcal {A}}(A_n,A_0)\).
Let \(\phi _1 \in \mathbf{C} ^{q_1}({\mathcal {A}})\) and \(\phi _2 \in \mathbf{C} ^{q_2}({\mathcal {A}})\), then each \(E_i \in {{\,\mathrm{NSOp}\,}}(q_1,q_2;q_1+q_21)\) determines a cochain \(\phi _1 \circ _i \phi _2 \in \mathbf{C} ^{q_1+q_21}({\mathcal {A}})\) as follows
which we can visualize using ncorollas
Lemma 2.7
Construction 2.5 extends to a morphism of operads
Proof
It suffices to verify the relations (I) and (II) of \({{\,\mathrm{NSOp}\,}}\). These are two simple computations and thus we only verify the first relation (I) as an example. Let \(\zeta := \zeta _{E_i} \circ _2 \zeta _{E_j}\) denote the lefthand side, then we compute
On the other hand, we compute the righthand side \(\zeta ' := \zeta _{E_{i1+j}} \circ _1 \zeta _{E_i}\) and obtain
\(\square \)
Remark 2.8
In appendix A we have added a generatorfree 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.
Example 2.9
Consider the indexed tree
then we compute \(\zeta _{I}\) and obtain
It is also possible to associate to an element of \({{\,\mathrm{mNSOp}\,}}\) an element of \(\text {Multi}\Delta \).
Lemma 2.10
Let \(X = [ I \circ _{n+1} m \circ _{n+1} \cdots \circ _{n+1} m] \in {{\,\mathrm{mNSOp}\,}}(q_{1},\ldots ,q_{n};q)\), then \(\zeta _{I,t}:[q_{t}]\longrightarrow [q]\) for \(t\in \langle n \rangle \) is independent of the representative I of X.
In this case, we write \(\zeta _{X}\).
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 such that \(X = X_{0,1} \circ _{n+1}m \circ _{n+1} m = X_{0,2} \circ _{n+1}m \circ _{n+1} m\), then by induction and Lemma 2.7 we have for \(t\in \langle n \rangle \) that
which proves the lemma. \(\square \)
Proposition 2.11
We have morphisms of operads
and its extension
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.
Example 2.12
Consider the indexed 2corolla and its permuted form as elements of \({{\,\mathrm{NSOp}\,}}(2,0,0;0)\), then they have the same image in \(\text {Multi}\Delta \). Note that this example holds for both \({{\,\mathrm{NSOp}\,}}\) and \({{\,\mathrm{mNSOp}\,}}\).
Hence, we can consider \({{\,\mathrm{mNSOp}\,}}\) as a finer operad than \(\text {Multi}\Delta \) and thus encoding more information.
2.5 The morphism \({{\,\mathrm{Brace}\,}}\longrightarrow {{\,\mathrm{NSOp}\,}}_s\)
In order to define the morphism \(\phi : {{\,\mathrm{Brace}\,}}\longrightarrow {{\,\mathrm{NSOp}\,}}_s\) properly we need to compile the colored operad \({{\,\mathrm{NSOp}\,}}\) into a graded noncolored operad
where an element \(x\in {{\,\mathrm{NSOp}\,}}(p_{1},\ldots ,p_{n};p)\) is graded as \(x= \sum _{i=1}^{r} (p_{i}1)  (p1) = 0\) (this is the suspended grading, whence the subscript) and \({{\,\mathrm{NSOp}\,}}_s(n)\) is the subspace generated by sequences of elements with constant grading. The composition on \({{\,\mathrm{NSOp}\,}}_s\) is derived from the composition of \({{\,\mathrm{NSOp}\,}}\) where it is set to 0 when the colors do not match. Note in particular that the \({\mathbb {S}}_{n}\)action on \({{\,\mathrm{NSOp}\,}}_s(n)\) is affected by this grading: permuting two vertices i and j introduces the signs \((1)^{(p_{i}1)(p_{j}1)}\).
Definition 2.13
Let \((T,I) \in {{\,\mathrm{NSOp}\,}}(p_{1},\ldots ,p_{n};p)\), then (T, I) is a coloring of T and we write \({{\,\mathrm{Clr}\,}}(T,p_{1},\ldots ,p_{n})\) as the set of all such colorings of T.
In order to define the sign \({{\,\mathrm{sgn}\,}}_{T}(I)\) for \(T\in {{\,\mathrm{Brace}\,}}(n)\), we use the morphism of operads
and base this definition on the sign \({{\,\mathrm{sgn}\,}}_{Q}(\zeta ,I)\) from [8, Def. 4.20].
Construction 2.14
We work with the following alphabet
for \(i=1,\ldots ,n\) and define the word
We define a second word \(J^{s}_{T}(I)\) having in the \(\zeta _{I,k}(i)\)th position \(i_{k}\) for \( 1 \le i \le q_{k}1\). Note that we start from position 1 for \(J^{s}_{T}(I)\) (instead of 0).
Definition 2.15
For \(I\in {{\,\mathrm{NSOp}\,}}(p_{1},\ldots ,p_{n};p)\) where we replace those \(p_{i} = 0\) by 2, we define \({{\,\mathrm{sgn}\,}}_{T}(I)\) as the sign of the shuffle transforming \(J^{s}(p_{1},\ldots ,p_{n})\) to \(J_{T}^{s}(I)\).
Theorem 2.16
We have a morphism of operads
Proof
Per definition of \({{\,\mathrm{sgn}\,}}_{T}(I)\) we see that \(\phi \) is equivariant. Hence, we only need to verify that \({{\,\mathrm{sgn}\,}}_{T\circ _{1}T'}(I\circ _{1}I') = {{\,\mathrm{sgn}\,}}_{T}(I) {{\,\mathrm{sgn}\,}}_{T'}(I')\) for \(T\in {{\,\mathrm{Brace}\,}}(n), T'\in {{\,\mathrm{Brace}\,}}(m)\) and \(I\in {{\,\mathrm{Clr}\,}}(T,p_{1},\ldots ,p_{n})\) and \(I'\in {{\,\mathrm{Clr}\,}}(T',p'_{1},\ldots ,p'_{m})\). This equation holds as we can decompose the shuffle \(\chi '': J^{s}(p'_{1},\ldots ,p'_{m},p_{2},\ldots ,p_{n}) \leadsto J_{T\circ _{i}T'}(I\circ _{i}I')\) into two shuffles
where \(\chi \) and \(\chi '\) are the corresponding shuffles determining \({{\,\mathrm{sgn}\,}}_{T}(I)\) and \({{\,\mathrm{sgn}\,}}_{T'}(I')\). \(\square \)
2.6 The morphism \({{\,\mathrm{F_{2}S}\,}}\longrightarrow {{\,\mathrm{mNSOp}\,}}_{st}\)
In order to define the morphism \( {\bar{\phi }}: {{\,\mathrm{F_{2}S}\,}}\longrightarrow {{\,\mathrm{mNSOp}\,}}_{st}\) properly, we again need to compile the colored operad \({{\,\mathrm{mNSOp}\,}}\) to obtain a graded noncolored operad
where an element \(x\in {{\,\mathrm{mNSOp}\,}}(q_{1},\ldots ,q_{n};q)\) is graded as \(deg(x) = \sum _{i=1}^{r} q_{i}  q\) (standard grading) and \({{\,\mathrm{mNSOp}\,}}_{st}(n)\) is generated by the sequences of constant grading. The composition on \({{\,\mathrm{mNSOp}\,}}_{st}\) is derived from the composition of \({{\,\mathrm{mNSOp}\,}}\) where it is set to 0 when the colors do not match. Note in particular that the \({\mathbb {S}}_{n}\)action on \({{\,\mathrm{mNSOp}\,}}_{st}(n)\) is affected by this grading: permuting two vertices i and j introduces the sign \((1)^{q_{i}q_{j}}\).
2.6.1 Colorings
Definition 2.17
Let \(X:=[I \circ _{n+1} m \circ _{n+1} \cdots \circ _{n+1} m ] \in {{\,\mathrm{mNSOp}\,}}(q_{1},\ldots ,q_{n};q)\) for I having \(n+k\) vertices, then X is a coloring of \(W \in {{\,\mathrm{F_{2}S}\,}}(n)\) if

Each vertex \(n+1,\ldots ,n+k\) has exactly two children in I

For \(u,v \in \langle n \rangle \) holds

(1)
\(u <_{X} v \iff W = \ldots {u} \ldots {v} \ldots {u}\ldots \)

(2)
\(u \vartriangleleft _{X} v \iff \) every occurrence of u in W is left of every occurrence of v in W.

(1)
We write \({{\,\mathrm{Clr}\,}}(W,q_{1},\ldots ,q_{n})\) for the set of all such colorings for W.
Remark 2.18
An element \(X\in {{\,\mathrm{Clr}\,}}(W,q_{1},\ldots ,q_{n})\) has \(n  deg(W) 1\) many m’s plugged in. Hence, \(X \in {{\,\mathrm{mNSOp}\,}}(q_{1},\ldots ,q_{n}; \sum _{i=1}^{n}(q_{i}1) + n  deg(W))\).
We give some examples.
Example 2.19
The following three elements of \({{\,\mathrm{mNSOp}\,}}\)
and
color respectively the words
Note however that not all elements of \({{\,\mathrm{mNSOp}\,}}\) color a word of \({{\,\mathrm{F_{2}S}\,}}\): the following set of elements
for \(r\in \{1,2\}\), colors no word in \({{\,\mathrm{F_{2}S}\,}}\) because the vertex plugged by m does not have two children.
Lemma 2.20
Definition 2.17 is welldefined, that is, it is independent of the chosen representative I of X.
Proof
Due to Lemma 2.4, both \(<_{X}\) and \(\vartriangleleft _{X}\) are welldefined. 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 X. Thus, suppose for some \(I_0 \in {{\,\mathrm{NSOp}\,}}\), such that both vertices k and \(k+1\) of I are plugged by m’s in X. Due to the relations in \({{\,\mathrm{mNSOp}\,}}\), X can equivalently be represented using . In this case, we have that vertices k and \(k+1\) each have exactly 2 children in I iff vertex k has exactly 3 children in \(I_0\) iff vertices k and \(k+1\) each have exactly 2 children in \(I'\). \(\square \)
We construct a word for every element of \({{\,\mathrm{mNSOp}\,}}\) satisfying the above criteria.
Construction 2.21
Let \(X:=[I \circ _{n+1} m \circ _{n+1} \cdots \circ _{n+1} m ] \in {{\,\mathrm{mNSOp}\,}}(q_{1},\ldots ,q_{n};q)\) such that each vertex \(a > n\) in I has exactly two children, then we construct a word \(W_{X} \in {{\,\mathrm{F_{2}S}\,}}(n)\) such that \(X \in {{\,\mathrm{Clr}\,}}(W_{X},q_{1},\ldots ,q_{n},q)\).

To every tree T we can associate a word \(W_{T} \in {{\,\mathrm{F_{2}S}\,}}(n+k)\) (see [8, §2.3]).

Suppose for \(X_{0}\in {{\,\mathrm{mNSOp}\,}}\) such that \(X_{0}\circ _{n+1}m =X\) we have an associated word \(W_{X_{0}}\in {{\,\mathrm{F_{2}S}\,}}(n+1)\), then let \(W_{X}\) be the word given by deleting all occurrences of \(n+1\). Then \(W_{X}\in {{\,\mathrm{F_{2}S}\,}}(n)\) because \(n+1\) had two children, so no degeneracy can occur.
We consider an example of this procedure.
Example 2.22
We consider the element
for some \(1\le i \le q_1\) 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.
Lemma 2.23
For \(X \in {{\,\mathrm{mNSOp}\,}}(q_{1},\ldots ,q_{n};q)\) we have \(X \in {{\,\mathrm{Clr}\,}}(W_{X},q_{1},\ldots ,q_{n})\) and if \(X \in {{\,\mathrm{Clr}\,}}(W,q_{1},\ldots ,q_{n})\), then \(W = W_{X}\).
Proof
This clearly holds for \(X = I \in {{\,\mathrm{NSOp}\,}}(q_{1},\ldots ,q_{n};q)\). Assume the lemma holds for \(X_{0} \in {{\,\mathrm{mNSOp}\,}}\) and \(X = X_{0} \circ _{n+1} m\), then \(W_{X_{0}} = W_{0} (n+1) W_{1} (n+1) W_{2} (n+1) W_{3}\) 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 \(X \in {{\,\mathrm{Clr}\,}}(W_{X},q_{1},\ldots ,q_{n})\).
Now reversely, if \(X \in {{\,\mathrm{Clr}\,}}(W,q_{1},\ldots ,q_{n})\) and \(a \vartriangleleft b\) are the two children of \(n+1\) in \(X_{0}\), then \(W = W_{0}W_{1}W_{2}W_{3}\) where \(W_{1}= {a} \ldots {a}, W_{2} = b \ldots b\) and \(W_{0}\) and \(W_{3}\) are possibly empty. In that case, \(X_{0} \in {{\,\mathrm{Clr}\,}}(W_{0} (n+1) W_{1} (n+1) W_{2} (n+1) W_{3},q_{1},\ldots ,q_{n},2)\) and thus by induction \(W_{X_{0}} = W_{0} (n+1) W_{1} (n+1) W_{2} (n+1) W_{3}\). Hence, \(W_{X} = W_{0}W_{1}W_{2}W_{3} = W\). \(\square \)
Lemma 2.24
Let \(X \in {{\,\mathrm{Clr}\,}}(V,q_{1},\ldots ,q_{n},q)\) and \(Y \in {{\,\mathrm{Clr}\,}}(W,q'_{1},\ldots ,q'_{m},q_{i})\), then there exists a unique \(U \in Ext(V,W,i)\) such that \(X\circ _{i} Y\in {{\,\mathrm{Clr}\,}}(U,q_{1},\ldots ,\)\(q'_{1},\ldots ,q'_{m},\ldots ,q_{n},q)\)
Proof
By Construction 2.21 we obtain a word \(U\in {{\,\mathrm{F_{2}S}\,}}(n+m1)\) such that \(Z := X \circ _{i} Y \in {{\,\mathrm{Clr}\,}}(U,\ldots )\). We show that \(U \in Ext(V,W,i)\): let \(U_{\alpha }\) be the word obtained from deleting from U occurrences of vertices not in the image of \(\alpha \) and eliminating consecutive repetitions (\(uu\mapsto u)\). It is easy to check that \(W = \cdots {u} \ldots {v} \ldots {u} \ldots \) iff \(U_{\alpha } = \cdots {\alpha (u)} \ldots {\alpha (v)} \ldots {\alpha (u)} \ldots \), and that all occurrences of u are left to those of v in W iff the same holds for \(\alpha (u)\) and \(\alpha (v)\) in \(U_{\alpha }\).
Let \(U_{\beta }\) be the word obtained from U by relabelling by \(\beta \) and eliminating consecutive repetitions. To verify that \(U_{\beta } = V\) is straight forward, except in the following case: \(U_{\beta } = \cdots {i} \ldots {\beta (u)} \ldots {i} \ldots \) where \(\beta (u) \ne i\). In this case, there exist \(v,v'\) such that \(U = \cdots {\alpha (v)} \ldots {u} \ldots {\alpha (v')} \ldots \). Now, we argue that there exists a vertex \(v''\) such that \(\alpha (v'') <_Z u\). If neither \(\alpha (v)\) nor \(\alpha (v')\) do, then U implies that \(\alpha (v) \vartriangleleft _Z u \vartriangleleft _Z \alpha (v')\). As \(\alpha (v)\) and \(\alpha (v')\) are part of the same subtree \(\alpha (Y)\), i.e. the image of Y under \(\alpha \) in Z, there must be some vertex a in the tree underlying \(\alpha (Y)\) (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 \(v''\) in Y such that \(\alpha (v'') =a <_Z u\). Thus, \(i <_X \beta (u)\) which verifies that \(V = \cdots i \ldots \beta (u) \ldots i \ldots \). This clearly also holds reversibly. \(\square \)
Lemma 2.25
Let \(U \in Ext(V,W,i)\) and \(Z\in {{\,\mathrm{Clr}\,}}(U,q_{1},\ldots ,q'_{1},\ldots ,q'_{m},\ldots ,q_{n},q)\), then there exist unique colorings \(X\in {{\,\mathrm{Clr}\,}}(V,q_{1},\ldots ,q_{n},q)\) and \(Y\in {{\,\mathrm{Clr}\,}}(W,q'_{1},\ldots ,q'_{m},q_{i})\) such that \(Z = X \circ _{i} Y\).
Proof
Let \(Z = [ I'' \circ _{n+m} m \circ _{n+m} \cdots \circ _{n+m}m]\) with \(l+k\) added m’s. The word W can be uniquely written as
where two subwords \(W_{j}\) and \(W_{j'}\) do not share any occurrence of the same number, and \(W_{j}\) is of the form \({a_{j}}\ldots {a_{j}}\). As Z is a coloring of U, we have that \(\alpha (a_{1})\vartriangleleft _{Z} \ldots \vartriangleleft _{Z} \alpha (a_{t})\) and no vertex of \({{\,\mathrm{Im}\,}}(\alpha )\) lies under any \(a_{j}\). In this case, there exists some vertex \(a\in \{n+m,\ldots ,n+m+k+l1\}\) such that \(a\le _{I''}\alpha (a_{j})\) which is \(\le _{I''}\)maximal for these conditions (otherwise when applying \(\beta \) to U we will not obtain V).
Let \(I'\) be the minimal subtree of \(I''\) on the root a containing \({{\,\mathrm{Im}\,}}(\alpha )\). By contracting this subtree to a point we obtain a tree I such that, after permutation of some vertices, we obtain \(I \circ _{i} I' = I''\). Consider also the permutation such that \(Z = [I'' \circ _{m+i} m \circ _{m+i} \cdots \circ _{m+i} m \circ _{n+m} m \circ _{n+m} \cdots \circ _{n+m} m]\).
It now suffices to show that \(X := [I \circ _{n+1} m\circ _{n+1} \cdots \circ _{n+1} m] \in {{\,\mathrm{Clr}\,}}(V,\ldots )\) and \(Y:= [I' \circ _{m+1} m \circ _{m+1} \cdots \circ _{m+1} m ] \in {{\,\mathrm{Clr}\,}}(W,\ldots )\), which is a straight forward computation using the facts \(X \circ _{i} Y =Z\), \(Z\in {{\,\mathrm{Clr}\,}}(U,\ldots )\) and \(U \in Ext(V,W,i)\). \(\square \)
2.6.2 Signs
In order to define a sign \({{\,\mathrm{sgn}\,}}_{W}(X)\) for \(W\in {{\,\mathrm{F_{2}S}\,}}(n)\) and \(X\in {{\,\mathrm{Clr}\,}}(W,q_{1},\ldots ,q_{n})\) we use the morphism of operads
which extends to
We base this definition of the sign on the sign \({{\,\mathrm{sgn}\,}}_{Q}(\zeta ,I)\) defined in [8, §4.7].
Lemma 2.26
Let \(X \in {{\,\mathrm{Clr}\,}}(W,q_{1},\ldots ,q_{n};q)\) for \(W\in {{\,\mathrm{F_{2}S}\,}}(n)\) and \(q_{i} > 0\), then \(\zeta _{X}\) is a coloring of W in the sense of [8, Def. 4.13], that is,

\(\zeta _{X} \in \text {Multi}\Delta _{+}\),

\(\bigcup _{i=1}^{n} {{\,\mathrm{Im}\,}}(\zeta _{X,i}) = [q]\)

For each \(a\in \langle n \rangle \) there exists a function \(\pi _{a}:[q_{a}] \longrightarrow W\) such that

(1)
The image of \(\pi _{a}\) is the set of occurrences of a in W,

(2)
For an \((i,a)\in W\), \(\zeta _{X,a}(\pi _{a}^{1}(i,a))\) is an interval

(3)
If \(W= \cdots {a}{b} \ldots \), then
$$\begin{aligned} \max \; \zeta _{X,a}(\pi _{a}^{1}(i,a)) = \min \; \zeta _{X,b}(\pi _{b}^{1}(i+1,b)) \end{aligned}$$

(1)
Proof
We first show it holds for \(X= (T,I) \in {{\,\mathrm{NSOp}\,}}\) 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 \(W = u W^1 u \ldots u W^k u\) where the subwords \(W_i\) represent the subtrees \(I^i\). Let \(\gamma _{i}:\langle k_{i} \rangle \hookrightarrow \langle n \rangle \) be the maps embedding the tree \(I^i_0\) onto \(I^i\) in I and \(I^i_0\in {{\,\mathrm{Clr}\,}}(W^i_0,\ldots )\), then they extend to a map \(\overline{\gamma _i}: W^i_0 \hookrightarrow W\) inserting \(W^i_0\) as \(W^i\) into W. By induction, the lemma holds for \(I^{i}_0 \in {{\,\mathrm{Clr}\,}}(W^i_0,\ldots )\). Let \((p_1,u),\ldots ,(p_{k+1},u)\) 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 \(X = [X_{0} \circ _{n+1} m]\) such that the lemma holds for \(X_{0} \in {{\,\mathrm{Clr}\,}}(W_{X_{0}},\ldots )\), then \(W_{X}\) is obtained from \(W_{X_{0}}\) by deleting all occurrences of \((n+1)\). As the vertex \(n+1\) has exactly two children in \(X_{0}\) and \(q_{n+1} = 2\), we still have that \(\bigcup _{i=1}^{n} {{\,\mathrm{Im}\,}}(\zeta _{X,i}) =\bigcup _{i=1}^{n+1} {{\,\mathrm{Im}\,}}(\zeta _{X_{0},i}) =[q]\) and that
for \(W= \cdots {a}{b} \ldots \). Hence, \(\zeta _{X}\) satisfies the lemma. \(\square \)
Let us define analogously the sign corresponding to the horizontal part of \({{\,\mathrm{sgn}\,}}_{Q}(\zeta _{X},I)\).
Construction 2.27
We work with the following alphabet
for \(i=1,\ldots ,n\) 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

\(J_{0,W}(X)\) consists of all \(0_{k}\) for k interposed in W, put in reverse \(\downarrow \)order.

\(J_{1,W}(X)\) has in the \(\zeta _{X,k}(i)\)th position \(i_{k}\) for \( 1 \le i \le q_{k}1\) for k interposed, and in the \(\zeta _{I,k}(i)\) position \(i_{k}\) for \(0 \le i \le q_{k}1\) for k not interposed. Note that we start from position 0.
Definition 2.28
For \(X\in {{\,\mathrm{mNSOp}\,}}(q_{1},\ldots ,q_{n};q)\) where we replace those \(q_{i} = 0\) by 2, we define \({{\,\mathrm{sgn}\,}}_{W}(X)\) as the sign of the shuffle transforming \(J(q_{1},\ldots ,q_{n})\) to \(J_{W}(X)\).
Example 2.29
Consider the words
and colorings
for which we calculate the words \(J_W(X)\) and \(J_{W'}(X')\) and their corresponding signs. In the first case, we have
which corresponds to the sign \({{\,\mathrm{sgn}\,}}_W(X) = (1)^{(q_21)(q_1j) + (q_31)(q_1i) + q_3q_2 +q_31}\). For the second case, we calculate
which corresponds to the sign \({{\,\mathrm{sgn}\,}}_{W'}(X') = (1)^{ (q_2+q_3)(q_1k) + q_1}\). Note in particular that in \(J_{W'}(X')\) the letter \(0_3\) is not taken to the front of the word as 3 is not interposed in \(W'\).
Lemma 2.30
Let \(X \in {{\,\mathrm{Clr}\,}}(W,q_{1},\ldots ,q_{n})\) and \(X'\in {{\,\mathrm{Clr}\,}}(W',q'_{1},\ldots ,q'_{m})\), and \(W''\in Ext(W,W',1)\), then \({{\,\mathrm{sgn}\,}}_{W}(X) {{\,\mathrm{sgn}\,}}_{W'}(X') = {{\,\mathrm{sgn}\,}}_{W''}(X\circ _{1}X') {{\,\mathrm{sgn}\,}}_{W,W',1}(W'')\).
Proof
We can assume that all \(q_{i}\) and \(q'_{j}\) are not zero. We can decompose \({{\,\mathrm{sgn}\,}}_{W}(X)\) in three components

Sign of the shuffle \(\sigma \) shuffling \(J_{0,W}(X)\) to \(0_{v_{1}}\ldots 0_{v_{k}}\) for \(v_{1}< \ldots < v_{k}\) the interposed vertices of X,

Sign of the shuffle \(\tau \) shuffling \(J_{1,W}(X)\) to concatenation of \(1_{i} \ldots (q_{i}1)_{i}\) for i interposed and \(0_{i} \ldots (q_{i}1)_{i}\) for i not interposed. We call this latter sequence \(J_{W}^{int}(q_{1},\ldots ,q_{n})\).

Sign of the shuffle \(\rho \) shuffling
$$\begin{aligned} 0_{v_{1}}\ldots 0_{v_{k}}J_{W}^{int}(q_{1},\ldots ,q_{n}) \leadsto J(q_{1},\ldots ,q_{n}) \end{aligned}$$
We add \('\) and \(''\) to denote the correspondings shuffles for \(X'\) and \(X'':= X \circ _{1} X'\).
First we have that \((1)^{\sigma ''} = (1)^{\sigma +\sigma '}{{\,\mathrm{sgn}\,}}_{W,W',1}(W'')\) per definition of \({{\,\mathrm{sgn}\,}}_{W,W',1}(W'')\). Further we clearly have \((1)^{\tau ''} = (1)^{\tau + \tau '}\) by simply applying them one after the other and renaming using \(\alpha \) and \(\beta ^{1}\)
as \(\zeta _{X''} = \zeta _{X} \circ _{1} \zeta _{X'}\). We also have that \((1)^{\rho ''}= (1)^{\rho +\rho '}\) because the length of the sequence \(J_{W'}^{int}(q'_{1},\ldots ,q'_{m})\) is \(q_{1}1\) if 1 is interposed, and \( q_{1}\) if 1 is not interposed. \(\square \)
2.6.3 The morphism
Proposition 2.31
We have a morphism of graded operads
Proof
By definition of \({{\,\mathrm{sgn}\,}}_{W}(X)\) the above linear maps are equivariant. By Lemmas 2.24, 2.25 and 2.30 they define a morphism of graded operads. \(\square \)
We make \({{\,\mathrm{mNSOp}\,}}_{st}\) into a dgoperad with the hochschild differential, then \(\phi \) will be a morphism of dgoperads.
Definition 2.32
Define for every \(q\in {\mathbb {N}}\) the element
which compile into an element of degree \(1\)
We consider the associated derivation
for \(X \in {{\,\mathrm{mNSOp}\,}}_{st}(n)\).
Proposition 2.33
\(\partial _{D}\) defines a differential making \({{\,\mathrm{mNSOp}\,}}\) into a dgoperad, for which holds
Proof
The first part follows directly if \(D \circ _{1} D =0\) 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 \(121\ldots 1k1\) for \(k\ge 1\).

For \(q_{1},q_{2}\in {\mathbb {N}}\), it is easy to compute that
in \({{\,\mathrm{mNSOp}\,}}(q_{1},q_{2};q_{1}+q_{2})\).

For \(q_{1},\ldots ,q_{k}\in {\mathbb {N}}\), we also have
and
$$\begin{aligned} \partial (121\ldots 1k1) =  2131\ldots 1 + \sum _{i=2}^{k1}(1)^{i} 1\ldots 1i(i+1)1\ldots 1 + (1)^{k} 1\ldots 1k \end{aligned}$$for which it is also a standard computation to see that \(\partial _{D}( {\bar{\phi }}(121\ldots 1k1)) = {\bar{\phi }}( \partial (121\ldots 1k1))\) (see [6, Thm 3]).
\(\square \)
Theorem 2.34
We have a morphism of dgoperads
Proof
This is the direct consequence of Propositions 2.31 and 2.33. \(\square \)
3 The Gerstenhaber–Schack complex for prestacks
Let \(({\mathcal {A}}, m, f, c)\) be a prestack over a small category \({\mathcal {U}}\) and let \(\mathbf{C} _{GS}({\mathcal {A}})\) be the associated Gerstenhaber–Schack complex as defined in [2] (see Sect. 3.1). In loc. cit., a homotopy equivalence \(\mathbf{C} _{GS}({\mathcal {A}}) \cong \mathbf {CC}({\mathcal {A}}!)\) is constructed with the Hochschild complex \(\mathbf {CC}({\mathcal {A}}!)\) of the Grothendieck construction \({\mathcal {A}}!\) of \({\mathcal {A}}\). Through homotopy transfer, this allows to endow the GScomplex with an \(L_{\infty }\)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 \({{\,\mathrm{Quilt}\,}}\subseteq {{\,\mathrm{F_{2}S}\,}}\otimes _H {{\,\mathrm{Brace}\,}}\) which he later extends to an operad \({{\,\mathrm{mQuilt}\,}}\) acting on the GScomplex. These operads are naturally endowed with \(L_{\infty }\)operations as desired. The action of \({{\,\mathrm{Quilt}\,}}\) on the GScomplex considered by Hawkins only involves the restriction functors f of the presheaf, the multiplication m being incorporated later on in \({{\,\mathrm{mQuilt}\,}}\). 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 \({{\,\mathrm{Quilt}\,}}\) in a fundamentally different way in relation to the GScomplex, but still allowing us to make use of the naturally associated \(L_{\infty }\)structure. In this section we capture the higher structure of \(\mathbf{C} _{GS}({\mathcal {A}})\) by introducing the operad \({{\,\mathrm{Patch}\,}}\subseteq {{\,\mathrm{mNSOp}\,}}\otimes _H {{\,\mathrm{NSOp}\,}}\) (see Sect. 3.3) over which the bicomplex \(\mathbf{C} ^{\bullet ,\bullet }({\mathcal {A}})\), of which \(\mathbf{C} _{GS}({\mathcal {A}})\) is the totalisation, is shown to be an algebra (see Theorem 3.24). Next, we construct a morphism \({{\,\mathrm{Quilt}\,}}\longrightarrow {{\,\mathrm{Patch}\,}}_s\) (see Proposition 3.27) as a restriction of
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 \({{\,\mathrm{Quilt}\,}}\) on \(\mathbf{End} (s\mathbf{C} _{GS}({\mathcal {A}}))\) only incorporates the restrictions. As far as the structure of both approaches goes, the auxiliary operad \({{\,\mathrm{Patch}\,}}\) we use is the counterpart of the operad \(\mathrm {ColorQuilt}\) from [8, Def. 4.6].
In Sect. 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 klinear categories. Let \({\mathcal {U}}\) be a small category.
Definition 3.1
A prestack \({\mathcal {A}}= ({\mathcal {A}},m,f,c)\) over \({\mathcal {U}}\) consists of the following data:

For every object \(U \in {\mathcal {U}}\), a klinear category \(({\mathcal {A}}(U), m^U, 1^U)\) where \(m^U\) is the composition of morphisms in \({\mathcal {A}}(U)\) and \(1^U\) encodes the identity morphisms of \({\mathcal {A}}(U)\).

For every morphism \(u:V \longrightarrow U\) in \({\mathcal {U}}\), a klinear functor \(f^u = u ^{*}: {\mathcal {A}}(U) \longrightarrow {\mathcal {A}}(V)\). For \(u= 1_U\) the identity morphism of U in \({\mathcal {U}}\), we require that \((1_U)^{*}= 1_{{\mathcal {A}}(U)}\).

For every couple of morphisms \(v: W \longrightarrow V, u: V \longrightarrow U\) in \({\mathcal {U}}\), a natural isomorphism
$$\begin{aligned} c^{u,v}: v^{*}u ^{*}\longrightarrow (uv)^{*}. \end{aligned}$$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 \(w:T \longrightarrow W\), \(v :W \longrightarrow V\) and \(u:V \longrightarrow U\):
$$\begin{aligned} c^{u,vw}(c^{v,w} \circ u ^{*}) = c^{uv,w}(w^{*}\circ c^{u,v}). \end{aligned}$$
Given such a prestack \({\mathcal {A}}\), we have an associated Gerstenhaber–Schack complex \(\mathbf{C} _{GS}({\mathcal {A}})\). In [2] this is defined as the totalisation of a bicomplex \(\mathbf{C} ^{\bullet ,\bullet }({\mathcal {A}})\). We first review some notations.
Notations. Let \(\sigma = (U_{0} \overset{u_{1}}{\rightarrow } U_{1} \rightarrow \ldots \overset{u_{p}}{\rightarrow } U_{p} )\) be a psimplex in the category \({\mathcal {U}}\), then we have two functors \({\mathcal {A}}(U_{p}) \longrightarrow {\mathcal {A}}(U_{0})\), namely
For each \(1 \le k \le p1\), denote by \(L_{k}(\sigma )\) and \(R_{k}(\sigma )\) the following simplices
and we consider the following natural isomorphisms
We write \(c^{\sigma ,k,A}= c^{\sigma ,k}(A)\) and \(\epsilon ^{\sigma ,k,A}= \epsilon ^{\sigma ,k}(A)\) for \(A\in {\mathcal {A}}(U_{p})\).
We also define a set \(P(\sigma )\) of formal paths from \(\sigma ^{\#}\) to \(\sigma ^{*}\) inductively. A formal path is finite sequence of couples \((\tau ,i)\) consisting of a simplex \(\sigma \) and a natural number i. We set
and
where \(\partial _{i}\) denotes the ith faceoperator of the nerve \(N_{p}({\mathcal {U}})\). Given such a formal path \(r=(r_{1},\ldots ,r_{p1})\) we define its sign
By interpreting the data \((\sigma ,i)\) as the natural isomorphism \(\epsilon ^{\sigma ,i}\), every formal path \(r \in P(\sigma )\) induces a sequence of natural isomorphisms \({\overline{r}} \in N_{p1}({{\,\mathrm{Fun}\,}}({\mathcal {A}}(U_p),{\mathcal {A}}(U_0)))\). Note that \(\epsilon ^{(u_1,u_2),1} = c^{u_1,u_2}\) and its associated sign is \(1\).
Let \({\mathbb {S}}_{t,p1}\) denote the set of \((t,p1)\)shuffles, then given a formal path \(r\in P(\sigma )\), a shuffle \(\beta \in {\mathbb {S}}_{t,p1}\) and a tuple \(a= (A_{0} \overset{a_{1}}{\leftarrow } A_{1} \leftarrow \cdots \overset{a_{t}}{\leftarrow } A_{t}) \in N_{t}({\mathcal {A}}(U_{p}))\) let \(\beta (a,r)\in N_{p1+t}({\mathcal {A}}(U_{0}))\) be its shuffle product with respect to evaluation of functors as defined in [2, Ex. 3.2, Ex. 3.4].
Here, we give a more explicit definition of \(\beta (a,r)\): first we construct inductively a sequence \((b_{1},\ldots ,b_{t+p1})\) which formally represents a sequence of morphisms in \(A(U_0)\). Every \(b_i\) is either of the form \((\tau ,a_i,A_{i1})\) or \((r_i,A_j)\) for \(\tau \) 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
then for \(1<i \le t+p1\), we have two cases: if \(b_{i+1} = (\tau , a_j, A_{j1})\) for some j, then define
If \(b_{i+1} = (r_k,A_j)\) for \(r_k = (\tau ,l)\), then define
Finally, we define \(\beta (a,r)\) as the realization \({\overline{b}}=(B_{0} \overset{\overline{b_1}}{\leftarrow } B_{1} \leftarrow \cdots \overset{\overline{b_t}}{\leftarrow } B_{p1+t}) \) of b where \(\overline{b_i} = \tau ^{\#}a_j\) if \(b_i = (\tau , a_j,A_j)\) and \(\overline{b_i} = \overline{r_k}(A_j)\) if \(b_i = (r_k, A_j)\).
Definition 3.2
Let \(p,q\ge 0\), then define
and set
The differential d on the GScomplex is defined for \(\theta \in \mathbf{C} ^{p,q}({\mathcal {A}})\) as
where \(d_{j}: \mathbf{C} ^{p,q}({\mathcal {A}}) \longrightarrow \mathbf{C} ^{p+j,q+1j}({\mathcal {A}})\) is defined as

$$\begin{aligned} d_{0}(\theta )^{\sigma }(A)(a) =&\, m^{U_{0}}(\sigma ^{*}(a_{1}), \theta ^{\sigma }(A_1,\ldots ,A_{q+1})(a_{2},\ldots ,a_{q+1})) \\&\,+ \sum _{i=1}^{q}(1)^{i} \theta ^{\sigma }(A_0,\ldots ,A_{i1},A_{i+1},\ldots ,A_{q+1})\\&\,(a_{1},\ldots ,m^{U_{p}}(a_{i},a_{i+1}),\ldots ,a_{1})\\&\,+ (1)^{q+1}m^{U_{0}}(\theta ^{\sigma }(A_0, \ldots , A_{q})(a_{1},\ldots ,a_{q}), \sigma ^{\#}(a_{q+1})) \end{aligned}$$

$$\begin{aligned} d_{1}(\theta )^{\sigma }(A)(a) =&\, (1)^{p+q+1} m^{U_{0}}(c^{\sigma ,1,A_{0}} , u_{1}^{*}(\theta ^{\partial _{0}\sigma }(A)(a))) \\&\,+ \sum _{i=1}^{p} (1)^{p+q+1+i} m^{U_{0}}(\theta ^{\partial _{i}\sigma }(A)(a), \epsilon ^{\sigma ,i,A_{q}}) \\&\,+ (1)^{q} m^{U_{0}}(c^{\sigma ,p,A_{0}} , \theta ^{\partial _{p+1}\sigma }(A)(u_{p+1}^{*}a_{1},\ldots ,u_{p+1}^{*}a_{q})) \end{aligned}$$

$$\begin{aligned} d_{j}(\theta )^{\sigma }(A)(a) = \sum _{\begin{array}{c} {r} \in P(R_{p}(\sigma ))\\ \beta \in {\mathbb {S}}_{qj+1,j1} \end{array}} (1)^{{r}+\beta +qj+1} m^{U_{p+j}}(c^{\sigma ,p,A_{0}},\theta ^{L_{p}(\sigma )}(B)(\beta (a,r)) \end{aligned}$$
for \(\sigma = (u_{1},\ldots ,u_{p+j})\in N_{p+j}({\mathcal {U}})(U_{0},U_{p+j})\), \(a = (a_{1},\ldots ,a_{qj+1})\) where \(a_{i} \in {\mathcal {A}}(U_{p+j})(A_{i},A_{i1})\) and such that B is the sequence of objects underlying \(\beta (a,r)\).
We will also be interested in the subcomplex \(\overline{\mathbf{C }}_{GS}({\mathcal {A}}) \subseteq \mathbf{C} _{GS}({\mathcal {A}})\) of normalized and reduced cochains which is shown to be quasiisomorphic to the GS complex (see [2, Prop. 3.16]). A simplex \(\sigma = (u_{1},\ldots ,u_{p})\) is reduced if \(u_i = 1_{U_i}\) for some \(1 \le i \le p\). A cochain \(\theta = \left( \theta ^{\sigma }(A)\right) _{\sigma ,A} \in \mathbf{C} _{GS}({\mathcal {A}})\) is reduced if \(\theta ^\sigma (A) = 0\) for every reduced simplex \(\sigma \). A simplex \(a=(a_{1},\ldots ,a_{q})\) in \({\mathcal {A}}(U)\) is normal if \(a_i= 1^{U}\) for some \(1 \le i \le q\). A cochain \(\theta \) is normalized if \(\theta ^{\sigma }(A)(a) = 0\) for every normal simplex a in \({\mathcal {A}}(U_p)\). We come back to this in Sect. 4.
Elements of the GS complex have a neat geometric interpretation as rectangles: for \(\theta \in \mathbf{C} ^{p,q}({\mathcal {A}})\) and the data \((\sigma ,A,a)\) from above, we can represent \(\theta ^\sigma (A)(a)\) 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 \(\beta (q)= i\), \(\beta (s) = s\) for \(s<i\) and \(\beta (s) = s+1\) for \(s\ge i\), and formal path \(r=((u_p,u_{p+1}),1)\). Note in particular that we can draw \(\beta (a,r)\) as follows
where \(b_s= a_{\beta ^{1}(s)}\) for \(s\ne i\), and \(b_i = c^{u_{p+1},u_{p+2}}(A_{i1})\).
We will use this rectangular interpretation as a guide in the next sections.
3.2 Endomorphism operad of a prestack
Although the GScomplex does not have partial compositions \(\circ _{i}\), its elements \(\theta = (\theta ^{\sigma }(A))_{(\sigma ,A)}\) consist of parts that lie in the endomorphism operad \(\mathbf{End} ({\mathcal {A}})\).
Definition 3.3
Let \({{\,\mathrm{Ob}\,}}({\mathcal {U}},{\mathcal {A}})\) be the set consisting of the triples \((U,A,A')\) for \(U \in {\mathcal {U}}\) and \(A,A'\in {\mathcal {A}}(U)\), then we define the \({{\,\mathrm{Ob}\,}}({\mathcal {U}},{\mathcal {A}})\)colored operad \(\mathbf{End} ({\mathcal {A}})\) as
with partial compositions defined by composition of linear maps.
Remark 3.4
Note that \(\theta ^{\sigma }(A) \in \mathbf{End} ({\mathcal {A}})((U_{p},A_{1},A_{0}),\ldots ,(U_{p},A_{q},A_{q1});(U_{0},\sigma ^{\#}A_{q},\sigma ^{*}A_{0}))\).
3.3 The operad \({{\,\mathrm{Patch}\,}}\)
In this section we define an \({\mathbb {N}}\times {\mathbb {N}}\)colored operad \({{\,\mathrm{Patch}\,}}\subseteq {{\,\mathrm{mNSOp}\,}}\times {{\,\mathrm{NSOp}\,}}\). 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 \({{\,\mathrm{Patch}\,}}((q_{1},p_{1}),\ldots ,(q_{n},p_{n});(q,p))\) consists of the elements \((X,J)\in {{\,\mathrm{mNSOp}\,}}(q_{1},\ldots ,q_{n};q)\times {{\,\mathrm{NSOp}\,}}(p_{1},\ldots ,p_{n};p)\) such that

(1)
\(a <_{J} b \Longrightarrow a \vartriangleleft _{X} b\)

(2)
\(a <_{X} b \Longrightarrow b \vartriangleleft _{J} a\)
Remark 3.6
Note that in order for \({{\,\mathrm{Patch}\,}}\) 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 righthand side, q inputs on top and a single output on respectively the bottom and the lefthand side
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
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.
Example 3.7
The following pair determines an element in \({{\,\mathrm{Patch}\,}}((3,5),(3,2),(2,1),(0,2);(6,7))\)
which we can draw as the following patchwork of rectangles
where the grey areas denote the open spaces.
Lemma 3.8
\({{\,\mathrm{Patch}\,}}\) is a suboperad of \({{\,\mathrm{mNSOp}\,}}\otimes _{H}{{\,\mathrm{NSOp}\,}}\).
Proof
Let \((X,J)\in {{\,\mathrm{Patch}\,}}((q_{1},p_{1}),\ldots ,(q_{n},p_{n});(q,p))\) and \((X',J')\in \)\( {{\,\mathrm{Patch}\,}}((q'_{1},p'_{1}),\ldots ,(q'_{m},p'_{m});(q_{i},p_{i}))\) and we set \(X'':= X \circ _{i} X'\) and \(J'':= J \circ _{i} J'\). Let \((\alpha ,\beta )\) be the extension of n by m at i, then for \(a,b\in \langle m \rangle \) we compute
and for \(c,d\notin {{\,\mathrm{Im}\,}}(\alpha )\) we compute
For \(c\notin {{\,\mathrm{Im}\,}}(\alpha )\) and \(b\in \langle m \rangle \), we have
and the same reasoning shows \(\alpha (b) <_{X''} c \Longrightarrow c \vartriangleleft _{J''} \alpha (b)\). Completely symmetrically, this also shows that \(c<_{J''} d\Longrightarrow c \vartriangleleft _{X''} d\) for \(c,d\in \langle n+m1 \rangle \). \(\square \)
We again compile the colored operad \({{\,\mathrm{Patch}\,}}\) to obtain a graded noncolored operad
where an element \(x\in {{\,\mathrm{Patch}\,}}((q_{1},p_{1}),\ldots ,(q_{n},p_{n});(q,p))\) is graded as
and \({{\,\mathrm{Patch}\,}}_s(n)\) is generated as a kmodule by the sequences of constant degree. Its composition is derived from \({{\,\mathrm{Patch}\,}}\) where it is set to 0 when the colors do not match. Note in particular that the \({\mathbb {S}}_{n}\)action on \({{\,\mathrm{Patch}\,}}(n)\) is affected by this grading: permuting two vertices i and j introduces a sign \((1)^{(q_{i}+p_{i}1)(q_{j}+p_{j}1)}\).
Lemma 3.9
\({{\,\mathrm{Patch}\,}}_s\) is a dgsuboperad of \(({{\,\mathrm{mNSOp}\,}}_{st} \otimes _{H}{{\,\mathrm{NSOp}\,}}_s,(\partial _{D},Id))\).
Proof
It suffices to see that the elements \((D_q,1) \in {{\,\mathrm{Patch}\,}}((q,p);(q+1,p))\) for every \(p,q\in {\mathbb {N}}\). \(\square \)
3.4 The morphism \({{\,\mathrm{Patch}\,}}_s \longrightarrow \mathbf{End} (s\mathbf{C} _{GS}({\mathcal {A}}))\)
In this section we make the GScomplex \(\mathbf{C} _{GS}({\mathcal {A}})\) of a prestack \({\mathcal {A}}\) into a \({{\,\mathrm{Patch}\,}}_s\)algebra. We do so by making its underlying bicomplex \(\mathbf{C} ^{\bullet ,\bullet }({\mathcal {A}})\) into a \({{\,\mathrm{Patch}\,}}\)algebra. We first fix some notations.
Definition 3.10
Let \(\sigma = (U_{0} \overset{u_{1}}{\rightarrow } U_{1} \rightarrow \ldots \overset{u_{p}}{\rightarrow } U_{p} )\) be a psimplex in the category \({\mathcal {U}}\) and \(\zeta : [p'] \longrightarrow [p]\) a nondecreasing map (or equivalently a nondecreasing sequence), then let \({\overline{\zeta }}\) be the reflection of \(\zeta \), that is,
and define
a \(p'\)subsimplex of \(\sigma \), where \(N_\bullet ({\mathcal {U}})\) denotes the nerve construction on \({\mathcal {U}}\).
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 \((X,J) \in {{\,\mathrm{Patch}\,}}\), 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 \(\sigma \) in \({\mathcal {U}}\) 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 \(\sigma (a,b)\) that we place between them. For the other vertical inputs \(1 \le i \le q_a\), we determine a simplex \(\sigma _a(i)\) to place on top of a at input i. To do so, we determine the set of leftmost 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 \(\sigma (a,b)\), 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 \({\mathcal {U}}\) the subsimplex corresponding to its horizontal inputs. Hence, using our auxiliary set, we contract the corresponding subsimplices and obtain \(\sigma (a,b)\) and \(\sigma _a(i)\).
Note that we have not yet treated the multiplications m. In order to do so, we have to add the following complexity. Let \(X = [ I \circ _{n+1} m \circ _{n+1} \cdots \circ _{n+1} m]\) where I is an indexed tree with \(n+k\) vertices, then we call a vertex a of I nonplugged 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 nonplugged is equivalent to stating \(a\le n\).
Definition 3.12
We define a function \( \mathop \downarrow : \langle n+k \rangle \longrightarrow [ n ]\) on the vertices of I which associates to every vertex a the closest nonplugged vertex in X under or equal to a, or 0 if no such vertex exists. Concretely,
We also set \(a \vartriangleleft _{J} 0\) for every vertex a and define
where p is the total number of inputs of J.
Remark 3.13
This is clearly independent of the representative I of X. Moreover, \(\mathop \downarrow \) is for the given representative I the identity on \(\langle n \rangle \).
Next, we determine the auxiliary set.
Definition 3.14
Consider a vertex a of I and let \(b_{1} \vartriangleleft _{I} \ldots \vartriangleleft _{I} b_{t}\) be the children of a in I lying in \(\langle n \rangle \) with \(i_{s} := I(a,b_{s})\). We then define
for \(i \in [ q_{a} ]\), and
and let \(\min L_{a}(i)\) and \(\min L(a,b_s)\) be respectively the set of \(<_{J}\)minimal elements of \(L_{a}(i)\) or \(L(a,b_s)\).
Remark 3.15
Remark that \(L_{a}(i),L(a,b_s) \subseteq \langle n \rangle \) and thus that it contains only vertices which are not plugged by m. By default, we will set the subsimplex underneath the plugged children of a as empty (see Definition 3.18).
Remark 3.16
Note that the condition appearing in the first case becomes superfluous in the others.******
Definition 3.17
Let a be a vertex of I and \(\min L_{a}(i) = \{ e_{1} \vartriangleleft _{J} \ldots \vartriangleleft _{J} e_{l} \}\), then we have the sequence of inequalities
and thus the nondecreasing sequence
from which we delete \(\zeta _{J,e_i}(p_{e_i})\) if \(\zeta _{J,e_i}(p_{e_i}) =\zeta _{J,e_{i+1}}(0)\) or \(\zeta _{J,e_l}(p_{e_l}) = \zeta _{J,a}(0)\). This defines a subsimplex \(\sigma _{a}(i)\) of \(\sigma \) by Definition 3.10.
Definition 3.18
Let a, b be vertices of I such that b is a child of a, and \(\min L(a,b) = \{ e_{1} \vartriangleleft _{J} \ldots \vartriangleleft _{J} e_{l} \}\), then we have the sequence of inequalities

If \(i = I(a,b)\) for some vertex \(b \in \langle n \rangle \) (nonplugged)
$$\begin{aligned} \zeta _{J, b}(p_b) \le \zeta _{J,e_{1}}(0) \le \zeta _{J,e_{1}}(p_{e_{1}}) \le \cdots \le \zeta _{J,e_{l}}(p_{e_{l}}) \le \zeta _{J,a}(0) \end{aligned}$$ 
If \(i = I(a,b)\) for some vertex \(b > n\) (plugged)
$$\begin{aligned} \zeta _{J, b}(p_b) = \zeta _{J, a}(0) \end{aligned}$$
and thus the nondecreasing sequence
from which we delete \(\zeta _{J,e_i}(p_{e_i})\) if \(\zeta _{J,e_i}(p_{e_i}) =\zeta _{J,e_{i+1}}(0)\) or \(\zeta _{J,e_l}(p_{e_l}) = \zeta _{J,a}(0)\). We also delete \(\zeta _{J,b}(p_b)\) if it equals \(\zeta _{J,e_1}(0)\). This defines a subsimplex \(\sigma (a,b)\) of \(\sigma \) by Definition 3.10.
We consider an example.
Example 3.19
Given the simplex \(\sigma =(u_1,\ldots ,u_8)\) and the following patchwork of rectangles
we analyse the case for rectangle 6: we have
and thus
Now, we can assemble for every element of \({{\,\mathrm{Patch}\,}}\) a concrete patchwork of elements of \(\mathbf{C} _{GS}({\mathcal {A}})\) where the first coordinate determines a vertical patching using the operadic structure and the second component determines the horizontal patching to fill in and align the corresponding simplices.
Construction 3.20
Given \(\left( X,J\right) \in {{\,\mathrm{Patch}\,}}\left( \left( q_{1},p_{1}\right) ,\ldots ,\left( q_{n},p_{n}\right) ;\left( q,p\right) \right) \) and \(\theta _{i} \in \mathbf{C} ^{p_{i},q_{i}}\left( {\mathcal {A}}\right) \), then we set \(\theta _{s} = m \in \mathbf{C} ^{0,2}\left( {\mathcal {A}}\right) \) for \(s = n+1,\ldots ,n+k\).
Let \(\sigma \) be a psimplex in \({\mathcal {U}}\) and \(A = \left( A_{0},\ldots ,A_{q}\right) \) \((q+1)\)tuple of objects in \({\mathcal {A}}\left( U_{p}\right) \), then we define for every vertex a in I
and for every \(i\in \langle q_{a} \rangle \) we make the compositions

If \(i = I\left( a,b\right) \) for some vertex b,
$$\begin{aligned} \Theta _{a} \circ _{i} \left( \sigma \left( a,b\right) ^{\#}\circ _1 \Theta _{b}\right) \end{aligned}$$ 
Otherwise,
$$\begin{aligned} \Theta _{a} \circ _{i} \sigma _{a}\left( i\right) ^{\#}\left( A_{\zeta _{I,a}\left( i1\right) }{,}A_{\zeta _{I{,}a}\left( i\right) }\right) \end{aligned}$$
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 \({{\,\mathrm{mNSOp}\,}}\). This follows directly from the associativity of the local composition \(m^U\) of the category \({\mathcal {A}}(U)\) for every \(U \in {\mathcal {U}}\). \(\square \)
Let us work out an example.
Example 3.22
Consider the patching (X, J) from Example 3.7. Let \(\theta _{1} \in \mathbf{C} ^{5,3}({\mathcal {A}}), \theta _{2} \in \mathbf{C} ^{2,3}({\mathcal {A}}), \theta _{3} \in \mathbf{C} ^{1,2}({\mathcal {A}})\) and \(\theta _{4} \in \mathbf{C} ^{2,0}({\mathcal {A}})\), then we compute \({\mathcal {L}}(X,J)(\theta _{1},\theta _{2},\theta _{3},\theta _{4}) \in \mathbf{C} ^{7,6}({\mathcal {A}})\). Given the simplex \((u_{1},\ldots ,u_{7})\in N_{7}({\mathcal {U}})(U_{0},U_{p})\) and the objects \((A_{0},\ldots ,A_{6}) \in {\mathcal {A}}(U_{p})\), we first compute
Then, given \((a_{1},\ldots ,a_{6})\) where \(a_{i} \in {\mathcal {A}}(U_{p})(A_{i},A_{i1})\), we finally compute
which we can draw as follows
Proposition 3.23
Proof
We can assume without loss of generality that \(a = n\) as \({\mathcal {L}}\) is clearly equivariant.
Let \((X'',J'') := (X,J) \circ _{n} (X',J')\), then we add \('\) or \(''\) to denote the notions associated to \((X',J')\) or \((X'',J'')\). Let I and \(I'\) be the underlying trees representing X and \(X'\) having respectively \(n+k\) and \(m+k'\) vertices, then let \((\alpha ,\beta )\) be the extension of \(n+k\) by \(m+k'\) at n. Let \(({\overline{\alpha }},{\overline{\beta }})\) be the extension of n by m at n.
We compute
and show that it equals
for \(\sigma \in N_{p}({\mathcal {U}})\) and \(A=(A_{0},\ldots ,A_{q})\) objects in \({\mathcal {A}}(U_{p})\).
It is clear that per construction the blocks involved are composed according to \(X'' = X \circ _{n} X'\). Hence it suffices to verify that they correspond to the blocks \(\Theta _x''\) in \({\mathcal {L}}(X'',J'')\) 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 \(\Theta ''_x\) is either \(\Theta _{\beta (x)}\), or \(\Theta _{\alpha ^{1}(x)}'\) evaluated at \(\sigma ' = \zeta _{J,n}(\sigma )\). Next, we verify the simplices \(\sigma ''_i(x)\). For its ith input, we have the following two cases:

If x does not lie in the image of \((X',J')\), then \(\sigma _{\beta x}(i) = \sigma ''_x(i)\) because if \(n \in \min L_{\beta x}(i)\) then it is replaced by \({\overline{\alpha }}(r')\) for \(r'\) the root of \(J'\) for which holds \(\zeta _{J'',{\overline{\alpha }}(r')}=\zeta _{J,n}\zeta _{J',r'}\).
As a result, in both (2) and (3) we have the term \(\Theta _{\beta x} \circ _i \sigma _{\beta x}(i)^{\#}\).

If x is part of \((X',J')\), i.e. \(x= \alpha (x')\) for some vertex \(x'\), then \(\min L''_{\alpha (x')}(i)\) is the union of \(\min L'_{x'}(i) \) and \(\min L_{n}(i')\) for some \(i'\). Hence, we obtain the concatenation of \(\sigma '_{x'}(i)\) for \(\sigma ' = \zeta _{J,n}(\sigma )\) and \(\sigma _n(i')\). As \(\zeta ''_{J''} = \zeta _{J}\circ _n \zeta _{J'}\), this corresponds exactly to \(\sigma ''_{x}(i)\).
Hence, the corresponding term in both calculations agrees.
Next, we calculate \(\sigma ''(x,b)\) 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 \(\sigma ''(x,b) = \sigma (\beta x,\beta b)\) or \(\sigma '(\alpha ^{1}x ,\alpha ^{1}b)\) for \(\sigma ' = \zeta _{J',n}(\sigma )\) due to the previous reasoning and thus the terms agree.

If b lies in the image of \((X',J')\), i.e. \(b =\alpha (b')\), and x does not, then \(b'\) is clearly the root of \(X'\). As a result, \(\sigma ''(x,b) = \sigma (\beta x, n)\) and thus the terms agree.

If x lies in the image of \((X',J')\), i.e. \(x = \alpha (x')\), and b does not, then \(\min L''_{\alpha (x')}(i)\) is the union of \(\min L'_{x'}(i) \) and \(\min L_{n}(i')\) for some \(i'\).
Hence, we obtain in (2) the concatenation of \(\sigma (n,\beta b)\) and \(\sigma '_{x'}(i)\) for \(\sigma ' = \zeta _{J,n}(\sigma )\), which corresponds exactly to \(\sigma ''(x,b)\).
In case either x or b is plugged, we possibly have to apply the functorial property of the restrictions, i.e. \(u^{*}\circ m^U = m^V \circ (u ^{*}\otimes u ^{*})\) for \(u:V \rightarrow U\) in \({\mathcal {U}}\), to pull down \(\Theta _{\beta x} = m^{U_{\zeta _{J,\beta x}(0)}}\) or \(\Theta '_{\alpha ^{1}x} = m^{U_{\zeta _{J,n}\zeta _{J',\alpha ^{1}x}(0)}}\). Specifically, in the following cases

Let \(\beta x\) lie on top of n in (X, J) and \(\mathop \downarrow x = \alpha (y)\) for some vertex y of \((X',J')\).
In this case, \(\Theta _{\beta x}= m^{U_{\zeta _{J,n}(0)}}\) occurs in (2) and \(\Theta ''_x = m^{U_{\zeta _{J,n}\zeta _{J',y}(0)}}\) occurs in (3). Using functoriality, in (2) we equivalently have \(\tau ^{\#}\circ m^{U_{\zeta _{J,n}(0)}} = m^{U_{\zeta _{J,n}\zeta _{J',y}(0)}} \circ ( \tau ^{\#}\otimes \tau ^{\#})\) for an appropriate simplex \(\tau \). As a result they agree. Next, it is clear from the drawing that \(\sigma ''_x(j)\) is the concatenation of \(\sigma _{\beta x}(j)\) and \(\tau \). Moreover, for some vertex b, we have \(\sigma ''(x,b)\) as the concatenation of \(\sigma (\beta x ,\beta b)\) and \(\tau \), except in the case that b is plugged as well. In the latter case, we can also pull \(\Theta _{\beta b}\) in (2) down to \(\Theta _{\beta x}\) and obtain \(m^{U_{\zeta _{J,n}\zeta _{J',y}(0)}} = \Theta ''_x = \Theta ''_b\) as in (3).

Let \(\beta x\) lie on top of n, but \(\mathop \downarrow x \notin {{\,\mathrm{Im}\,}}(\alpha )\).
Again, we can pull down \(\Theta _{\beta x}\) in (2) past both functors \(\tau ^{\#}\) and \(\sigma (y,n)^{\#}\) and obtain \(m^{U_{\zeta _{J,\beta \mathop \downarrow y}(0)}}= \Theta ''_x\). The same reasoning as before also holds for the functors \(\sigma ''_x(j)\) and \(\sigma ''(x,b)\) in (3) and its counterparts \(\sigma _{\beta x}(j)\) and \(\sigma (\beta x ,\beta b)\) in (2).

The case where x lies in the image of \((X',J')\) such that \(\mathop \downarrow x \notin {{\,\mathrm{Im}\,}}(\alpha )\), is analogous to the previous one.
This finishes the proof. \(\square \)
Theorem 3.24
We obtain a morphism of dgoperads
Proof
The map \({\mathcal {L}}: {{\,\mathrm{Patch}\,}}\longrightarrow \mathbf{End} (\mathbf{C} ^{\bullet ,\bullet }({\mathcal {A}}))\) is clearly equivariant and thus it is a morphism of operads due to Proposition 3.23. Hence, the induced map \({\mathcal {L}}: {{\,\mathrm{Patch}\,}}_s \longrightarrow \mathbf{End} (s\mathbf{C} _{GS}({\mathcal {A}}))\) is a morphism of graded operads. Moreover, it is a morphism of dgoperads as \({\mathcal {L}}(D,1) = d_{0}\). \(\square \)
3.5 The morphism \({{\,\mathrm{Quilt}\,}}\longrightarrow {{\,\mathrm{Patch}\,}}_s\)
In [8], Hawkins defines a suboperad \({{\,\mathrm{Quilt}\,}}\subseteq {{\,\mathrm{F_{2}S}\,}}\otimes _{H}{{\,\mathrm{Brace}\,}}\) for which \({{\,\mathrm{Quilt}\,}}(n)\) is the free kmodule generated by \((W,T) \in {{\,\mathrm{F_{2}S}\,}}(n)\times {{\,\mathrm{Tree}\,}}(n)\) such that

(1)
\(W= \ldots {u} \ldots {v} \ldots \Longrightarrow u \not >_{T} v;\)

(2)
\(W= \ldots {u} \ldots {v}\ldots {u} \ldots \Longrightarrow v \vartriangleleft _{T} u.\)
Here, \(deg(W,T) := deg(W)\) and the boundary operator is \(\partial (W,T) := (\partial W,T)\).
Insightfully, elements of \({{\,\mathrm{Quilt}\,}}\) can also be drawn as a stacking of rectangles in the plane, as extensively explained in [8, §3.2]. We will use \({{\,\mathrm{Quilt}\,}}\) 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 \({{\,\mathrm{Quilt}\,}}\) on their side. We give an example.
Example 3.25
We consider an example from [8, Ex. 3.2] and flip it on its side as follows
Note the double line above rectangle 4: this reflects the fact that 3 is not interposed, otherwise the corresponding word would be 142434.
By definition, we have \({{\,\mathrm{Patch}\,}}_s \subseteq {{\,\mathrm{mNSOp}\,}}_s \otimes _H {{\,\mathrm{NSOp}\,}}_{st}\). In this section, we will construct a morphism of operads \({{\,\mathrm{Quilt}\,}}\longrightarrow {{\,\mathrm{Patch}\,}}_s\) as a restriction of
Lemma 3.26
Let \(Q=(W,T) \in {{\,\mathrm{Quilt}\,}}\), \(X \in {{\,\mathrm{Clr}\,}}(W,q_{1},\ldots ,q_{n})\) and \(I \in {{\,\mathrm{Clr}\,}}(T,p_{1},\ldots ,p_{n})\), then \((X,I) \in {{\,\mathrm{Patch}\,}}\).
Proof
Let \(u, v \in \langle n \rangle \), if \(u<_{I} v\), then \(u<_{T}v\) and thus \(W \ne \cdots {v} \ldots {u} \ldots \) and thus every occurrence of u in W is left of every occurrence of v in W. Hence, \(u \vartriangleleft _{X} v\).
The other way around, if \(u <_{X} v\), then \(W = \cdots {u} \ldots {v} \ldots {u} \ldots \) and thus \(v \vartriangleleft _{T} u\) which is equivalent to \(v \vartriangleleft _{I} u\). \(\square \)
We obtain a morphism of graded operads
defined as
where the sign \((1)^{\sigma }\) is defined as the Koszul sign obtained from switching
This is the consequence of the Hadamard product of two graded operads
where we have switched the order of the inputs.
Note in particular that this sign corresponds to the sign defined in [8, §4.7] and that we write \({{\,\mathrm{sgn}\,}}_Q(X,J) := {{\,\mathrm{sgn}\,}}_W(X) {{\,\mathrm{sgn}\,}}_T(J) (1)^{\sigma }\).
As a direct consequence of Lemma 3.9 we have the following.
Proposition 3.27
We have a morphism of dgoperads
Corollary 3.28
We have a morphism of dgoperads
Proof
Immediate from Theorem 3.24 and Proposition 3.27. \(\square \)
Our action of \({{\,\mathrm{Quilt}\,}}\) on the GScomplex 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 \(c^{u,v}: v ^{*}u ^{*}\longrightarrow (uv)^{^{*}}\).
4 Incorporating twists
The morphism \(R:{{\,\mathrm{Quilt}\,}}\longrightarrow \mathbf{End} (s\mathbf{C} _{GS}({\mathcal {A}}))\) from Corollary 3.28 only involves the multiplication m and the functors f of the data of a prestack \(({\mathcal {A}},m,f,c)\). In this section, we will incorporate the twists c by adding a formal element with certain relations, resulting in the bounded powerseries operad \({{\,\mathrm{Quilt_{b}}\,}}[[c]]\). In Sect. 4.4, we extend R above to a morphism \(R_c: {{\,\mathrm{Quilt_{b}}\,}}[[c]] \longrightarrow \mathbf{End} (s\mathbf{C} _{GS}({\mathcal {A}}))\) (see Theorem 4.17). In Hawkins’ approach to the presheaf case, the initial action of \({{\,\mathrm{Quilt}\,}}\) on \(\mathbf{End} (s\mathbf{C} _{GS}({\mathcal {A}}))\), which only involves the restriction maps f, is later extended in order to incorporate the multiplications m. As far as the structure of both approaches goes, our operad \({{\,\mathrm{Quilt_{b}}\,}}[[c]]\) is the counterpart of the operad \(\mathrm {mQuilt}\) from [8, Def. 5.2].
In [8, §7.1], Hawkins constructs a morphisms \(L_{\infty } \longrightarrow {{\,\mathrm{Quilt}\,}}\) (see Sect. 4.2). In Sect. 4.2, we establish a more involved morphism \(L_{\infty } \longrightarrow {{\,\mathrm{Quilt_{b}}\,}}[[c]]\) (see Theorem 4.10) by extending to an infinite series of higher components incorporating the element c.
Putting Theorems 4.10 and 4.17 together, we have thus endowed \(s\mathbf{C} _{GS}({\mathcal {A}})\) with an \(L_{\infty }\)structure. In the case of presheaves, this coincides on reduced and normalised cochains with the \(L_{\infty }\)structure from [8, Thm. 7.13].
In the final Sect. 4.5 we briefly discuss the relation of this structure with the deformation theory of the prestack \({\mathcal {A}}\).
4.1 Powerseries operads
In order to obtain an \(L_{\infty }\)structure incorporating twists, we will make use of operads of formal power series.
Definition 4.1
Let \({\mathcal {O}}\) be a graded operad, then define \({\mathcal {O}}[x]\) as the graded operad generated by \({\mathcal {O}}\) and an element x of degree t and define the subspaces
which is welldefined as there are no relations on x. Define
with componentwise \({\mathbb {S}}_{n}\)action and write their elements as \(\sum _{r\ge 0} Q_{r}\) for \(Q_{r} \in {\mathcal {O}}[x](n,r)\). For every \(1\le k \le n\) the composition of \({\mathcal {O}}[x]\) descends to a map
which extends to a composition map
We call an element \(\sum _{r\ge 0} Q_{r} \in {\mathcal {O}}[[x]]\) bounded if the set \(\{ deg(Q_{r}) : r \ge 0\}\subseteq {\mathbb {Z}}\) is bounded. Let \({\mathcal {O}}_{b}[[x]]\) be the \({\mathbb {S}}\)submodule of bounded series which is graded by the series with coefficients of constant degree.
Lemma 4.2

(1)
\({\mathcal {O}}[[x]]\) is an operad.

(2)
\({\mathcal {O}}_{b}[[x]]\) is a graded suboperad of \({\mathcal {O}}[[x]]\).

(3)
We have a sequence of injective operad morphisms
$$\begin{aligned} {\mathcal {O}}\hookrightarrow {\mathcal {O}}[x] \hookrightarrow {\mathcal {O}}_{b}[[x]] \hookrightarrow {\mathcal {O}}[[x]] \end{aligned}$$
Proof
These are straightforward computations. \(\square \)
We call \({\mathcal {O}}[[x]]\) the operad of powerseries with coefficients in \({\mathcal {O}}\) and \({\mathcal {O}}_{b}[[x]]\) the operad of bounded powerseries with coefficients in \({\mathcal {O}}\).
Definition 4.3
Consider \({{\,\mathrm{Quilt}\,}}[x]\), \({{\,\mathrm{Quilt}\,}}_b[[x]]\) and \({{\,\mathrm{Quilt}\,}}[[x]]\) for x a 0ary element \(x\in {{\,\mathrm{Quilt}\,}}(0)\) of degree \(1\), then let \({{\,\mathrm{Quilt}\,}}[c]\), \({{\,\mathrm{Quilt}\,}}_b[[c]]\) and \({{\,\mathrm{Quilt}\,}}[[c]]\) be their respective quotients under the following relations

(1)
\(\partial (c) =0\)
 (2)

(3)
\((W,T)\circ _{i} c = 0\) if i has more than two children in T or i is repeated in W (that is, it has a child in W).
Remark 4.4
In Definition 4.3, (1) determines that c encodes a natural transformation, (2) embodies the cocycle condition and (3) determines the form of c. The letter c will always stand for the twist subject to its relations, and should not be confused with a free variable.
On inspection of \({{\,\mathrm{mQuilt}\,}}\) from [8, Def. 5.2], we see that our conditions on c are a subset of those imposed on m in \({{\,\mathrm{mQuilt}\,}}\). Hence, we obtain a morphism \({{\,\mathrm{Quilt}\,}}[c] \longrightarrow {{\,\mathrm{mQuilt}\,}}\) sending c to m.
4.2 The morphism \(L_{\infty }\longrightarrow {{\,\mathrm{Quilt}\,}}\)
In [8, Thm 7.8] a morphism \(L_{\infty }\longrightarrow {{\,\mathrm{Quilt}\,}}: l_n \longmapsto L_n^0\) is defined by setting
In particular, this means that for every \(n\ge 2\) the equation
holds.
An important feature which we will need, is that we can write \(L_{n}^{0}\) as the antisymmetrization of elements \(P_{n}^{0}\). Namely, we set
then we have
In fact, the \(L_{\infty }\)relations translate to the following
where \(\langle  \rangle \) denotes ‘free kmodule generated by−’.
4.3 The morphism \(L_{\infty }\longrightarrow {{\,\mathrm{Quilt_{b}}\,}}[[c]]\)
Next we will define more involved \(L_{\infty }\)operations incorporating c.
Definition 4.5
For \(n+r\ge 2\) we define
where \((1)^{y_{1},\ldots ,y_{r}}\) denotes the sign of the (r, n)shuffle defined by \((y_{1},\ldots ,y_{r})\). Using these we set
Remark 4.6
Note that \(L_{n}^{r}\) live in \({{\,\mathrm{Quilt}\,}}[c]\).
Let us compute \(P_n^r\) for some low n and r.
Example 4.7
In case no elements c are added, we obtain the original \(P_n^0\)
Similar to how we drew elements from \({{\,\mathrm{mNSOp}\,}}\) as trees with vertices plugged by m, elements of \({{\,\mathrm{Quilt}\,}}[c]\) can be drawn as quilts with rectangles plugged by c. For example, we have
or
Note that depending on where we plug the elements c a sign is added.
This enables us to define the following.
Definition 4.8
Set
where
(a derivation by an element) which will be the new differential.
Remark 4.9
Note that \(L_{n}\) are bounded because their components have constant degree \(n2\). Hence, \(L_{n}\) live in \({{\,\mathrm{Quilt_{b}}\,}}[[c]]\).
The main theorem of this section is the following.
Theorem 4.10
The map
defines a morphism of dgoperads.
First we need some lemmas.
Lemma 4.11
For \(r \ge 2\) we have \(L^{r}_{0} = 0\).
Proof
Due to relation (3) in Definition 4.3 of c, there cannot be a vertical composition of c. Hence, \(L_{0}^{r} =0\) for \(r \ge 3\). \(L_{0}^{2}=0\) due to condition 2 of the definition of c. \(\square \)
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 \(L_{\infty }\)equation of \((L^{0}_{n})_{n}\) for higher \(L_{n}^{r}\).
Lemma 4.13
For \(n\ge 1\) and \(r\ge 0\), we have
Proof
By applying \(\partial \) and using Eq. (5) and \(\partial (c)=0\), we deduce
Given \(1 \le y_{1}< \ldots < y_{r} \le n+r\) and \(1 \le x \le p\), we have a subdivision into two groups where \(\circ _{y_{s}s+1}c\) is inserted into either \(P^{0}_{p}\) or \(P^{0}_{q}\). Hence, if we are also given a permutation \(\sigma \in {\mathbb {S}}_{n}\), then we show that there exists unique integers \(i+j=r, k+l=n+1\), indices \(1 \le z_{1}< \ldots < z_{i} \le k+i\) and \(1 \le z'_{1}< \ldots < z'_{j} \le j+l\), permutations \(\tau \in {\mathbb {S}}_{k+i},\tau '\in {\mathbb {S}}_{j+l}\) and a shuffle \(\chi \in Sh_{k1,l}\) such that
In this case, we obtain
By applying the definition of \(L_n\), this proves the lemma.
We show that Eq. (6) holds: the set \(y_{1}< \ldots < y_{r}\) splits into three subsets

\(z_{1}^{1}< \ldots< z_{i_{1}}^{1} \text { such that } z^{1}_{t} < x\),

\(z_{1}^{2}< \ldots < z_{i_{2}}^{2} \text { such that } z^{2}_{t} > x+q1\),

\(z_{1}^{'0}< \ldots< z_{j}^{'0} = \{y_{1}< \ldots< y_{r}\} \setminus \{ z_{1}^{1}< \ldots< z_{i_{1}}^{1}, z_{1}^{2}< \ldots < z_{i_{2}}^{2}\}\).
from which we define
We then compute
where the sign appears because we move the c’s corresponding to \(z_{t}^{'0}\) past j c’s corresponding to \(z^{'0}_{t}\) and also i c’s past \(P_{q}^{0}\). Note that if we know x and \(i_{1}\), then given \(i,j,k,l,z_{1},\ldots ,z_{i},z'_{1},\ldots ,z'_{j}\) we can uniquely determine \(p,q,r,n,y_{1},\ldots ,y_{r}\).
We also compute the sign of \(y_{1},\ldots ,y_{r}\): let \(\theta \) and \(\theta '\) be the shuffles such that
then we obtain that
On the other hand,
and thus we have
Given \(\sigma \in {\mathbb {S}}_{n}\), let \(b_{1}<\ldots <b_{l}\subseteq \langle k+l1\rangle \) such that \(\sigma (b_{t}) \in \{xi_{1},\ldots ,xi_{1}+l1\}\) and let \(a_{1}<\ldots< a_{k_1} := \langle k+l1 \rangle \setminus \{b_{1}< \ldots < b_{l} \}\). We then define the \((k1,l)\)shuffle \(\chi = (a_{1}< \ldots< a_{k1},b_{1}< \ldots < b_{l})\) and
It is then easy to see that
Note that \(\tau \) and \(z_{1},\ldots ,z_{i}\) determine x and \(i_{1}\) uniquely. Hence, given \(z_{1},\ldots ,z_{i},z'_{1},\ldots ,z'_{j},k,l\) we can uniquely determine \(y_{1},\ldots ,y_{r},p,q,n,r\) in the above manner. In order to show that Eq. (6) holds, we only need to verify the corresponding signs: let \(\tau _{0}\) be the permutation such that
and thus we have that \((1)^{\sigma +\chi } {=} (1)^{\tau _{0}{+}\tau '+l(kx+i_{1})}\). On the other hand, we have that \(\tau \) corresponds to
and thus we have
As such, we can compute
and
which completes the proof. \(\square \)
Lemma 4.14
\(L_{n}\) are skew symmetric and \(\partial '\) is a differential making \({{\,\mathrm{Quilt_{b}}\,}}[[c]]\) into a dgoperad.
Proof
It is clear from the definition of \(L_{n}^{r}\) that they are skew symmetric and thus also \(L_{n}\).
Per definition \(\partial _{L_{1}}\) is a derivation by construction and so is \(\partial \), and thus so is \(\partial '\).
It is clear from the definition of c that \(\partial '(c)= 0\). Hence, we only need to show that \(\partial ' \partial '(Q) = 0\) for every \(Q\in {{\,\mathrm{Quilt}\,}}\). Using Lemmas 4.11 and 4.13 we first prove that \(\partial L_{1} = L_{1} \circ _{1} L_{1}\). Namely, we compute
Now let us compute
we compute separately
and
adding them gives
Next, we compute \(\partial _{L_{1}}\partial _{L_{1}} Q\). As \(L_{1}\) has only a single input and has degree \(1\), we have for \(i\ne j\) that \(Q \circ _{i} L_{1} \circ _{j} L_{1} =  Q \circ _{j} L_{1} \circ _{i} L_{1}\). Hence, by also using using Eq. (9), we obtain
As \(\partial \partial Q =0\) we thus obtain \(\partial '\partial ' Q= 0\). \(\square \)
Proof of Theorem 4.10
We need to show for every \(n\ge 2\) that the equation
holds, which is equivalent to
This is equivalent to showing for every \(r\ge 0\) that the equation
holds, which follows from Lemma 4.13 and \(L^{i}_{0} = 0\) for \(i\ge 0\) (Lemma 4.11). \(\square \)
Remark 4.15
Under the natural morphism \({{\,\mathrm{Quilt}\,}}[c] \longrightarrow {{\,\mathrm{mQuilt}\,}}\) sending c to m our \(L_{\infty }\)structure corresponds to the \(L_{\infty }\)structure from [8, Thm. 7.13], that is, we have the commutative diagram
where \({{\,\mathrm{Quilt}\,}}_{b}[[m]]\) denotes the quotient of the operad of bounded powerseries by the relations on m in \({{\,\mathrm{mQuilt}\,}}\).
4.4 The morphism \({{\,\mathrm{Quilt_{b}}\,}}[[c]] \longrightarrow \mathbf{End} (s\mathbf{C} _{GS}({\mathcal {A}}))\)
In this section, we make (the suspension of) \(\mathbf{C} _{GS}({\mathcal {A}})\) into a \({{\,\mathrm{Quilt_{b}}\,}}[[c]]\)algebra.
The morphism \(R: {{\,\mathrm{Quilt}\,}}\longrightarrow \mathbf{End} (s\mathbf{C} _{GS}({\mathcal {A}}))\) naturally extends to a morphism of graded operads \(R_{c}:{{\,\mathrm{Quilt}\,}}[c] \longrightarrow \mathbf{End} (s\mathbf{C} _{GS}({\mathcal {A}}))\) by sending c to \(c\in \mathbf{C} _{GS}({\mathcal {A}})\) as the axioms of \({{\,\mathrm{Quilt}\,}}[c]\) correspond respectively to c being a natural transformation, the cocycle condition of c and \(c\in \mathbf{C} ^{2,0}({\mathcal {A}})\). Next, we will show that it further extends to the operad of bounded power series.
Lemma 4.16
Let \(\theta _{1},\ldots ,\theta _{n}\in \mathbf{C} _{GS}({\mathcal {A}})\) where \(\theta _{i} \in C^{p_{i},q_{i}}({\mathcal {A}})\) and \(Q \in {{\,\mathrm{Quilt}\,}}[c](n,r)\) of degree t, then \(R_{c}(Q)(\theta _{1},\ldots ,\theta _{n}) = 0\) if \(r > \sum _{i=1}^{n}q_{i}  t\).
Proof
This follows from \(\mathbf{C} _{GS}({\mathcal {A}})\) having only nonnegative bidegree. Namely, given \(Q = Q' \circ _{i_{1}} c \circ _{i_{2}} \ldots \circ _{i_{r}} c\) for a certain \(Q'\in {{\,\mathrm{Quilt}\,}}(n+r)\), the bidegree of \(R_{c}(Q')\) is \((\sum _{i=1}^{n}p_{i}+ 2r (n+r1), \sum _{i=1}^{n}q_{i}  deg(Q'))\). As \(deg(Q') = deg(Q) + r = t +r\), we have that
proving the lemma. \(\square \)
Theorem 4.17
For \({\mathcal {A}}= ({\mathcal {A}},m,f,c)\) a linear prestack over \({\mathcal {U}}\), the map
defined as \(R_{c}(Q) = R(Q)\) for \(Q\in {{\,\mathrm{Quilt}\,}}\) and \(R_{c}(c) = c\), is a morphism of dgoperads.
Proof
The representation \(R_{c}\) extends to \({{\,\mathrm{Quilt_{b}}\,}}[[c]]\) due to Lemma 4.16.
We verify that \(R_{c}\partial 'Q = \partial _{d}R_{c}Q \) for \(Q\in {{\,\mathrm{Quilt}\,}}\). It suffices to verify that \(R_{c}L_{1}^{j} = d_{j}\) as \(\partial \) corresponds to \(\partial _{d_{0}} \). Let \(\theta \in \mathbf{C} ^{p,q}({\mathcal {A}})\) and \(\sigma \in N_{p+j}({\mathcal {U}})\) and note that we write \(\theta \) for the degree \(p+q\) of \(\theta \) in \(\mathbf{C} _{GS}({\mathcal {A}})\).
Step 2 \(R_c L_1^1 = d_1\)
We can write \(d_1 = \sum _{s=0}^{p+1}(1)^{p+q+1+s}d_1^s\) where \(d_1^s\) names the sth component of \(d_1\). We write out the lefthand side
where . There exists two colorings \((X_0,J_0)\) and \((X_{p+1},J_{p+1})\) of the quilt \(Q = (W,T)\) in \(P_2^0(c,\theta )\) given by the patchworks
which correspond respectively to \(d_1^0\) and \(d_1^{p+1}\). We verify their signs: the sign \({{\,\mathrm{sgn}\,}}_Q(X_0,J_0)= {{\,\mathrm{sgn}\,}}_W(X_0) {{\,\mathrm{sgn}\,}}_T(J_0) (1)^{ q}\) is determined by two shuffles
and thus \({{\,\mathrm{sgn}\,}}_Q(X_0,J_0) = (1)^{p1 +q}\). In the second case, we have \({{\,\mathrm{sgn}\,}}(X_{p+1},J_{p+1}) = {{\,\mathrm{sgn}\,}}_{W}(X_{p+1}) {{\,\mathrm{sgn}\,}}_{T}(J_{p+1})(1)^q \) determined by the two shuffles
Hence, \({{\,\mathrm{sgn}\,}}_Q(X_{p+1},J_{p+1}) = (1)^q\).
There exists p colorings \((X_i,J_i)_{i=1}^p\) of \(P_{2}^{0}\) occuring in the second term of (11), given by the patchworks
for \(i= 1,\ldots ,p\). They correspond to \(d_{1}^{p+1i}\) and we verify their signs: we have \({{\,\mathrm{sgn}\,}}_Q(X_i,J_i) = {{\,\mathrm{sgn}\,}}_W(X_i){{\,\mathrm{sgn}\,}}_T(J_i)\) determined by the two shuffles
Hence, we obtain the sign
Step 2 \( R_c L_1^j = d_j\)
First, we name the terms of \(d_j\) and write
such that
Next, we note that the only nonvanishing term of \(L_{1}^{j}\) is given by \(\left( 1\right) ^{\left( j1\right) }P_{j+1}^{0} \circ _{1} c \circ _{2} c \circ _{2} \ldots \circ _{2} c\) as c cannot have any children vertically. Moreover, the only nonvanishing quilts are of the form
for any binary tree T on \(j1\) vertices whose numbering is compatible with the word component W of \(Q_T\). Hence, the possible colorings of \(Q_{T}\) are
for \(t_{1}<\ldots <t_{j1}\) in \(\langle q \rangle \) and J an indexed tree coloring T.
As a result, we have that
which sums over the terms
We finish the proof by showing that a formal path r and a shuffle \(\beta \in {\mathbb {S}}_{q+1j,j1}\) correspond uniquely to such a binary tree T and a coloring \(\left( X,J'\right) \) of \(Q_{T}\) such that
Given a formal path \(r=\left( r_{1},\ldots ,r_{j1}\right) = \left( \left( \tau _1,i_1\right) ,\ldots ,\left( \tau _{j1},i_{j1}\right) \right) \) we first define T and its coloring J as trees with vertex set \(\{3,\ldots ,j+1\}\) 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 \(\left( (\tau _1,i_1),\ldots ,(\tau _{j2},i_{j2})\right) \), in order to add vertex \(j+1\) we have three cases

(1)
If \(i_{j1}< i_{j2}\), then set \(j \vartriangleleft _{J} (j+1)\) and start over with \((i_{j1},i_{j+1})\)

(2)
If \(i_{j1} = i_{j2}\) or \(i_{j1} = i_{j2}+1\), then let \((j,j+1)\in E_{T}\) and set resp. \(J(j,j+1)= 2\) or \(= 1\).

(3)
If \(i_{j1} > i_{j2}+1\), then set \((j+1) \vartriangleleft _J j\) and start over with \((i_{j1},i_{j+1}1)\)
which we can draw as follows

(1)
Clearly, this process is reversible: given (T, J) we obtain a unique sequence \(r\in P\left( R_p(\sigma )\right) \). By identifying the shuffle \(\beta \in Sh_{q+1j,j1}\) and X via \(t_{l} = \beta (qj+1+l)\), we clearly obtain that \({\mathcal {L}}(X,I')(\theta )^{\sigma } = d_j(\theta )^{\sigma }(\beta ,r)\).
The remaining work is to verify the signs: \({{\,\mathrm{sgn}\,}}_{Q_T}(X,J')\) consists of three components
where \((1)^{\sigma } = (1)^q\) as it corresponds to the shuffle
The sign \({{\,\mathrm{sgn}\,}}_{W'}(X)\) corresponds to the shuffle transforming the word
for \(t= qj+1\), into the word
We observe that shuffling the second part (2)
almost corresponds to the shuffle \(\beta \). However, there is in every interval \(\beta (s)_2,\ldots , (\beta (s+1)1)_2\) exactly one element too many. We remedy this by moving \(1_{(j2)}\) one place to the right, then \(1_{(j3)}\) two place, and so on. As such, its corresponding sign is \((1)^{\beta + \sum _{i=1}^{j2} i } = (1)^{\beta + \frac{(j2)(j1)}{2}}\). Next, we shuffle
whose sign is \((1)^{(j1)(q+2)}\). Hence, we obtain that
Next, we determined \({{\,\mathrm{sgn}\,}}_{T'}(J')\) 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 \(\chi : [C_J] \leadsto 1_3 \ldots 1_{(j+1)}\) is \((1)^{r + j1}\). As a consequence, we obtain that
and thus that
Hence, we have
which completes the proof.
We compute \(\chi \) inductively: we have that \([C_J] = A 1_{(j+1)} B\) for certain words A and B and let \(\chi _0\) denote the shuffle \(AB \leadsto 1_3 \ldots 1_j\). By induction we know that \((1)^{\chi _0}= (1)^{r_0 + j2}\) for the formal path \(r_0= (r_1,\ldots ,r_{j2})\) and \((1)^{r} = (1)^{r_0 + i_{j1}}\) where \(r_{j1} = (\sigma , i_{j1})\). Moreover, we have \((1)^{\chi } = (1)^{\chi _0 + B}\) where B denotes the length of B. We determine B. First, observe that the sequence associated to two indexed trees
is respectively \(1_{h4}1_{h3}\) and \(1_{h3}1_{h4}\). Thus, let \(3 = v_{1}<_{T} \ldots <_{T} v_{t} = j+1\) 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 \(i_{j1}\) of \(r_{j1}=(\partial _{i_{j1}}\sigma ,i_{j1})\) is exactly \((1+l)+k\),

Length of B is \(l+k\)
and thus by induction we obtain
where the last equality follows from induction. \(\square \)
Remark 4.18
In the case of presheaves, when looking at the subcomplex \(\overline{\mathbf{C }}_{GS}({\mathcal {A}})\) of normalised and reduced cochains, the map \(R_{c}\) factorises through \({{\,\mathrm{Quilt}\,}}_{b}[[m]]\) as m is sent to the identity \(1^{u,v}:u^{*}v^{*}= (vu)^{*}\). Only in the case of normalised and reduced cochains does the identity satisfy all the relations on m.
Note however that our induced morphism \({{\,\mathrm{mQuilt}\,}}\longrightarrow {{\,\mathrm{Quilt}\,}}_{b}[[m]] \longrightarrow \mathbf{End} (s\overline{\mathbf{C }}_{GS}({\mathcal {A}}))\) does not correspond to the \({{\,\mathrm{mQuilt}\,}}\)algebra structure [8, Thm. 5.6].
However, the resulting \(L_{\infty }\)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 \({{\,\mathrm{mQuilt}\,}}\)algebra structure also induces a Gerstenhaberalgebra structure on cohomology [8, Thm. 6.11]. Writing down the relevant quilts, it is also easy to see that both \({{\,\mathrm{mQuilt}\,}}\)algebra structures (ours and [8, Thm. 5.6]) induce the same Gerstenhaberalgebra structure on cohomology.
4.5 Deformations of prestacks
Let k be a field of characteristic 0. Let \(({\mathbf {C}}, d, (L_n)_{n \ge 2})\) be an \(L_{\infty }\)algebra. By definition, the Maurer–Cartan equation for \(\theta \in {\mathbf {C}}\) is given by
We consider the set of (degree 1) Maurer–Cartan elements \({MC}({\mathbf {C}}) = \{ \theta \in {\mathbf {C}}^1 \,\,  \,\, MC(\theta ) = 0\}\) and for the appropriate notion of gauge equivalence (see [9]), we consider
This gives rise to a functor \({\underline{MC}}_{{\mathbf {C}}}: \mathrm {Art} \longrightarrow \mathsf {Set}: (R, {\mathfrak {m}}) \longmapsto {\underline{MC}}({\mathfrak {m}} \otimes {\mathbf {C}})\) on the category \(\mathrm {Art}\) of Artin local kalgebras.
Consider the GScomplex \(({\mathbf {C}}_{GS}({\mathcal {A}}), d)\) of a prestack \(({\mathcal {A}},(m+f+c))\). In [2, Thm. 3.19] it is shown that normalised reduced 2cocycles in \({\mathbf {C}}_{GS}({\mathcal {A}})\) correspond to first order deformations of \({\mathcal {A}}\), that is, deformations in the direction of \(R = k[\epsilon ]\). More precisely, for \((m', f', c')\in \mathbf{C} _{GS}^{2}({\mathcal {A}})\), we have that \(({\mathcal {A}}[\epsilon ],m+m'\epsilon ,f+f'\epsilon ,c+c'\epsilon )\) is a firstorder deformation of \(({\mathcal {A}},m,f,c)\) if and only if \(d(m', f', c')=0\) and \((m', f', c')\) is normalised and reduced. Further, it is shown in loc. cit. that the cohomology of the GScomplex classifies the firstorder deformations of \({\mathcal {A}}\) up to equivalence.
Putting Theorems 4.10 and 4.17 together, \(s\mathbf{C} _{GS}({\mathcal {A}})\) is endowed with an \(L_{\infty }\)structure which can be used to obtain higher order versions of these results.
For normalised, reduced cochains, it will be convenient to express the MCequation in terms of the unsymmetrised components \((P_{n})_{n\ge 1}\) from Definition 4.5. The following characteristicfree expression of the MCequation should be seen as the counterpart of the equation \(MC(\theta ) = d(\theta ) + \theta \{ \theta \}\) 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 \(\mathbf{End} (s\mathbf{C} _{GS}({\mathcal {A}}))\).
Proposition 4.19
For a reduced and normalised cochain \(\theta = (m',f',c') \in s\mathbf{C} _{GS}^{1}({\mathcal {A}})\), we have
In particular, the MCequation is quartic.
Proof
First note that as \(\theta  =1\), that \(Q^{\sigma }(\theta ,\ldots ,\theta ) = (1)^{\sigma }Q(\theta ,\ldots ,\theta )\). Moreover, as we can consider \(c\in \mathbf{C} ^{2,0}({\mathcal {A}})\), the MCequation consists of \(d_0( \theta ')\) and summations of \(Q(\theta _{1},\ldots ,\theta _{n})\) for quilts Q of degree \(n2\) and \(\theta _{i} \in \mathbf{C} _{GS}^{2}({\mathcal {A}})\). Let \(Q=(W,T)\) be such a quilt, part of some \(P_{n}^{0}\), then \(W= 12\ldots 2\). For \(n\ge 3\), we know that 3 is also a child in T of 1. As 1 can have at most two children in T, for \(n \ge 4\), we thus have 4 and 3 are children of 2 in W, i.e. \(W= 12324\ldots 2\).
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 \(Q(\theta ,\ldots ,\theta ) =0\) for \(n\ge 4\).
As c is not necessarily reduced, more quilts are possible. Note that we write \(\theta [i]\) to refer to the element \(\theta \) inserted in vertex i. The above reasoning still applies and we can once more apply this reasoning to obtain for \(n\ge 6\) that \(W= 12324546\ldots 42\) 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 \({\mathcal {A}}(U)\) and it is plugged into \(m'[2]\) which is normalised, whence we obtain that \(Q(\theta _{1},\ldots ,\theta _{n}) =0\) for \(n \ge 6\). Hence, \(P_n^0(\theta _{1},\ldots ,\theta _{n}) =0\) for \(n \ge 6\).
Combining the above reasonings, we have that \(P_{n}(\theta ,\ldots ,\theta ) = 0\) for \(n> 5\). In the case \(n=5\) there has to be at least one c present and thus the nonzero terms are contained in \(P_{4}\). \(\square \)
Proposition 4.20
For a prestack \(({\mathcal {A}},m,f,c)\) and \(\theta = (m', f', c') \in s\mathbf{C} _{GS}^1({\mathcal {A}})\), we have that \(({\mathcal {A}},m+m',f+f',c+c')\) is a prestack if and only if \(MC(\theta ) = 0\) and \(\theta \) is normalised and reduced.
Proof
Using the fact that \(\theta = (m', f', c') \in \mathbf{C} _{GS}^{2}({\mathcal {A}})\) is reduced and normalised, we compute \(MC(\theta )\) and look at each component \(MC(\theta )_{[p,q]} \in \mathbf{C} ^{p,q}({\mathcal {A}})\) 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 \(\theta = (\theta ^\sigma (A))_{\sigma ,A}\), we omit writing the set of objects A explicitly where possible in order to lighten the equations below.
For \((p,q) = (0,3)\), let \( U \in {\mathcal {U}}\) and \(A=(A_0,A_1,A_2,A_3)\) objects in \({\mathcal {A}}(U)\), then we compute
For \((p,q) = (1,2)\), let \(u:U_0 \rightarrow U_1\) in \({\mathcal {U}}\) and \(A=(A_0,A_1,A_2)\) objects in \({\mathcal {A}}(U_1)\), then we compute
and thus
For \((p,q)= (2,1)\), let \(\sigma =(U_0 \overset{u_1}{\rightarrow } U_1 \overset{u_2}{\rightarrow } U_2)\) be a 2simplex in \({\mathcal {U}}\) and \(A=(A_0,A_1)\) objects in \({\mathcal {A}}(U_2)\), then we compute
and thus
Lastly, for \((p,q) = (3,0)\), let \(\sigma =(U_0 \overset{u_1}{\rightarrow } U_1 \overset{u_2}{\rightarrow } U_2 \overset{u_3}{\rightarrow } U_3)\) be a 3simplex in \({\mathcal {U}}\) and \(A_0\) an object in \({\mathcal {A}}(U_2)\), then we compute
and thus
These computations show that \(MC(\theta ) = 0\) if and only if \(({\mathcal {A}},m+m',f+f',c+c')\) is a prestack. \(\square \)
Remark 4.21
Note that for the functor condition we only need the cubic part and for the twists the full quartic part of the equation.
Based upon Proposition 4.20, with some more work taking gauge equivalence into account, one may extend [2, Thm. 3.19] to higher order deformations. Recall that for \((R, {\mathfrak {m}}) \in \mathrm {Art}\) an Rdeformation of a klinear prestack \(({\mathcal {A}},m,f,c)\) is an Rlinear prestack \((R \otimes _k {\mathcal {A}}, {\bar{m}}, {\bar{f}}, {\bar{c}})\) of which the algebraic structure reduces to that of \({\mathcal {A}}\) modulo \({\mathfrak {m}}\), and an equivalence of deformations is an isomorphism between the deformed prestacks which reduces to the identity morphism. Let \(\mathrm {Def}_{{\mathcal {A}}}: \mathrm {Art} \longrightarrow \mathsf {Set}\) be the deformation functor of \({\mathcal {A}}\) with \(\mathrm {Def}_{{\mathcal {A}}}(R, {\mathfrak {m}})\) the set of Rdeformations of \({\mathcal {A}}\) up to equivalence of deformations. The following theorem, of which the proof will appear elsewhere, expresses that the deformation theory of \({\mathcal {A}}\) is controlled by the \(L_{\infty }\)algebra \(s\mathbf{C} _{GS}({\mathcal {A}})\).
Theorem 4.22
Let \(({\mathcal {A}},m,f,c)\) be a prestack. There is a natural isomorphism of functors \(\mathrm {Art} \longrightarrow \mathsf {Set}\):
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Acknowledgements
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.
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This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 817762). The second named author holds a Ph.D. fellowship of the Research Foundation—Flanders (FWO). The third named author was partially supported by the HSE University Basic Research Program.
Appendix A: Generatorfree description of the morphism \({{\,\mathrm{NSOp}\,}}\longrightarrow \hbox {Multi}\Delta \)
Appendix A: Generatorfree description of the morphism \({{\,\mathrm{NSOp}\,}}\longrightarrow \hbox {Multi}\Delta \)
In this appendix, we provide a generatorfree description of the morphism \({{\,\mathrm{NSOp}\,}}\longrightarrow \text {Multi}\Delta \) from Lemma 2.5. Although per construction we have a morphism of operads induced from the generators \(E_i\) of \({{\,\mathrm{NSOp}\,}}\), we consider it valuable in practice to have an explicit definition.
For I an indexed tree in \({{\,\mathrm{NSOp}\,}}(q_{1},\ldots ,q_{n};\sum _{i=1}^n q_i  n+1)\), we call \(l_I =\sum _{i=1}^n q_i  n+1\) the number of leaves of I. Given a number \(m\in {\mathbb {N}}\), we call the interval
the numbering set of leaves of I.
Construction A.1
For I an indexed tree in \({{\,\mathrm{NSOp}\,}}(q_{1},\ldots ,q_{n};\sum _{i=1}^n q_i n +1)\), let \(I_j\) be the maximal subtree of I with root j. Let u be the root of I with children \(u_1 \vartriangleleft _I \ldots \vartriangleleft _I u_k\) which have index \(i_j:= I(u,u_j)\), then I decomposes as follows
Given a number \(m\in {\mathbb {N}}\), we define a nondecreasing map \(\zeta (I,m): [q_u] \longrightarrow [m,m+l_I]\) as follows
which determines where the leaves of the root of I are placed in its numbering set of leaves \([m,m+l_I]\).
In order to define the tuple \(\zeta _I = (\zeta _{I,1},\ldots ,\zeta _{I,n}) \in \text {Multi}\Delta (q_{1},\ldots ,q_{n};\sum _{i=1}^nq_i  n + 1)\), we run inductively through the tree I from root u to leaves setting
and for vertex a with child b and index \(i:=I(a,b)\) we set
with \(l_I = \sum _{i=1}^nq_i  n + 1\).
Note that Construction 2.5 corresponds to the above construction applied to the generators \(E_i\). As such, if the generatorfree description defines a morphism of operads, they coincide.
Proposition A.2
Construction A.1 defines a morphism of operads \({{\,\mathrm{NSOp}\,}}\longrightarrow \text {Multi}\Delta \).
Proof
Let \(I\in {{\,\mathrm{NSOp}\,}}(q_{1},\ldots ,q_{n}; q)\) and \(I' \in {{\,\mathrm{NSOp}\,}}(q'_{1},\ldots ,q'_{m}; q')\) and consider their composition \(I'':= I \circ _i I'\). We will show that
As in Construction A.1, let \(I_j, I'_j\) and \(I''_j\) be the maximal subtrees of respectively \(I,I'\) and \(I''\) with root j. Due to equivariance, we can assume that a vertex j belongs to the subtree \(I_i\) iff \(j\ge i\). We then have three cases to consider which we depict diagrammatically as follows

(1)
For \(j<i\), if we contract the subtree \(I_i\) and \(I''_i\) to a single vertex with number of leaves \(l_{I_i} = l_{I''_i}\), we obtain the same indexed tree \(I\setminus I_i = I'' \setminus I''_i\) in which \(I_j\) and \(I''_j\) coincide. Thus, it is easy to see that
$$\begin{aligned} \zeta _{I'',j} = \zeta _{I,j} \end{aligned}$$ 
(2)
If \(j\ge i\) and j lies in the image \({{\,\mathrm{Im}\,}}(I')\) of \(I'\) in \(I''\), then \(I''_j\) consists of the subtree \(I'_{ji+1}\) with a sequence of subtrees \(I_{t},\ldots ,I_{t'}\) placed on top.
In this case, \(\zeta _{I',ji+1}\) determines where the leaves of j are placed in \(I'\) and \(\zeta _{I,i}\) determines where the leaves of i in I are placed. As a result, we see that
$$\begin{aligned} \zeta _{I'',j} = \zeta _{I,i}\circ \zeta _{I',ji+1} \end{aligned}$$which determines where the leaves of j are put in \(I''\).

(3)
If \(j\ge i\) and j does not lie in the image of \(I'\) in \(I''\), then \(jm+1\) lies in I above i. In this case, the subtrees \(I_{jm+1}\) and \(I''_j\) are equal. If the parent a of j in \(I''\) does not lie in \({{\,\mathrm{Im}\,}}(I')\), then we clearly have
$$\begin{aligned} \zeta _{I'',j} = \zeta _{I,jm+1}. \end{aligned}$$If a lies in \({{\,\mathrm{Im}\,}}(I')\), then \(\zeta _{I'',a}=\zeta _{I,i}\circ \zeta _{I',ai+1}\) due to the previous case.
Moreover, the index \(I''(a,j)\) then equals \(I(i,j)  \zeta _{I',ai+1}(0)\). Hence, we have that
$$\begin{aligned} \zeta _{I'',j} = \zeta _{I,jm+1} \end{aligned}$$
\(\square \)
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Dinh Van, H., Hermans, L. & Lowen, W. Operadic structure on the Gerstenhaber–Schack complex for prestacks. Sel. Math. New Ser. 28, 47 (2022). https://doi.org/10.1007/s00029022007591
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DOI: https://doi.org/10.1007/s00029022007591
Keywords
 Prestack
 Hochschild cohomology
 Gerstenhaber–Schack complex
 Operad
 Brace operations
 \(L_{\infty }\)algebra