\documentclass[ALCO]{cedram} \usepackage{amssymb,mathrsfs,enumitem,amsmath,amsthm} %\usepackage[mathscr]{euscript} \usepackage{graphicx} \usepackage{tipa} \usepackage{transparent} \usepackage{import} \usepackage{bbm} \newcommand\cyrillic[1]{ {\fontencoding{OT2}\fontfamily{wncyr}\selectfont #1} } \newcommand\mathcyr[1]{\text{\cyrillic{#1}}} \newcommand\Sha{\textnormal{\mathcyr{Sh}}} \usepackage{hyperref} \usepackage[OT2,OT1]{fontenc} \usepackage{tikz-cd} \usepackage{tikz} \usetikzlibrary{shapes,fit,positioning,calc,matrix} \tikzset{ optree/.style={scale=.5,thick,grow'=up,level distance=10mm,inner sep=1pt}, comp/.style={draw=none,circle,fill,line width=0,inner sep=0pt}, dot/.style={draw,circle,fill,inner sep=0pt,minimum width=3pt}, circ/.style={draw,circle,inner sep=1pt,minimum width=4mm}, emptycirc/.style={draw,circle,inner sep=1pt,minimum width=2mm}, root/.style={level distance=10mm,inner sep=1pt}, leaf/.style={draw=none,circle,fill,line width=0,inner sep=0pt}, nodot/.style={draw,circle,inner sep=1pt}, } \newcommand{\ac}{\scriptstyle \text{\rm !`}} \newcommand{\calA}{\mathcal{A}} \newcommand\calB{\mathcal{B}} \newcommand\calC{\mathcal{C}} \newcommand\calD{\mathcal{D}} \newcommand\calE{\mathcal{E}} \newcommand\calF{\mathcal{F}} \newcommand\calG{\mathcal{G}} \newcommand\calH{\mathcal{H}} \newcommand\calI{\mathcal{I}} \newcommand\calJ{\mathcal{J}} \newcommand\calK{\mathcal{K}} \newcommand\calL{\mathcal{L}} \newcommand\calM{\mathcal{M}} \newcommand\calN{\mathcal{N}} \newcommand\calO{\mathcal{O}} \newcommand\calP{\mathcal{P}} \newcommand\calQ{\mathcal{Q}} \newcommand\calR{\mathcal{R}} \newcommand\calS{\mathcal{S}} \newcommand\calT{\mathcal{T}} \newcommand\calU{\mathcal{U}} \newcommand\calV{\mathcal{V}} \newcommand\calW{\mathcal{W}} \newcommand\calX{\mathcal{X}} \newcommand\calY{\mathcal{Y}} \newcommand\calZ{\mathcal{Z}} \DeclareMathOperator{\Set}{\mathsf{Set}} \DeclareMathOperator{\Vect}{\mathsf{Vect}} \DeclareMathOperator{\dgVect}{\mathsf{dgVect}} \DeclareMathOperator{\sfC}{\mathsf{C}} \DeclareMathOperator{\rmR}{\mathrm{R}} \DeclareMathOperator{\Gr}{\mathsf{Gr}} \DeclareMathOperator{\CGr}{\mathsf{CGr}} \DeclareMathOperator{\GrCol}{\mathsf{GrCol}} \DeclareMathOperator{\Red}{Red} \DeclareMathOperator{\Tw}{Tw} \DeclareMathOperator{\Iso}{Iso} \DeclareMathOperator{\Aut}{Aut} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\Fun}{Fun} \DeclareMathOperator{\Des}{Des} \DeclareMathOperator{\Rec}{\mathsf{Rec}} \DeclareMathOperator{\grCom}{\mathsf{grCom}} \DeclareMathOperator{\grLie}{\mathsf{grLie}} \DeclareMathOperator{\grGerst}{\mathsf{grGerst}} \DeclareMathOperator{\grGrav}{\mathsf{grGrav}} \DeclareMathOperator{\grHyperCom}{\mathsf{grHyperCom}} \DeclareMathOperator{\grBV}{\mathsf{grBV}} \newcommand\kk{\Bbbk} \newcommand{\ptwo}[2]{\ensuremath{ \vcenter{\hbox{\xymatrix@R=.4pc@C=.9pc{ %@R=.7mm@C=1mm{ *+[o][F-]{#1}\ar@{-}[r]&*+[o][F-]{#2} }}}}} \title{Reconnectads} \author[\initial{V.} Dotsenko]{\firstname{Vladimir} \lastname{Dotsenko}} %% Do not include any other information inside \author's argument! %% Other author data have special commands for them: \address{ Institut de Recherche Math\'ematique Avanc\'ee, UMR 7501\\ Universit\'e de Strasbourg et CNRS\\ 7 rue Ren\'e-Descartes\\ 67000 Strasbourg CEDEX\\ France} %% e-mail address \email{vdotsenko@unistra.fr} \author[\initial{A.} Keilthy]{\firstname{Adam} \lastname{Keilthy}} %% Do not include any other information inside \author's argument! %% Other author data have special commands for them: \address{Department of Mathematical Sciences\\ Chalmers Technical University and the University of Gothenburg\\ SE-412 96 Gothenburg\\ Sweden} %% e-mail address \email{keilthy@chalmers.se} \author[\initial{D.} Lyskov]{\firstname{Denis} \lastname{Lyskov}} %% Do not include any other information inside \author's argument! %% Other author data have special commands for them: \address{National Research University Higher School of Economics\\ 20 Myasnitskaya street\\ Moscow 101000\\ Russia} %% e-mail address \email{ddl2001@yandex.ru} %% Acknowledgements are not a footnote in %% \author, but are given apart: \thanks{The first author was supported by Institut Universitaire de France, by Fellowship USIAS-2021-061 of the University of Strasbourg Institute for Advanced Study through the French national program ``Investment for the future'' (IdEx-Unistra), and by the French national research agency (project ANR-20-CE40-0016). The second author was supported by the ``Postdoctoral program in Mathematics for researchers from outside Sweden'' (project KAW 2020.0254). Results of Section 6 were obtained under support of the grant RSF 22-21-00912 of Russian Science Foundation.} % Key words and phrases: \keywords{Feynman categories, graph associahedra, Koszul duality, toric varieties} %% Mathematical classification (2010) %% This will not be printed but can be useful for database search \subjclass{14M25, 18M70, 18M75} \begin{document} \begin{abstract} We introduce a new operad-like structure that we call a reconnectad; the ``input'' of an element of a reconnectad is a finite simple graph, rather than a finite set, and ``compositions'' of elements are performed according to the notion of the reconnected complement of a subgraph. The prototypical example of a reconnectad is given by the collection of toric varieties of graph associahedra of Carr and Devadoss, with the structure operations given by inclusions of orbits closures. We develop the general theory of reconnectads, and use it to study the ``wonderful reconnectad'' assembled from homology groups of complex toric varieties of graph associahedra. \end{abstract} \maketitle \section{Introduction} In this paper, we define and study a new algebraic structure for which we propose the term \emph{reconnectad}\footnote{The intended pronunciation is \textipa{[rIk@nE"ktA:d]} with the stress on the last syllable, as if it were a French word.}. It captures a certain self-similarity of stratifications of toric varieties whose dual polytopes are the so called graph associahedra, originally defined by Carr and Devadoss in \cite{MR2239078}. Their original goal was to find convex polytopes that would give tilings of Coxeter complexes of general Coxeter groups in the same way the associahedra of Stasheff give tilings of the Deligne--Mumford compactifications $\overline{\calM}_{0,n}(\mathbb{R})$ of the moduli space of real projective lines with marked points. The answer found in \cite{MR2239078}, originating in the theory of wonderful models for subspace arrangements due to De Concini and Procesi \cite{MR1366622}, turned out to be quite remarkable. Particular cases of that construction lead to well known families of polytopes: the associahedra \cite{MR0158400,MR146227}, the cyclohedra \cite{MR1295465}, and the permutahedra~\cite{MR305800}, and the general notion of a graph associahedron has been studied in a wide range of contexts, including algebraic combinatorics, complex algebraic geometry, polyhedral geometry, theoretical computer science, toric topology etc. See, for example, a highly non-exhaustive list of references \cite{MR4176852,MR3340192,MR2997505,MR3439304,MR2252112}. To mention two remarkable concrete examples of applications, face posets of certain graph associahedra are responsible for the ``correct'' combinatorics behind the algebraic structures arising in Floer homology \cite{MR2764887,MR3342676}, and the intersection theory on complex toric varieties of stellahedra is shown to be invaluable for studying combinatorial invariants of matroids, see \cite{MR4477425,Stellar}. \\ Specifically for the associahedra themselves, the corresponding complex toric varieties were studied in \cite{MR4072173} where the natural nonsymmetric operad structure on the homology of those varieties was studied; it was found that many properties of that nonsymmetric operad $\mathsf{ncHyperCom}$ are remarkably similar to the known properties of the symmetric operad $\mathsf{HyperCom}$ obtained as the homology of the operad of complex Deligne-Mumford compactifications $\overline{\calM}_{0,n}(\mathbb{C})$, controlling what is known as the tree level part of a cohomological field theory \cite{MR1702284}. One important deficiency, however, is that the operad $\mathsf{ncHyperCom}$ is not cyclic, meaning that there is no meaningful notion of a compatible scalar product on an algebra over that operad. To deal with this problem, the second author of this paper proposed to view the operadic structure on toric varieties of associahedra in a wider context. This paper is the first step of this bigger programme: we exhibit a way to organise toric varieties of all possible graph associahedra in a remarkable operad-like structure. Classically, components of operads are indexed by finite sets in a functorial way, so that automorphisms of finite sets acts on components. In the situation that we consider, components are indexed by simple connected graphs in a functorial way, and structure operations arise from inclusions of orbit closures of the tori; those orbit closures are products of toric varieties of smaller graph associahedra which is the self-similarity we referred to above. The combinatorics of orbit closures is closely related to the notion of the reconnected complement of a subgraph in a graph, hence the terminology that we propose. \\ The motivation of De Concini and Procesi was to construct a compactification of the complement of the given subspace arrangement by gluing in a normal crossing divisor. In our case, the hyperplane arrangement in question is the arrangement of coordinate hyperplanes, and hence we already are dealing with the complement of a divisor with normal crossings. However, something bizarrely remarkable happens: even in this extremely simple situation, there are other nontrivial choices of compactifications that turn out to be notable algebraic varieties. To give a simple example, if one chooses the \emph{maximal building set} and blows up all possible intersections, in other words, if one considers the case of the complete graph, the resulting varieties, first studied by Procesi \cite{MR1252661}, are found in the work Losev and Manin \cite{MR1786500}; they encode a certain version of the notion of a cohomological field theory \cite{LosevPolyubin,MR3112505}, and which were recently shown in \cite{DSVV} to play the role analogous to that played by the spaces $\overline{\calM}_{0,n+1}(\mathbb{C})$ in an analogue of the BV formalism arising in topological quantum mechanics~\cite{Lysov}. The operad-like structure on these varieties is very close to that of that describing permutads of Loday and Ronco \cite{MR2995045}, though without a total ordering of the underlying set.\\ A closely related though different operad-like structure in the context of graph associahedra was recently defined by Forcey and Ronco in \cite{jlms.12596} using the formalism of operadic categories of Batanin and Markl \cite{MR3406537}. Our approach is different in several ways. First, the formalism of \cite{jlms.12596} imposes a total ordering of the set of vertices and thus is closer to ``shuffle reconnectads''; second, our approach allows us to view reconnectads as monoids in a certain monoidal category, similarly to how it can be done for operads. The advantage of such approach is that there is a very powerful existing range of ideas and methods available, and we are able to use those ideas and methods in a very meaningful way. If one wishes to place our approach under the umbrella of a general formalism for studying generalisations of operads, Feynman categories of Kaufmann and Ward \cite{MR3636409} furnish an example of such a formalism; however, for us, several other more concrete approaches turn out to be available. \\ It is worth mentioning that our work may be viewed as a shadow of a much more general theory developed by Coron \cite{CoronLattices} for the full generality of geometric lattices and their building sets. However, since unlike the \emph{op. cit.}, our focus is on a very particular case (of Boolean lattices and their graphical building sets), a lot of general statements become much more concrete, some generally very complicated objects become much simpler, and, as a result, some elegant combinatorial patterns shine through. We hope that our work will help in further understanding of the Feynman category of built lattices of \cite{CoronLattices}, and in placing constructions from toric geometry like the Bergman fan of a matroid \cite{MR2191630} in the categorical context.\\ An intriguing question that is for the moment left outside the scope of this paper is to give a generalisation of the Batalin-Vilkovisky operad $\mathsf{BV}$ in the context of reconnectads, and to compute its homotopy quotient by the circle action, at least on the algebraic level, generalising some of the results of \cite{DSVV}. This task that is not obvious for a number of reasons. First, it is absolutely crucial for reconnectads to have no nontrivial elements associated to the empty graph, which is where the BV operator would be expected to appear. Second, for the triangle graph there are two contradicting wishes of what the corresponding component of the BV reconnectad should be: viewing the triangle as a cycle relates it to path graphs and the $\mathrm{ncBV}$ operad of \cite{MR4072173}, while viewing it as a complete graph relates it to the $\mathrm{tBV}$ twisted associative algebra of \cite{DSVV}, and reconciling those two relationships is not an easy task. We hope to address this question elsewhere.\\ The paper is organised as follows. In Section \ref{sec:recollections}, we recall some necessary background information. In Section \ref{sec:graphical-vars}, we present three equivalent constructions of toric varieties of graph associahedra, including an interpretation as a ``graphical Grassmannian'' which allows us to give a new explicit description of the stratification by toric orbits (Theorem \ref{th:orbits}). In Section \ref{sec:reconnectads}, we give several equivalent definitions of a reconnectad, identify two known particular cases, and define the ``commutative reconnectad''. In Section \ref{sec:algebra-reconnectads}, we develop a wide range of methods for studying algebraic reconnectads (reconnectads whose components are vector spaces). Finally, in Section \ref{sec:wonderful}, we define the gravity reconnectad and determine its presentation by generators and relations (Theorem \ref{th:gravpresent}), obtain a presentation by generators and relations of the ``wonderful reconnectad'' formed by the collection of homologies of all complex toric varieties of graph associahedra (Theorem \ref{th:homologyhyper}), and give an algebraic and a geometric proof of Koszul duality between these reconnectads and of the Koszul property of both of them. Geometrically, the reconnectadic Koszul duality is implemented by the compactifications to open orbits of the torus action on these varieties (Proposition \ref{prop:delignekoszul}); this is analogous to the celebrated result of Getzler \cite{MR1363058}.\\ \section{Background, notations, and recollections}\label{sec:recollections} By $\sfC$ we denote a category that has all coproducts (which we denote $\oplus$) and an initial object (which we denote by $0$), and is equipped with a symmetric monoidal structure (for which we denote the monoidal product by $\otimes$ and the unit object by $1_{\sfC}$) that distributes over coproducts. In most applications, $\sfC$ will be either the category of ``spaces'' (topological spaces, projective varieties, etc) or the category of ``modules'' (vector spaces, chain complexes, etc). For a finite set $I$ and a family of objects $\{V_i\}_{i\in I}$ of $\sfC$, the unordered monoidal product of these objects is defined by \[ \bigotimes_{i\in I} V_i:=\left(\bigoplus_{(i_1,\ldots,i_n) \text{ a total order on } I} V_{i_1}\otimes\cdots\otimes V_{i_n}\right)_{\Aut(I)}. \] In cases where we work with ``modules'', we assume that all of them are defined over a field $\kk$ of zero characteristic. All chain complexes are graded homologically, with the differential of degree~$-1$. To handle suspensions of chain complexes, we introduce a formal symbol~$s$ of degree~$1$, and define, for a graded vector space~$V$, its suspension $sV$ as $\kk s\otimes V$. \subsection{Toric varieties} Let us give a short summary of basics of toric varieties, referring the reader to \cite{MR495499,MR1234037} for more details. We denote by $\mathbb{G}_m$ the algebraic group $\mathop{\mathrm{Spec}}\left(\kk[x,x^{-1}]\right)$, i.e. the multiplicative group $\kk^\times$. An \emph{algebraic torus} is a product of several copies of $\mathbb{G}_m$. A \emph{toric variety} is a normal algebraic variety $X$ that contains a dense open subset $U$ isomorphic to an algebraic torus, for which the natural torus action on $U$ extends to an action on $X$. Toric varieties may be constructed from the combinatorial data of a \emph{lattice} (free finitely generated Abelian group) $N$ and a \emph{fan} (collection of strongly convex rational polyhedral cones closed under taking intersections and faces) $\Sigma$ in $N\otimes_{\mathbb{Z}}\mathbb{R}$. Each cone in a fan gives rise to an affine variety, the affine spectrum of the semigroup algebra of the dual cone. Gluing these affine varieties together according to face maps of cones gives an algebraic variety denoted $X(\Sigma)$ and called the toric variety associated to the fan $\Sigma$. It is known that a toric variety $X(\Sigma)$ is projective if and only if $\Sigma$ is a normal fan of a convex polytope $\calP$, uniquely determined from $\Sigma$ up to normal equivalence. In such situation, we also use the notation $X(\calP)$ instead of $X(\Sigma)$. The variety $X(\calP)$ is smooth if and only if $\calP$ is a \emph{Delzant polytope}, that is a polytope for which the slopes of the edges adjacent to each given vertex form a basis of the lattice~$N$. For a complex toric variety $X_{\mathbb{C}}(\calP)$ corresponding to an $n$-dimensional Delzant polytope $\calP$, the Betti numbers of $X_{\mathbb{C}}(\calP)$ are given by the coefficients $h_i$ of the $h$-polynomial of $\calP$ \[ \sum_{i=0}^n h_it^i=\sum_{i=0}^n f_i(t-1)^i, \] where $f_i$ denotes the number of faces of $\calP$ of dimension $i$. \subsection{Wonderful compactifications of subspace arrangements}\label{sec:rec-wond} We also give a short summary of basics of (projective) wonderful compactifications of subspace arrangements in the particular case of the the coordinate subspace arrangements, referring the reader to \cite{MR1366622,MR2038195,MR2746338} for details as well as for the general theory. Let $I$ be a finite set. We shall consider the vector space $\kk^I$, and the collection of coordinate subspaces $C_T:=\{x_t=0\colon t\in T\}$ in that vector space. The Boolean lattice $2^I$ of all subsets of $I$ ordered by inclusion can be identified with the poset of all sums of the subspaces $C_T$, ordered by reverse inclusion. A \emph{building set} of $2^I$ is a collection $G$ of subsets of $2^I\setminus\{\emptyset\}$ such that for each $x\in 2^I$ the natural map \[ \prod_{g\in \max(G\cap [\emptyset,x])}[\emptyset,g]\to [\emptyset,x] \] sending a tuple of elements to their union is an isomorphism of posets. This definition is valid in full generality of atomic lattices \cite{MR2038195}, and in the case of $2^I$ is equivalent to containing all singletons $\{i\}$ and containing, together with any two elements $x,y\in 2^I$ with $x\cap y\ne\emptyset$, their union $x\cup y$. We shall only consider building sets that contain the whole set $I$. Since every building set contains all singletons, for each building set $G$, the complement in $\kk^I$ of the union of the subspaces $C_T$ for $T\in G$ is the algebraic torus $\mathbb{G}_m^I$. Since $\mathbb{G}_m^I=(\kk^\times)^I$, we have the maps \[ \mathbb{G}_m^I/\mathbb{G}_m\to\mathbb{P}(\kk^I/(\kk^T)^\bot) \] for all $T\in G$, and therefore a map \begin{equation}\label{eq:torusmap} \mathbb{G}_m^I/\mathbb{G}_m\to\mathbb{P}(\kk^I)\times\prod_{T\in G} \mathbb{P}(C_T) . \end{equation} The \emph{projective wonderful compactification} $\overline{Y}_G$ of $\mathbb{G}_m^I/\mathbb{G}_m$ is the closure of the image of $\mathbb{G}_m^I/\mathbb{G}_m$ under the map \eqref{eq:torusmap}. It is known that for every building set $G$, $\overline{Y}_G$ is a smooth projective irreducible variety. The natural projection map $\pi\colon \overline{Y}_G\to \mathbb{P}(\kk^I)$ is surjective, and restricts to an isomorphism on $\mathbb{G}_m^I/\mathbb{G}_m$. Additionally, the complement $\overline{Y}_G\setminus (\mathbb{G}_m^I/\mathbb{G}_m)$ is a divisor with normal crossings. Its irreducible components $D_T$ are in one-to-one correspondence with elements $T\in G\setminus\{I\}$, and we have \[ \pi^{-1}(\mathbb{P}(C_T))=\bigcup_{T\leq T'} D_{T'} \ . \] Intersections of the divisors $D_T$ may be described combinatorially using the notion of a nested set. For a fixed building set $G$, a subset $\tau$ of $G$ is said to be \emph{nested} if for any elements $T_1,T_2,\ldots,T_k\in \tau$ which are pairwise incomparable (as subsets of $I$, that is by inclusion), we have $T_1\cup T_2\cup \cdots\cup T_k\notin G$. It is known that for $\tau\subset G$, the intersection $D_\tau=\bigcap_{T\in \tau} D_T$ is non-empty if and only if $\tau$ is nested. The collection of all nested sets forms a simplicial complex $\tilde{N}(I,G)$ with the set of vertices $G$. Topologically, $\tilde{N}(I,G)$ is a cone with apex $\{I\}$, and the link of $\{I\}$ is denoted $N(I,G)$ and called the \emph{nested set complex} with respect to $G$. A closely related notion is the \emph{augmented nested set complex} obtained by adding to $N(I,G)$ one $-1$-simplex $\varnothing$. For our purposes, it will be convenient to realize the augmented nested set complex as the set $N^+(I,G)$ of all nested sets containing~$I$ as an element; removing $I$ from such a nested set gives a bijection with the augmented nested set complex as described above. \begin{exam} Let $G$ be the maximal building set of $2^{\{1,2\}}$, that is, \[G=\{\{1\},\{2\},\{1,2\}\}.\] Then \[ N(\{1,2\},G)=\{ \{\{1\}\}, \{\{2\}\}, \{\{1\},\{2\}\}, \] while \[ N^+(\{1,2\},G)=\{ \{\{1,2\}\}, \{\{1\},\{1,2\}\}, \{\{2\},\{1,2\}\}, \{\{1\},\{2\},\{1,2\}\} \} \] \end{exam} Let us recall a useful way to visualise elements of $N^+(I,G)$ using labelled rooted trees going back to \cite{MR2191630,MR2487491}. \begin{defi}\label{def:lambda} Suppose that $\tau\in N^+(I,G)$. We associate to $\tau$ a rooted tree $\mathbb{T}_\tau$ whose vertices have additional labels by disjoint subsets of $I$. The rooted tree structure is defined as follows. We take $\tau$ as the set of vertices, and say that a vertex $T$ is a parent of another vertex $T'$ if in the restriction of the order of $2^I$ to $\tau$ the element $T$ covers the element $T'$. To define extra labels, we associate to each vertex $T$ the subset \[ \lambda(T):=T\setminus \bigcup_{T'\in \tau, T'\subset T} T' . \] \end{defi} \begin{exam} Let $G$ be the maximal building set of $2^{\{1,2,3,4\}}$. In Figure~\ref{fig:nested-tree}, we give an example of the labelled rooted tree $\mathbb{T}_\tau$ corresponding to the element \[\tau=\{\{1\},\{3,4\},\{1,2,3,4\}\}\] of $N^+(I,G)$. For example, the root vertex of the tree corresponds to $\{1,2,3,4\}\in\tau$, and the label of that vertex is $\{2\}=\{1,2,3,4\}\setminus(\{1\}\cup \{3,4\})$. (We always depict rooted trees in the way that the root is at the bottom.) \begin{figure} \caption{Rooted tree associated to a nested set} \label{fig:nested-tree} \[ \tau=\{\{1\},\{3,4\},\{1,2,3,4\}\} \qquad \Leftrightarrow \qquad \mathbb{T}_\tau=\vcenter{\hbox{\begin{tikzpicture}[scale=0.7] \draw[thick] (0,0.5)--(-1,2); \draw[thick] (0,0.5)--(1,2); \draw[fill=white, thick] (0,0.5) circle [radius=15pt]; \draw[fill=white, thick] (-1,2) circle [radius=15pt]; \draw[fill=white, thick] (1,2) circle [radius=20pt]; \node at (0,0.5) {\scalebox{1}{$\{2\}$}}; \node at (-1,2) {\scalebox{1}{$\{1\}$}}; \node at (1,2) {\scalebox{1}{$\{3,4\}$}}; \node (n) at (0,-0.5) {}; \end{tikzpicture}}} \] \end{figure} \end{exam} \subsection{Graphs}\label{sec:graphs-def} By a \emph{graph} we shall mean what is usually referred to as a finite simple graph. In other words, the datum of a graph is a pair $\Gamma=(V_\Gamma,E_\Gamma)$, where $V_\Gamma$ is a (possibly empty) finite set of \emph{vertices} and $E_\Gamma\subset V_\Gamma^2$ is a symmetric irreflexive relation on $V_\Gamma$ (two vertices $v_1,v_2$ such that $(v_1,v_2)\in E_\Gamma$ are said to be \emph{connected by an edge}). We shall frequently use the following particular examples of graphs: \begin{itemize} \item the empty graph $(\emptyset,\emptyset)$, which we denote simply by $\emptyset$, \item the complete graph $K_n$ on the vertex set $\{1,\ldots,n\}$, whose edges are all possible pairs $(i,j)$ with $i\ne j$, \item the stellar graph $S_n$ on the vertex set $\{0,\ldots,n\}$, whose edges are all possible pairs $(0,i)$ with $1\le i\le n$, \item the path graph $P_n$ on the vertex set $\{1,\ldots,n\}$, whose edges are all possible pairs $(i,i+1)$ with $1\le i\le n-1$, \item the cycle graph $C_n$ on the vertex set $\mathbb{Z}/n\mathbb{Z}$, whose edges are all possible pairs $(i,i+1)$ with $i\in\mathbb{Z}/n\mathbb{Z}$. \end{itemize} The \emph{disjoint union of two graphs} is defined by the formula \[ \Gamma\sqcup \Gamma' = (V_\Gamma\sqcup V_{\Gamma'}, E_\Gamma\sqcup E_{\Gamma'}). \] We shall denote by $C_\Gamma\subset V_\Gamma\times V_\Gamma$ the minimal equivalence relation containing~$E_\Gamma$. If every two vertices of $\Gamma$ belong to the same equivalence class of $C_\Gamma$, we say that our graph $\Gamma$ is \emph{connected}, otherwise, $\Gamma$ coincides with the disjoint union of its \emph{connected components} $\mathrm{Conn}(\Gamma)$: \[ \Gamma=\bigsqcup_{\Gamma'\in\mathrm{Conn}(\Gamma)}\Gamma'. \] Recall that for a subset $V$ of $V_\Gamma$, the corresponding \emph{induced subgraph} is the graph $\Gamma_V$ whose vertex set is $V$ and whose edges are precisely the edges that exist in $\Gamma$. \section{Graphical varieties}\label{sec:graphical-vars} In this section, we shall discuss a remarkable collection of algebraic varieties associated to graphs. It relies on a beautiful combinatorial construction that emerged independently in \cite{MR2239078,MR2487491,MR2470574,MR2252112}. \subsection{Graph associahedra, graphical building sets, graphical nested sets}\label{sec:graph-assoc} Let $\Gamma$ be a graph. We define \emph{graphical building set} $G_\Gamma$ of the Boolean lattice $2^{V_\Gamma}$ as the set of all subsets $V\subseteq V_\Gamma$ for which the induced graph $\Gamma_V$ is connected (in the context of graph associahedra such subsets are often referred to as the tubes of~$\Gamma$, following \cite{MR2239078}). The \emph{graph associahedron} $\calP\Gamma$ is the convex polyhedron obtained as follows. Take a simplex with the vertex set $V_\Gamma$, and index each of its faces by the vertices it misses. Then truncate the faces corresponding to elements of $G_\Gamma$, in the increasing order of dimension. It follows from \cite[Th.~2.6]{MR2239078} that the combinatorial type of $\calP\Gamma$ does not depend on the way in which the partial order by dimension is refined to a total order. As an example, suppose that $\Gamma$ is the path graph $P_3$. Then the vertices of the simplex are the two-dimensional subsets $\{1,2\}$, $\{1,3\}$, and $\{2,3\}$, of which $\{1,3\}\notin G_\Gamma$. Thus, we should truncate two vertices of the triangle, obtaining the pentagon, which is the classical Stasheff associahedron $K^2$. For the two connected graphs on three vertices, the corresponding polyhedra (in this case, polygons) are displayed in Figure~\ref{fig:hedra}. \begin{figure}[ht] \begin{center} \includegraphics[scale=0.25]{figures/cyclohedron.pdf} \end{center} \caption{Graph associahedra for the path $P_3$ and the cycle $C_3$} \label{fig:hedra} \end{figure} In the context of graph associahedra, one often uses the combinatorics of ``tubings''; however, the precise definition of that notion varies throughout the literature, for instance the tubings of \cite{MR2239078} are elements of the nested set complex $N(V_\Gamma,G_\Gamma)$, while tubings of \cite{jlms.12596} are elements of the realisation $N^+(V_\Gamma,G_\Gamma)$ of the augmented nested set complex. Both notions arise naturally in different aspects of the story; to simplify the notation, we denote $N(V_\Gamma,G_\Gamma)$ by $N(\Gamma)$ and $N^+(V_\Gamma,G_\Gamma)$ by $N^+(\Gamma)$. It follows easily from the definition that both $N(\Gamma)$ and $N^+(\Gamma)$ consist of sets $\{T_1,\ldots,T_r\}$ of subsets of $V_\Gamma$ for which each graph $\Gamma_{T_i}$ is connected and for all $i\ne j$ either one of the subsets $T_i$ and $T_j$ is a subset of the other or $\Gamma_{T_i\cup T_j}=\Gamma_{T_i}\sqcup \Gamma_{T_j}$; the only difference is that in the case of $N(\Gamma)$ we require that all $T_i$ are proper subsets of $V_\Gamma$, and in the case of $N^+(\Gamma)$ one of them must be equal to $V_\Gamma$. Two examples of nested sets in $N^+(\Gamma)$ for two different graphs $\Gamma$ are displayed in Figure~\ref{fig:nested}. \begin{figure}[ht] \centering \includegraphics[width=0.7\linewidth]{figures/tubings.pdf} \caption{Examples of nested sets in $N^+(\Gamma)$} \label{fig:nested} \end{figure} It is also prudent to note that in \cite{MR4425832} the notion of a ``nesting'' of a graph are used, where edges rather than vertices are in the spotlight; in our terminology, those correspond to the nested sets of the derived graph \cite{MR262097}. \subsection{Toric varieties of graph associahedra and wonderful compactifications} In the context of toric geometry, each truncation of a face of a polytope $\calP$ corresponds to a certain blow up of the toric variety $X(\calP)$. In our case, the simplex with the vertex set $V_\Gamma$ corresponds to the projective space $\mathbb{P}(\kk^{V_\Gamma})$, and the toric variety $X(\calP\Gamma)$ can be obtained from the latter by iterated blow ups centered at coordinate subspaces $\mathbb{P}(U_T)$ corresponding to elements of $G_\Gamma$, in the increasing order of dimension. \begin{prop}\label{prop:wondtoric} For the graphical building set $G_\Gamma$ of the Boolean lattice $2^{V_\Gamma}$, the corresponding projective wonderful compactification $\overline{Y}_{G_\Gamma}$ of $\mathbb{G}_m^{V_\Gamma}/\mathbb{G}_m$ is naturally isomorphic to the toric variety $X(\calP\Gamma)$. \end{prop} \begin{proof} Recall that $\overline{Y}_{G_\Gamma}$ is the closure of the image of the map \[ \mathbb{G}_m^{V_\Gamma}/\mathbb{G}_m\to\mathbb{P}(\kk^{V_\Gamma})\times\prod_{T\in G_\Gamma} \mathbb{P}(C_T). \] The codomain of this map has the obvious action of $\mathbb{G}_m^{V_\Gamma}/\mathbb{G}_m$, and the map is equivariant, so the algebraic torus acts on $\overline{Y}_{G_\Gamma}$ with an open orbit, therefore the variety $\overline{Y}_{G_\Gamma}$ is toric. Moreover, the projection from $\overline{Y}_{G_\Gamma}$ on each individual factor $\mathbb{P}(\kk^{V_\Gamma}/g^\bot)$ is surjective, and hence by \cite[Prop.~8.1.4]{MR2394437}, the polytope corresponding to the toric variety $\overline{Y}_{G_\Gamma}$ is the Minkowski sum of simplices corresponding to elements of $G_\Gamma$. The same is known \cite{MR2487491} for each graph associahedron $\calP\Gamma$. \end{proof} \subsection{Graphical Grassmannians} Let us give an equivalent description of the toric variety $X(\calP\Gamma)$ as the parameter space of certain collections of subspaces in a vector space; for that reason, we shall call that parameter space a ``graphical Grassmannian''. This notion is very close to that of a type A brick manifold \cite{MR4072173,MR3512647} in the case where $\Gamma$ is a path graph, and so we shall use the symbol $\calB$ to emphasize that relationship. \begin{defi} Let $\Gamma$ be a graph. The \emph{graphical Grassmannian} $\calB(\Gamma)$ parametrises collections $\{H_T\}_{T\in G_\Gamma}$ of subspaces of $\kk^{V_\Gamma}$ satisfying the following properties: \begin{itemize} \item $H_T\subset \kk^{T}\subset \kk^{V_\Gamma}$, \item $\dim H_T=|T|-1$, \item if $T\subset T'$, then $H_T\subset H_{T'}$. \end{itemize} \end{defi} \begin{prop} The graphical Grassmannian $\calB(\Gamma)$ is naturally isomorphic to the toric variety $X(\calP\Gamma)$. \end{prop} \begin{proof} In view of Proposition \ref{prop:wondtoric}, it is enough to identify $\calB(\Gamma)$ with the projective wonderful compactification $\overline{Y}_{G_\Gamma}$ of $\mathbb{G}_m^{V_\Gamma}$. We shall see that this is a simple consequence of projective duality, exactly as in \cite[Th.~5.2.1]{MR4072173}. The variety $\overline{Y}_{G_\Gamma}$ is the closure of the image of the map \[ \mathbb{G}_m^{V_\Gamma}/\mathbb{G}_m\to\mathbb{P}(\kk^{V_\Gamma})\times\prod_{T\in G_\Gamma} \mathbb{P}(C_T). \] Using the canonical basis of $\kk^{V_\Gamma}$, we may identify the linear dual $(\kk^{V_\Gamma})^*$ with $\kk^{V_\Gamma}$. Under this identification, a point of $\mathbb{P}(C_T)$ is identified with a hyperplane in $\kk^{T}$, so a point in $\prod_{T\in G_\Gamma} \mathbb{P}(C_T)$ gives rise to a collection of subspaces $\{H_T\}_{T\in G_\Gamma}$ satisfying the first two conditions. Moreover, the third condition is also satisfied because of the compatibility of the canonical basis of $\kk^{V_\Gamma}$ with the canonical bases of all $\kk^T$. The procedure we described is clearly invertible: to each point of the graphical Grassmannian, we may associate a point in $\prod_{T\in G_\Gamma} \mathbb{P}(C_T)$. Moreover, if point of $\kk^{V_\Gamma}$ associated to $H_{V_\Gamma}$ is in the complement of the coordinate hyperplanes, all other subspaces $H_V$ are reconstructed uniquely as $H_{V_\Gamma}\cap\kk^{V}$, so the open piece of the graphical Grassmannian maps isomorphically to the open piece of $\overline{Y}_{G_\Gamma}$, and hence the image of the inverse map is precisely $\overline{Y}_{G_\Gamma}$. \end{proof} The torus action is completely transparent in this equivalent description of our varieties. Indeed, the torus $(\mathbb{G}_m)^{V_\Gamma}$ acts on $\kk^{V_\Gamma}$ by diagonal matrices, and this action leads to the action on the collections $\{H_T\}_{T\in G_\Gamma}$ in the obvious way. Clearly, the diagonally embedded torus $\mathbb{G}_m\subset (\mathbb{G}_m)^{V_\Gamma}$ acts trivially, so we have the action of the quotient torus $(\mathbb{G}_m)^{V_\Gamma}/\mathbb{G}_m$. \subsection{Torus orbits} \label{sec:toric-orbits} From the point of view of toric geometry, the varieties $X(\calP\Gamma)$ have natural stratifications where the open boundary strata are toric orbits with stabilizers given by the tori corresponding to faces of $\calP\Gamma$. From the point of view of wonderful compactifications, the varieties $\overline{Y}_{G_\Gamma}$ have natural stratifications where the closed boundary strata are intersections $D_\tau$ of the divisors $D_T$ for all nested sets $\tau$. Our previous comparisons of these two approaches show that the two stratifications are equivalent. We shall now give a direct description of the open strata in the language of graphical Grassmannians, generalising the stratification of brick manifolds given in \cite[Def.5.1.4]{MR4072173}; since we do not restrict ourselves to the boundary, it is more natural to use the augmented nested set complex $N^+(\Gamma)$ to index the strata. Suppose that $\tau\in N^+(\Gamma)$. We define a variety $\calB(\Gamma,\tau)$ as the set of collections $\{H_T\}_{T\in G_\Gamma}\in \calB(\Gamma)$ satisfying two conditions, both expressed in terms of the tree $\mathbb{T}_\tau$ and the labels $\lambda(T)$ of its vertices introduced in Definition \ref{def:lambda}: \begin{itemize} \item For each $T\in\tau$ with the parent $T'$ in $\mathbb{T}_\tau$ and for each $v\in \lambda(T')$ such that $T\cup\{v\}\in G_\Gamma$, we have \[ H_{T\cup\{v\}}=\kk^{T}. \] \item For each $T\in\tau$ and each $v_1\ne v_2\in \lambda(T)$, if for some $T'\subset T\setminus\{v_1,v_2\}$ we have \[ T'\in \tau\cup\{\emptyset\}\text{ and } T'\cup\{v_1\}\cup\{v_2\}\in G_\Gamma, \] then the subspace $H_{T\cup\{v_1\}\cup\{v_2\}}$ is different from $\kk^{T\cup\{v_1\}}$ and from $\kk^{T\cup\{v_2\}}$. \end{itemize} Note that the first condition is a ``boundary'' condition and that the second one is an ``open'' condition. As a sanity check, let us consider $\tau=\{V_\Gamma\}$. In this case, the first condition is empty, and the second condition, once restricted to edges, is easily seen to describe precisely the open piece discussed above ($H_{V_\Gamma}$ is in the complement of all coordinate hyperplanes), as expected. \begin{theorem}\label{th:orbits} The subsets $\calB(\Gamma,\tau)$ for $\tau\in N^+(\Gamma)$ describe the stratification of $\calB(\Gamma)$ by torus orbits. \end{theorem} \begin{proof} Let us first show that \[ \calB(\Gamma)=\bigcup_{\tau\in N^+(\Gamma)} \calB(\Gamma,\tau). \] Given a collection of subspaces parametrised by $\calB(\Gamma)$, let consider all $T\in G_\Gamma$ which satisfy $H_T=\kk^{T'}$ for some $T'\subset T$ and which are maximal such, that is, there does not exist $T\subset S\in G_\Gamma$ such that $H_S=\kk^{S'}$ with $T\not\subset S'$. We claim we must have $H_S=\kk^{T'}$ for all $S\in G_\Gamma$ with $S=T'\cup\{v_S\}$ for some $v_S\in V_\Gamma$. Otherwise, let us take such $S$ and consider $H_{T\cup S}$. Since $H_S\neq H_T$, we must have $H_{T\cup S}=H_S+H_T= \kk^{T'}+\kk w_S$, where $w_S\in \kk^{T'}\oplus \kk e_{v_S}$. But this implies $V_{T\cup S}=\kk^{T'}\oplus \kk e_{v_S}=\kk^{S}$, contradicting the maximality of $T$. Running through all such $H_T$, we obtain a collection \[ \tilde\tau:=\{T'\colon H_T=\kk^{T'} \text{ for some maximal } T\}. \] We claim that $\tilde\tau\in N(\Gamma)$, so $\tau:=\tilde\tau\sqcup\{V_\Gamma\}\in N^+(\Gamma)$. Suppose that there exist $T_1,T_2\in\tilde\tau$ that are incomparable and satisfy $\Gamma_{T_1\cup T_2}\ne\Gamma_{T_1}\sqcup\Gamma_{T_2}$. Then $H_{T_1\cup T_2}$ contains $\kk^{T_1}$ and $\kk^{T_2}$. But \[ \dim (\kk^{T_1}+\kk^{T_2}) = |T_1|+|T_2|-|T_1\cap T_2| \] while \[ \dim H_{T_1\cup T_2} = |T_1|+|T_2|-|T_1\cap T_2|-1, \] so we have a contradiction, and therefore $\tau:=:\tilde\tau\sqcup\{V_\Gamma\}\in N^+(\Gamma)$. The collection of subspaces $\{H_T\}$ clearly satisfies the first condition defining $\calB(\Gamma,\tau)$. If it fails to satisfy the second condition, then one can find $T\in\tau$, $v_1\ne v_2\in \lambda(T)$, and $T'\subset T\setminus\{v_1,v_2\}$ with \[ T'\in \tau\cup\{\emptyset\}\text{ and } T'\cup\{v_1\}\cup\{v_2\}\in G_\Gamma, \] so that $H_{T\cup\{v_1\}\cup\{v_2\}}$ coincides with $\kk^{T\cup\{v_1\}}$. In that case, we can find a maximal such $T$, in the sense of the previous paragraph, which would imply that we have $T\cup\{v_1\}\in \tau$, which is a contradiction, since in this case $v_1\notin \lambda(T)$. We next show that the subsets $\calB(\Gamma,\tau)$ for various $\tau\in N^+(\Gamma)$ are disjoint, so that \[ \calB(\Gamma)=\bigsqcup_{\tau\in N^+(\Gamma)} \calB(\Gamma,\tau). \] Suppose we have a collection $\{H_T\}$ contained in $\calB(\Gamma,\tau_1)\cap \calB(\Gamma,\tau_2)$. The argument will depend on whether $\tau_1\cup \tau_2\in N^+(\Gamma)$. If $\tau_1\cup \tau_2\notin N^+(\Gamma)$, there exist $T_1\in\tau_1$ and $T_2\in\tau_2$ that are incomparable and $\Gamma_{T_1}\sqcup \Gamma_{T_2}\neq\Gamma_{T_1\cup T_2}$. In this case, there are vertices $\{v_1,v_2\}$ such that $v_1\in T_1\setminus T_2$, $v_2\in T_2\setminus T_1$, and $T_1\cup\{v_2\}\in G_\Gamma$, $T_2\cup \{v_1\}\in G_\Gamma$. The first condition for $\calB(\Gamma,\tau_1)$ implies that $H_{T_1\cup\{v_2\}}=\kk^{T_1}$. Note that the first condition defining the subset $\calB(\Gamma,\tau)$ also forces $H_{T'}=\kk^{T'\setminus\{v\}}$ for all $T'\in G_\Gamma$ such that $v\subset T'\subset T\cup\{v\}$. Therefore, $H_{(T_1\cap T_2)\cup\{v_1\}\cup\{v_2\}}=\kk^{(G_1\cap G_2)\cup \{v_1\})}$. Similarly, $H_{T_2\cup\{v_1\}}= \kk^{T_2}$, and so \[ H_{(T_1\cap T_2)\cup\{v_1\} \cup \{v_2\}}=\kk^{(T_1\cap T_2)\cup\{v_2\})}, \] which is a contradiction. Suppose that $\tau_1\cup\tau_2\in N^+(\Gamma)$. Let $T\in G_\Gamma$ belong to $\tau_1\setminus\tau_2$. Because of the nested set condition for $\tau_1\cup\tau_2$, there exists at least one vertex $v$ for which $T\cup\{v\}\in G_\Gamma$ and the set $T\cup\{v\}$ is disjoint from every set of $\tau_2$ that does not contain $T$ as a subset. Thus, the first condition for $\calB(\Gamma,\tau_2)$ implies that $H_{T\cup\{v\}}=\kk^{T}$, which means that $T\in \tau_2$, a contradiction. Finally, we note that the first condition defining the subset $\calB(\Gamma,\tau)$ implies that, once the second condition is applicable, the subspace $H_{T\cup\{v_1\}\cup\{v_2\}}$ different from $\kk^{T\cup\{v_1\}}$ and from $\kk^{T\cup\{v_2\}}$ always contains $\kk^{T}$, which gives a $\mathbb{G}_m$ of choices for such subspace. This easily implies that the subsets $\calB(\Gamma,\tau)$ are toric orbits. In fact, examining the second condition defining the subset $\calB(\Gamma,\tau)$, it is immediate to describe the stabiliser: \[ \mathrm{Stab}(\calB(\Gamma,\tau))=\left(\prod_{T\in\tau} \mathbb{G}_m\right)/\mathbb{G}_m\subset \left(\prod_{T\in\tau} \mathbb{G}_m^{\lambda(T)}\right)/\mathbb{G}_m\cong(\mathbb{G}_m)^{V_\Gamma}/\mathbb{G}_m, \] where the product is over all the diagonal inclusions $\mathbb{G}_m\subset\mathbb{G}_m^{\lambda(T)}$. \end{proof} It follows from either of the descriptions of our varieties that the closure of each stratum is isomorphic to a product of graphical varieties for smaller graphs, and so inclusions of closed strata provide the collection of all graphical varieties with an operad-like structure. The notion that is instrumental for describing this structure is that of a reconnected complement of a subgraph, which we shall now recall. \begin{defi}[reconnected complement] Let $V\in 2^{V_\Gamma}$. The \emph{reconnected complement} of $V$ in $\Gamma$, denoted $\Gamma^*_V$, is the graph obtained from $\Gamma$ by deleting some vertices and adding some new edges as follows. Its vertex set is $V_\Gamma\setminus V$, and its edge set is the union of the set of edges connecting vertices from $V_\Gamma\setminus V$ in $\Gamma$ and the set of all pairs $(v_1,v_2)\in (V_\Gamma\setminus V)^2$ such that $v_1\ne v_2$ and there is a path in $\Gamma$ that connects $v_1$ to $v_2$ and only uses vertices of $V$ along the way. \end{defi} \begin{figure}[ht] \centering \includegraphics[width=0.9\linewidth]{figures/reconnectedcomplement.pdf} \caption{Examples of reconnected complements} \end{figure} In the context of graph associahedra, Devadoss and Carr \cite{MR2239078} proved in \cite[Th.~2.9]{MR2239078} that facets of $\calP\Gamma$ are in one-to-one correspondence with elements of $G_\Gamma$ and that the facet corresponding to $V_\Gamma\ne T\in G_\Gamma$ is combinatorially equivalent to the product $\calP\Gamma^*_T\times\calP\Gamma_T$. In the language of graphical Grassmannian, this corresponds to the following picture. Among the toric orbits we described above, the orbits of codimension one are the orbits $\calB(\Gamma,\tau)$ where $\tau=\{T,V_\Gamma\}$ for some $V_\Gamma\ne T\in G_\Gamma$, and we expect that for such $\tau$ we have \[ \overline{\calB(\Gamma,\tau)}\cong \calB(\Gamma^*_T)\times \calB(\Gamma_T). \] Let us construct a map \[ \psi_{\Gamma,T}\colon \calB(\Gamma^*_T)\times \calB(\Gamma_T)\to \calB(\Gamma) \] whose image coincides with $\overline{\calB(\Gamma,\tau)}$. Suppose that \[ \{H'_S\}_{S\in G_{\Gamma^*_T}} \in \calB(\Gamma^*_T),\quad \{H''_S\}_{S\in G_{\Gamma_T}} \in \calB(\Gamma_T). \] We define the image of this pair as the element $\{H_S\}_{S\in G_{\Gamma}} \in \calB(\Gamma)$ defined by the formula \begin{equation}\label{eq:composition-toric} H_S = \begin{cases} H'_{S \setminus T} \oplus \kk^{S\cap T}, & \text{if } S \not\subset T,\\ \qquad H''_S, & \text{if } S \subset T. \end{cases} \end{equation} By a direct inspection, the image of this map is the closed stratum $\overline{\calB(\Gamma,\tau)}$. Considering how smaller closed strata are obtained of iterations of such maps is precisely what leads one to the definition of a reconnectad in the following section. Let us note that, though this last remark used the reconnected complement of $T\in G_\Gamma$, the definition is available for any $V\in 2^{V_\Gamma}$, not necessarily belonging to~$G_\Gamma$, and we shall later use it in full generality. \begin{rema} The combinatorics of reconnected complements was used in the recent work of Forcey and Ronco \cite{jlms.12596} to define a strict operadic category in the sense of Batanin and Markl \cite{MR3406537}. The formalism we develop below is very close to that one, except for two important differences. Firstly, we do not require the set of vertices of a graph to carry a linear order, and when we impose a linear order to define the related ``shuffle'' formalism, this will constrain the existing morphisms. Secondly, operads for the operadic category of \cite{jlms.12596} are defined by taking as the starting point the reconnected complements for $T\in G_\Gamma$, and then building all possible composition maps as composites of these. Our approach will arrive at these operations from two other definitions which exhibit clearly defined categorical constructions that are not apparent at a first glance for the approach of \cite{jlms.12596}. \end{rema} \section{Reconnectads}\label{sec:reconnectads} In this section, we shall define and study a new operad-like structure responsible for stratifications of graphical varieties by toric orbits. We begin with the definition of a graphical collection, generalising the notion of a species of structures valued in the category $\sfC$, see \cite{MR1629341,MR633783}. \begin{defi}[graphical collection] The \emph{groupoid of connected graphs} $\CGr$ is the category whose objects are connected simple graphs and whose morphisms are graph isomorphisms. A \emph{graphical collection} with values in $\sfC$ is a functor $\calF\colon \CGr\to\sfC$ satisfying $\calF(\emptyset)=1_{\sfC}$. All graphical collections with values in $\sfC$ form a category $\GrCol_{\sfC}$, where morphisms are natural transformations. \end{defi} \subsection{The monad of nested sets} Operads can be thought of as algebras over the monad of trees. We shall now define the monad of nested sets on the category of graphical collections. \begin{defi}[nested set endofunctor] The \emph{nested set endofunctor} $\calN$ on the category of graphical collections is defined as follows. Let $\calX$ be a graphical collection. The graphical collection $\calN(\calX)$ has the components \[ \calN(\calX)(\Gamma):=\bigoplus_{\tau\in N^+(\Gamma)}\bigotimes_{T\in \tau}\calX((\Gamma_T)^*_{T\setminus\lambda(T)}). \] Note that according to the definition of $\lambda(T)$, the set $T\setminus\lambda(T)$ in this formula can be rewritten as $\bigcup_{T'\in \tau, T'\subset T} T'$. It is also useful to note that the set of vertices of the graph $(\Gamma_T)^*_{T\setminus\lambda(T)}$ is the non-empty set $\lambda(T)$. \end{defi} Let us explain how to give $\calN$ a natural structure of a monad. The unit is the natural inclusion $\calX\to\calN(\calX)$ corresponding to $\tau=\{V_\Gamma\}$. The natural maps \[ \calN(\calN(\calX))\to\calN(\calX) \] come from the slogan saying that ``a nested set of nested sets is a nested set''. More precisely, suppose that $\tau\in N^+(\Gamma)$. We would like to define a map \[ \bigotimes_{T\in \tau}\calN(\calX)((\Gamma_T)^*_{T\setminus\lambda(T)})\to \calN(\calX)(\Gamma). \] For that, we note that the left hand side is the sum of tensor products over all possible nested sets of connected graphs $(\Gamma_T)^*_{T\setminus\lambda(T)}$ on the vertex sets $\lambda(T)$, and to each such collection of nested sets one can canonically associate a nested set of $\Gamma$ by joining together the subsets which violate the condition $\Gamma_{U\cup V}=\Gamma_U\sqcup\Gamma_V$, thus obtaining a map of the required form. (In terms of trees $\mathbb{T}_\tau$ associated to nested sets, this corresponds to grafting of trees, which can be used to establish the associativity required by the monad axioms.) In particular, for every $\tau=\{T,V_\Gamma\}$ with $V_\Gamma\ne T\in G_\Gamma$, the corresponding tree $\mathbb{T}_\tau$ has $\lambda(T)=T$ and $\lambda(V_\Gamma)=V_\Gamma\setminus T$, and moreover, \[ (\Gamma_T)^*_{\emptyset}=\Gamma_T, \quad (\Gamma_{V_\Gamma})^*_{V_\Gamma\setminus (V_\Gamma\setminus T)}=(\Gamma)^*_T, \] so we find a summand $\calX(\Gamma^*_T)\otimes\calX(\Gamma_T)$ in $\calN(\calX)$. This observation will shine through in the following sections; for now we use this particular kind of nested sets in an example. \begin{exam} Let us consider the graph $\Gamma$ depicted in the top left corner of Figure~\ref{fig:nested set composition}. We choose the subset $T=\{1,2,3\}$ for which the induced graph $\Gamma_T$ is isomorphic to $K_3$ and the reconnected complement $\Gamma^*_T$ is isomorphic to $P_3$. \begin{figure}[ht] \centering \def\svgwidth{\columnwidth} \import{figures}{infinitesimalcomposition.pdf_tex} \caption{Nested sets of nested sets and compositions} \label{fig:nested set composition} \end{figure} For the nested set $\tau=\{T,V_\Gamma\}$, there is a term in $\calN(\calN(\calX))$ corresponding to the tensor product of the term of $\calN(\calX)(\Gamma^*_T)$ associated to the nested set $\{\{4\},\{6\},\{4,5,6\}\}$ and the term of $\calN(\calX)(\Gamma_T)$ associated to the nested set $\{\{1,2\},\{1,2,3\}\}$. Joining these together gives us the nested set $\{\{4\},\{1,2\},\{1,2,3\},\{1,2,3,6\},\{1,2,3,4,5,6\}\}$ of $\Gamma$ depicted in the bottom right corner of Figure~\ref{fig:nested set composition}. \end{exam} \begin{defi}[reconnectad, monadic definition] A \emph{reconnectad} is an algebra over the nested set monad. Concretely, it is a graphical collection $\calF$ equipped with structure maps \[ \calN(\calF)\to\calF \] compatible with the monad structure on $\calN$ in the usual sense. \end{defi} Our definition of a reconnectad leads to an immediate definition of the free reconnectad generated by a graphical collection $\calX$. \begin{defi}[free reconnectad] The \emph{free reconnectad} generated by a graphical collection $\calX$ is the graphical collection $\calN(\calX)$ with the structure maps \[ \calN(\calN(\calX))\to\calN(\calX) \] defined above to encode the monad structure. \end{defi} The following proposition, which will be very useful for us, is straightforward from the definition. \begin{prop}\label{prop:grEnd-monoidal} Suppose that $\sfC$ and $\sfC'$ are two symmetric monoidal categories satisfying all the assumptions we impose on a symmetric monoidal category, and that $\phi\colon \sfC\to \sfC'$ is a symmetric monoidal functor. Then for each reconnectad $\calF$ in $\sfC$, we obtain a reconnectad $\phi(\calF)$ in $\sfC'$. In particular, for a reconnectad $\calF$ in topological spaces, the graphical collection $H_{\bullet}(\calF,\kk)$ is a reconnectad in $\dgVect$. \end{prop} Our definition of a reconnectad is not very easy to unwrap to handle various particular cases. We shall now present two equivalent definitions that will allow us to view reconnectads in a much more concrete way. \subsection{The monoidal category of graphical collections} There is an obvious symmetric monoidal structure on the category of graphical collections called the \emph{Hadamard product}. It is defined by the formula \[ (\calF\underset{H}{\otimes}\calG)(\Gamma):=\calF(\Gamma)\otimes\calG(\Gamma). \] The graphical collection $\mathbb{I}$ with $\mathbb{I}(\Gamma)=1_{\sfC}$, equipped with the trivial action of the group $\Aut(\Gamma)$, is the unit of this monoidal structure. For our purposes, another (highly non-symmetric) monoidal structure on the category of graphical collections will play an important role. It is defined as follows. \begin{defi}[reconnected product] The \emph{reconnected product} of two graphical collections $\calF$ and $\calG$ is the graphical collection $\calF\circ_{\rmR}\calG$ defined by the formula \[ (\calF\circ_{\rmR}\calG)(\Gamma):=\bigoplus_{V\subset V_\Gamma}\calF(\Gamma^*_V)\otimes\bigotimes_{\Gamma'\in\mathrm{Conn}(\Gamma_V)} \calG(\Gamma'). \] Here, as in Section \ref{sec:graphs-def}, $\mathrm{Conn}(\Gamma_V)$ denotes the set of connected components of the graph $\Gamma_V$. \end{defi} It turns out that the product we defined makes the category of graphical collections into a monoidal category. \begin{prop}\label{prop:assoc} The category $\GrCol_{\sfC}$ equipped with the reconnected product $\circ_{\rmR}$ is a monoidal category whose unit is the graphical collection $\mathbbm{1}$ defined by \[ \mathbbm{1}(\Gamma)= \begin{cases} 1_{\sfC}, & \text{if } \Gamma=\emptyset, \\ 0, & \text{otherwise.} \end{cases} \] \end{prop} \begin{proof} Let us allow ourselves to evaluate each graphical collection on not necessarily connected graphs by putting \[ \calG(\Gamma_1\sqcup\Gamma_2):=\calG(\Gamma_1)\otimes\calG(\Gamma_2). \] This permits us to write the definition of the reconnected product in a more compact way \[ (\calF\circ_{\rmR}\calG)(\Gamma)=\bigoplus_{V\subset V_\Gamma}\calF(\Gamma^*_V)\otimes\calG(\Gamma_V). \] Notice that this convention does not lead to a contradiction when evaluating the reconnected product on disconnected graphs: we have \begin{align*} (\calF\circ_{\rmR}\calG)(\Gamma_1\sqcup\Gamma_2) &= (\calF\circ_{\rmR}\calG)(\Gamma_1)\otimes (\calF\circ_{\rmR}\calG)(\Gamma_2)\\ &= \bigoplus_{V_1\subset V_{\Gamma_1}}\calF((\Gamma_1)^*_{V_1})\otimes\calG((\Gamma_1)_{V_1})\otimes \bigoplus_{V_2\subset V_{\Gamma_2}}\calF((\Gamma_2)^*_{V_2})\otimes\calG((\Gamma_2)_{V_2}), \end{align*} which, thanks to the symmetry isomorphisms of $\sfC$, the property $\calG(\emptyset)=1_{\sfC}$, and the properties \begin{gather*} (\Gamma_1\sqcup\Gamma_2)^*_{V}=(\Gamma_1)^*_{V\cap V_{\Gamma_1}}\sqcup (\Gamma_2)^*_{V\cap V_{\Gamma_2}},\\ (\Gamma_1\sqcup\Gamma_2)_{V}=(\Gamma_1)_{V\cap V_{\Gamma_1}}\sqcup (\Gamma_2)_{V\cap V_{\Gamma_2}}, \end{gather*} is isomorphic to \[ \bigoplus_{V\subset V_{\Gamma_1\sqcup\Gamma_2}}\calF((\Gamma_1\sqcup\Gamma_2)^*_{V})\otimes\calG((\Gamma_1\sqcup\Gamma_2)_{V}). \] Because of that, we have \begin{align*} (\calF\circ_{\rmR}(\calG\circ_{\rmR}\calH))(\Gamma) &= \bigoplus_{V\subset V_\Gamma}\calF(\Gamma^*_V)\otimes(\calG\circ_{\rmR}\calH)(\Gamma_V)\\ &= \bigoplus_{V\subset V_\Gamma}\calF(\Gamma^*_V)\otimes\bigoplus_{U\subset V}\calG((\Gamma_V)^*_U)\otimes\calH((\Gamma_V)_U) \end{align*} and \begin{align*} ((\calF\circ_{\rmR}\calG)\circ_{\rmR}\calH)(\Gamma) &= \bigoplus_{U\subset V_\Gamma}(\calF\circ_{\rmR}\calG)(\Gamma^*_U)\otimes\calH(\Gamma_U)\\ &= \bigoplus_{U\subset V_\Gamma}\bigoplus_{W\subset V_\Gamma\setminus U} \calF((\Gamma^*_U)^*_W)\otimes\calG((\Gamma^*_U)_W)\otimes \calH(\Gamma_U), \end{align*} which, if we denote $V:=U\sqcup W$, becomes \[ \bigoplus_{U\subset V_\Gamma}\bigoplus_{U\subset V\subset V_\Gamma} \calF((\Gamma^*_U)^*_{V\setminus U})\otimes\calG((\Gamma^*_U)_{V\setminus U})\otimes \calH(\Gamma_U), \] and the associativity isomorphism follows from the obvious properties \[ (\Gamma^*_U)^*_{V\setminus U}=\Gamma^*_V, \quad (\Gamma^*_U)_{V\setminus U}=(\Gamma_V)^*_U, \quad (\Gamma_V)_U=\Gamma_U \] that hold for any $U\subset V\subset V_\Gamma$. Moreover, the associativity isomorphisms are immediately seen to satisfy the axioms of a monoidal category. Finally, we note that we have $\Gamma_\emptyset=\emptyset$ and $\Gamma^*_\emptyset=\Gamma$ and, dually, we have $\Gamma_{V_\Gamma}=\Gamma$ and $\Gamma^*_{V_\Gamma}=\emptyset$. This immediately implies the isomorphisms \[ \calF\circ_{\rmR}\mathbbm{1}\cong\mathbbm{1}\circ_{\rmR}\calF\cong\calF, \] and the necessary compatibility of these isomorphisms with the monoidal structure is verified by direct inspection. \end{proof} This result ensures that the following definition of a reconnectad makes sense; it is straightforward to see that it is equivalent to the monadic definition. \begin{defi}[reconnectad, monoidal definition] A \emph{reconnectad} is a monoid in the monoidal category of graphical collections equipped with the reconnected product $\circ_{\rmR}$. \end{defi} For a reconnectad $\calF$, a connected graph $\Gamma$, and a subset $V$ of $V_\Gamma$, we shall denote by $\mu_V^\Gamma$ the restriction of the structure map $(\calF\circ_{\rmR}\calF)(\Gamma)\to \calF(\Gamma)$ to the summand $\calF(\Gamma^*_V)\otimes\bigotimes_{\Gamma'\in\mathrm{Conn}(\Gamma_V)} \calF(\Gamma')$. In the particular case when $V_\Gamma\ne T\in G_\Gamma$, we shall use the notation $\circ_T^\Gamma$ for $\mu_T^\Gamma$; this distinction should help the reader to navigate between the connected and the disconnected situation. Additionally, when $V=\{v\}$, we write $\circ_v^\Gamma$ instead of $\circ_{\{v\}}^\Gamma$ to simplify the notation slightly. One immediate consequence of the monoidal definition is that whenever one can talk about monoids, one can also talk about comonoids, and so the notion of a \emph{coreconnectad} arises naturally. We shall denote by $\Delta_V^\Gamma$ the composition of the structure map $\calG(\Gamma)\to (\calG\circ_{\rmR}\calG)(\Gamma)$ of a coreconnectad $\calG$ with the projection onto the summand $\calG(\Gamma^*_V)\otimes\bigotimes_{\Gamma'\in\mathrm{Conn}(\Gamma_V)} \calG(\Gamma')$ of $(\calG\circ_{\rmR}\calG)(\Gamma)$. \subsection{The coloured operad encoding reconnectads} In the case of operads, one can use the operations $\circ_i$, often referred to as infinitesimal compositions, or partial compositions, to give an equivalent definition. Even though this viewpoint somewhat obscures the fact that operads are associative monoids, it allows one to view operads as algebras over a coloured operad, which has its advantages. The following proposition is proved by direct inspection (using the properties of restrictions and reconnected complements used in the proof of Proposition~\ref{prop:assoc}). \begin{prop}\label{prop:infini} The datum of a reconnectad on a graphical collection $\calF$ is equivalent to the datum of infinitesimal compositions \[ \circ^\Gamma_T\colon\calF(\Gamma^*_T)\otimes\calF(\Gamma_T)\to\calF(\Gamma) \] for all elements $T\ne V_\Gamma$ of the graphical building set $G_\Gamma$; these operations must satisfy the following properties: \begin{itemize} \item (unit axiom) Under the identifications \[ \Gamma^*_{\emptyset}=\Gamma,\quad \Gamma^*_{V_\Gamma}=\emptyset,\quad 1_{\sfC}\otimes X\cong X\otimes1_{\sfC}\cong X, \] we have $\circ^\Gamma_\emptyset=\circ^\Gamma_{V_\Gamma}=\mathrm{id}_{\calF(\Gamma)}$. \item (parallel axiom) For all $T_1,T_2\in G_\Gamma$ such that $\Gamma_{T_1\cup T_2}=\Gamma_{T_1}\sqcup \Gamma_{T_2}$, the diagram \[\begin{tikzcd} \calF(\Gamma^*_{T_1\cup T_2}) \otimes \calF(\Gamma_{T_1}) \otimes \calF(\Gamma_{T_2}) \arrow{r}{\circ_{T_1}^{\Gamma^*_{T_2}}} \arrow[swap]{d}{\circ_{T_2}^{\Gamma^*_{T_1}}} & \calF(\Gamma^*_{T_2}) \otimes \calF(\Gamma_{T_2}) \arrow{d}{\circ_{T_2}^{\Gamma}} \\ \calF(\Gamma^*_{T_1}) \otimes \calF(\Gamma_{T_1})\arrow{r}{\circ_{T_1}^{\Gamma}} & \calF(\Gamma) \end{tikzcd} \] commutes. \item (consecutive axiom) For all $T_1,T_2\in G_\Gamma$ with $T_1\subset T_2$, the diagram \[\begin{tikzcd} \calF(\Gamma^*_{T_2})\otimes \calF((\Gamma_{T_2})^*_{T_1}) \otimes \calF(\Gamma_{T_1}) \arrow{r}{\mathrm{id} \otimes \circ_{T_1}^{\Gamma_{T_2}}} \arrow[swap]{d}{\circ_{T_2\setminus T_1}^{\Gamma^*_{T_1}}\otimes \mathrm{id}} & \calF(\Gamma^*_{T_2}) \otimes\calF(\Gamma_{T_2}) \arrow{d}{\circ_H^{\Gamma}} \\ \calF(\Gamma^*_{T_1}) \otimes \calF(\Gamma_{T_1}) \arrow{r}{\circ_{T_1}^{\Gamma}} & \calF(\Gamma) \end{tikzcd} \] commutes. \item(equivariance) For every $T\in G_\Gamma$ and every automorphism $\alpha \in \mathrm{Aut}(\Gamma)$ the diagram \[\begin{tikzcd} \calF(\Gamma^*_{T}) \otimes \calF(\Gamma_{T}) \arrow{r}{\circ_T^{\Gamma}} \arrow[swap]{d}{\phi} & \calF(\Gamma) \arrow{d}{\phi} \\ \calF(\Gamma^*_{\alpha(T)}) \otimes \calF(\Gamma_{\alpha(T)}) \arrow{r}{\circ_{\alpha(T)}^{\Gamma}} & \calF(\Gamma) \end{tikzcd} \] commutes. \end{itemize} \end{prop} This proposition immediately implies that the maps $\psi_{\Gamma,T}$ defined by Formula~\eqref{eq:composition-toric} give the collection of all toric varieties of graph associahedra the structure of a reconnectad, thus introducing the central example of a reconnectad that motivated our work. \begin{defi}[wonderful reconnectad]\label{def:wond} The \emph{wonderful reconnectad} is the graphical collection $\calW$ with \[ \calW(\Gamma):=\calB(\Gamma)=X(\calP\Gamma) \] and with the structure operations \[ \circ^\Gamma_T\colon\calW(\Gamma^*_T)\otimes\calW(\Gamma_T)\to\calW(\Gamma) \] given by $\circ^\Gamma_T:=\psi_{\Gamma,T}$. \end{defi} We shall use Proposition \ref{prop:infini} in conjunction with the notion of a groupoid coloured operad of Petersen \cite{MR3134040} as follows, mimicking the approach to modular operads of Ward \cite{MR4425832}, see also \cite{DSVV}. \begin{defi}\label{def:groupoidcoloured} We define the $\CGr$-coloured operad $\Rec=\calT(E)/(R)$ as follows. It is generated by the elements \[ E\big(\Gamma; \Gamma^*_T, \Gamma_T\big) :=\\ \left\{ \Aut(\Gamma^*_T)\times \Aut(\Gamma_T)\times \begin{aligned}\begin{tikzpicture}[optree] \node{} child { node[circ]{$\circ^\Gamma_T$} child { edge from parent node[left,near end]{\tiny$\Gamma^*_T$} } child { edge from parent node[right,near end]{\tiny$\Gamma_T$} } edge from parent node[right,near start]{\tiny$\phantom{i}\Gamma$} } ; \end{tikzpicture}\end{aligned} \right\}\ , V_\Gamma\ne T\in G_\Gamma, \] with the regular $\Aut(\Gamma^*_V)\times \Aut(\Gamma_V)$-action and with the $\Aut(\Gamma)$-action given by \[\left(\circ_V^\Gamma,\mathrm{id},\mathrm{id}\right)^{\phi}=\left(\circ_{\phi(V)}^\Gamma,\phi|_{\Gamma^*_V},\phi|_{\Gamma_V}\right)\ .\] The quadratic relations $R$ are the ones given in Proposition \ref{prop:infini}. \end{defi} We are now in the position to give another equivalent definition of a reconnectad. \begin{defi} A reconnectad is an algebra over the $\CGr$-coloured operad $\Rec$. \end{defi} One immediate consequence of this definition is that all the standard constructions for algebras over operads are available for reconnectads. It is also useful to note that the $\CGr$-coloured operad $\Rec$ has an obvious diagonal making it a Hopf $\CGr$-coloured operad. In particular, the category of reconnectad has a symmetric monoidal structure: that structure is the Hadamard product equipped with the obvious composition maps. \begin{rema}\label{rem:Feynman} Expressing certain operadic structures as algebras over groupoid coloured operads is essentially equivalent to talking about operads over a Feynman category \cite{MR3636409}. Let $\Gr$ denote the category whose objects are simple (not necessarily connected) graphs and whose morphisms are generated by graph isomorphisms and the morphisms \[ \phi^\Gamma_V \colon \Gamma_V\sqcup\Gamma^*_V \to \Gamma \] that are associated to the datum of a graph $\Gamma$ and a choice of a subset $V\subset V_\Gamma$. The groupoid of connected graphs $\CGr$ is a full subcategory of $\Gr$. By a direct inspection, the triple $(\CGr,\Gr,\imath)$, where $\imath$ is the inclusion $\CGr\to\Gr$ defines the datum of a Feynman category, and reconnectads are operads over this Feynman category. \end{rema} \subsection{Particular types of graphs}\label{sec:part-types} If we restrict ourselves to various families of graphs, we may recognize known algebraic structures in the guise of reconnectads. Recall that a twisted associative algebra is a symmetric collection (a functor from the groupoid of finite sets to $\sfC$) which is a monoid with respect to the Cauchy monoidal structure \[ (\calF\cdot\calG)(I):=\bigoplus_{I=J\sqcup K}\calF(J)\otimes\calG(K). \] on symmetric collections. A twisted associative algebra $\calA$ is said to be connected if $\calA(\emptyset)=1_{\sfC}$. \begin{prop}\label{prop:LM} Suppose that we restrict ourselves to the full subcategory of collections supported on complete graphs. The datum of a reconnectad in that category is the same as the datum of a connected twisted associative algebra. \end{prop} \begin{proof} Note that the datum of a graphical collection supported on complete graphs is obviously the same as the datum of a symmetric collection whose evaluation on the empty set is $1_{\sfC}$: indeed, a complete graph carries as much information as its set of vertices. Moreover, for a complete graph $\Gamma$ and every $V\subset V_\Gamma$, the graph $\Gamma_V$ is connected and complete, and the graph $\Gamma^*_V$ is also complete, so the monoidal structure on our category corresponds precisely to the Cauchy monoidal structure. \end{proof} Recall that a nonsymmetric operad is a nonsymmetric collection (a functor from the groupoid of finite totally ordered sets to $\sfC$) which is a monoid with respect to the composition product \[ (\calF\circ\calG)(I):=\bigoplus_{k\ge 0}\bigoplus_{I=I_1+\cdots+I_k}\calF(\{1,\ldots,k\})\otimes\calG(I_1)\otimes\cdots\otimes G(I_k). \] on nonsymmetric collections. A nonsymmetric operad $\calO$ is said to be reduced if $\calO=0$ and connected if $\calO(\{\mathrm{pt}\})=1_{\sfC}$. Let us call a nonsymmetric operad \emph{mirrored} if it comes from a functor from the groupoid quotient by the $\mathbb{Z}/2\mathbb{Z}$-action reversing the order. Components of such an operad have actions of the group $\mathbb{Z}/2\mathbb{Z}$ for which the generator $\sigma$ of that group satisfies, for all elements $f$ and $g$ of arities $p$ and $q$ respectively, the property \[ \sigma(f\circ_i g)=\sigma(f)\circ_{p-i+1}\sigma(g). \] \begin{prop}\label{prop:NS} Suppose that we restrict ourselves to the full subcategory of collections supported on path graphs (that is, on the type $A$ Dynkin diagrams). The datum of a reconnectad in that category is the same as the datum of a connected reduced mirrored nonsymmetric operad. \end{prop} \begin{proof} Let us consider the assignment to each nonempty finite totally ordered set $I$ the set of \emph{gaps} \[ \mathrm{Gap}(I)=\{(i_1,i_2)\colon i_1,i_2\in I, \{i\in I \colon i_11$ and choose a maximal (by inclusion) element $T$ from this set. Let $v$ be the minimal (with respect to the order on $V_\Gamma$) vertex of $\lambda(T)$. There is a unique minimal (by inclusion) element $T'$ containing $v$ that is compatible with the nested set $\omega$: it is the union of $\{v\}$ with all sets $S$ to which $\{v\}\cup S\in G_\Gamma$. We add $T'$ to $\omega$, and repeat the same procedure until we obtain an element of $|N^+(\Gamma)|$ for which $|\lambda(T)|=1$ for all $T$, in other words, an element of $N^+_{|V_\Gamma|}(\Gamma)$. We denote that element by $\omega^{\mathrm{ind}}$ and call it the induced maximal nested set. \begin{lemma}\leavevmode \begin{enumerate} \item For any $\tau\in N^+_{|V_\Gamma|}(\Gamma)$, $\tau^{\mathrm{red}}$ is a normal monomial. \item For any $\omega\in N^+(\Gamma)$, we have $(\omega^{\mathrm{ind}})^{\mathrm{red}}\subset\omega$. \item For any normal monomial $\omega\in N^+(\Gamma)$, we have $(\omega^{\mathrm{ind}})^{\mathrm{red}}=\omega$. \end{enumerate} \end{lemma} \begin{proof} Let us start with the first assertion. Suppose that $\tau^{\mathrm{red}}$ is not a normal monomial, that is, it is divisible by a leading term of our relations. We note that it is enough to consider the particular case where $\tau$ has the maximal possible number of descents, that is $|V_\Gamma|-2$: removing the elements that contain the outer element set of the leading term, or are contained in the inner element of the leading term, or are disjoint from either of them will not impact the argument. We are therefore left with the case where the result of reduction is a quadratic monomial; let us show that it is normal. There are two possibilities: $V_{\tau,\max(V_\Gamma)}=V_\Gamma$ and $V_{\tau,\max(V_\Gamma)}\ne V_\Gamma$. In the latter case, $V_{\tau,\max(V_\Gamma)}$ is never deleted in the reduction process, and hence is the only nontrivial element of $\tau^{\mathrm{red}}$, which is therefore a normal monomial. In the former case, let $T$ be the only nontrivial element of $\tau^{\mathrm{red}}$, and let $v\in\lambda(T)$. As $T$ survived the reduction process, there is a vertex $w