1079 The homological degree of an element $a \in \bc_*^A(J)$ |
1079 The homological degree of an element $a \in \bc_*^A(J)$ |
1080 is the sum of the blob degree and the internal degree. |
1080 is the sum of the blob degree and the internal degree. |
1081 |
1081 |
1082 We first define $\bc_0^A(J)$ as a vector space by |
1082 We first define $\bc_0^A(J)$ as a vector space by |
1083 \begin{equation*} |
1083 \begin{equation*} |
1084 \bc_0^A(J) = \DirectSum_{\substack{\{J_i\}_{i=1}^n \\ \mathclap{\bigcup_i J_i = J}}} \left(\{J_i\}, \Tensor_{i=1}^n (\CD{J_i \to I} \tensor A) \right). |
1084 \bc_0^A(J) = \DirectSum_{\substack{\{J_i\}_{i=1}^n \\ \mathclap{\bigcup_i J_i = J}}} \Tensor_{i=1}^n (\CD{J_i \to I} \tensor A). |
1085 \end{equation*} |
1085 \end{equation*} |
1086 (That is, for each division of $J$ into finitely many subintervals, |
1086 (That is, for each division of $J$ into finitely many subintervals, |
1087 we have the tensor product of chains of diffeomorphisms from each subinterval to the standard interval, |
1087 we have the tensor product of chains of diffeomorphisms from each subinterval to the standard interval, |
1088 and a copy of $A$ for each subinterval.) |
1088 and a copy of $A$ for each subinterval.) |
1089 The internal degree of an element $(f_1 \tensor a_1, \ldots, f_n \tensor a_n)$ is the sum of the dimensions of the singular chains |
1089 The internal degree of an element $(f_1 \tensor a_1, \ldots, f_n \tensor a_n)$ is the sum of the dimensions of the singular chains |
1090 plus the sum of the homological degrees of the elements of $A$. |
1090 plus the sum of the homological degrees of the elements of $A$. |
1091 The differential is defined just by the graded Leibniz rule and the differentials on $\CD{J_i \to I}$ and on $A$. |
1091 The differential is defined just by the graded Leibniz rule and the differentials on $\CD{J_i \to I}$ and on $A$. |
1092 |
1092 |
1093 Next, |
1093 Next, |
1094 \begin{equation*} |
1094 \begin{equation*} |
1095 \bc_1^A(J) = \DirectSum_{\substack{\{J_i\}_{i=1}^n \\ \mathclap{\bigcup_i J_i = J}}} \DirectSum_{T \in \cT_{1,n}} \left(\{J_i\}, T, \Tensor_{i=1}^n (\CD{J_i \to I} \tensor A) \right). |
1095 \bc_1^A(J) = \DirectSum_{\substack{\{J_i\}_{i=1}^n \\ \mathclap{\bigcup_i J_i = J}}} \DirectSum_{T \in \cT_{1,n}} \Tensor_{i=1}^n (\CD{J_i \to I} \tensor A). |
1096 \end{equation*} |
1096 \end{equation*} |
1097 \end{defn} |
1097 \end{defn} |
1098 |
1098 |
|
1099 \begin{figure}[!ht] |
|
1100 \begin{equation*} |
|
1101 \mathfig{0.7}{associahedron/A4-vertices} |
|
1102 \end{equation*} |
|
1103 \caption{The vertices of the $k$-dimensional associahedron are indexed by binary trees on $k+2$ leaves.} |
|
1104 \label{fig:A4-vertices} |
|
1105 \end{figure} |
|
1106 |
|
1107 \begin{figure}[!ht] |
|
1108 \begin{equation*} |
|
1109 \mathfig{0.7}{associahedron/A4-faces} |
|
1110 \end{equation*} |
|
1111 \caption{The faces of the $k$-dimensional associahedron are indexed by trees with $2$ vertices on $k+2$ leaves.} |
|
1112 \label{fig:A4-vertices} |
|
1113 \end{figure} |
|
1114 |
1099 \newcommand{\tm}{\widetilde{m}} |
1115 \newcommand{\tm}{\widetilde{m}} |
1100 \newcommand{\ttm}{\widetilde{\widetilde{m}}} |
1116 |
1101 |
1117 Let $\tm_1(a) = a$. |
1102 Define $\ttm_k$ by |
1118 |
|
1119 We now define $\bdy(\tm_k(a_1 \tensor \cdots \tensor a_k))$, first giving an opaque formula, then explaining the combinatorics behind it. |
|
1120 \begin{align} |
|
1121 \notag \bdy(\tm_k(a_1 & \tensor \cdots \tensor a_k)) = \\ |
|
1122 \label{eq:bdy-tm-k-1} & \phantom{+} \sum_{i=1}^k (-1)^{\sum_{j=1}^{i-1} \deg(a_j)} \tm_k(a_1 \tensor \cdots \tensor \bdy a_i \tensor \cdots \tensor a_k) + \\ |
|
1123 \label{eq:bdy-tm-k-2} & + \sum_{\ell=1}^{k-1} \tm_{\ell}(a_1 \tensor \cdots \tensor a_{\ell}) \tensor \tm_{k-\ell}(a_{\ell+1} \tensor \cdots \tensor a_k) + \\ |
|
1124 \label{eq:bdy-tm-k-3} & + \sum_{\ell=1}^{k-1} \sum_{\ell'=0}^{l-1} \tm_{\ell}(a_1 \tensor \cdots \tensor m_{k-\ell + 1}(a_{\ell' + 1} \tensor \cdots \tensor a_{\ell' + k - \ell + 1}) \tensor \cdots \tensor a_k) |
|
1125 \end{align} |
|
1126 The first set of terms in $\bdy(\tm_k(a_1 \tensor \cdots \tensor a_k))$ just have $\bdy$ acting on each argument $a_i$. |
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1127 The terms appearing in \eqref{eq:bdy-tm-k-2} and \eqref{eq:bdy-tm-k-3} are indexed by trees with $2$ vertices on $k+1$ leaves. |
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1128 Note here that we have one more leaf than there arguments of $\tm_k$. |
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1129 (See Figure \ref{fig:A4-vertices}, in which the rightmost branches are helpfully drawn in red.) |
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1130 We will treat the vertices which involve a rightmost (red) branch differently from the vertices which only involve the first $k$ leaves. |
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1131 The terms in \eqref{eq:bdy-tm-k-2} arise in the cases in which both |
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1132 vertices are rightmost, and the corresponding term in $\bdy(\tm_k(a_1 \tensor \cdots \tensor a_k))$ is a tensor product of the form |
|
1133 $$\tm_{\ell}(a_1 \tensor \cdots \tensor a_{\ell}) \tensor \tm_{k-\ell}(a_{\ell+1} \tensor \cdots \tensor a_k)$$ |
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1134 where $\ell + 1$ and $k - \ell + 1$ are the number of branches entering the vertices. |
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1135 If only one vertex is rightmost, we get the term $$\tm_{\ell}(a_1 \tensor \cdots \tensor m_{k-\ell+1}(a_{\ell' + 1} \tensor \cdots \tensor a_{\ell' + k - \ell}) \tensor \cdots \tensor a_k)$$ |
|
1136 in \eqref{eq:bdy-tm-k-3}, |
|
1137 where again $\ell + 1$ is the number of branches entering the rightmost vertex, $k-\ell+1$ is the number of branches entering the other vertex, and $\ell'$ is the number of edges meeting the rightmost vertex which start to the left of the other vertex. |
|
1138 For example, we have |
1103 \begin{align*} |
1139 \begin{align*} |
1104 \ttm_k(a_1 \tensor \cdots \tensor a_k) & = m_k(a_1 \tensor \cdots \tensor a_k) \\ |
1140 \bdy(\tm_2(a \tensor b)) & = \left(\tm_2(\bdy a \tensor b) + \tm_2(a \tensor \bdy b)\right) + \\ |
1105 \ttm_k(a_1 \tensor \cdots \tensor a_{k-1} \tensor z) & = z \tensor \tm_{k-1}(a_1 \tensor \cdots \tensor a_{k-1}) \\ |
|
1106 \intertext{and} |
|
1107 \ttm_k(a_1 \tensor \cdots \tensor a_{k-2} \tensor z \tensor a_k) & = z \tensor \tm_{k-2}(a_1 \tensor \cdots \tensor a_{k-2}) \tensor a_k. |
|
1108 \end{align*} |
|
1109 |
|
1110 Let $\tm_1(a) = a$. |
|
1111 |
|
1112 Then define |
|
1113 \begin{align*} |
|
1114 \bdy(\tm_k(a_1 \tensor \cdots \tensor a_k)) & = \sum_{j=1}^{k} \tm_k(a_1 \tensor \cdots \tensor \bdy a_j \tensor \cdots \tensor a_k) + \\ |
|
1115 & z \perp \sum_{q=2}^{k-1} \sum_{p=1}^{k-q+2} \ttm_{k-q+1}(a_1 \tensor \cdots a_{p-1} \tensor \ttm_q(a_p \tensor \cdots \tensor a_{p+q-1}) \tensor a_{p+q} \tensor \cdots \tensor a_{k+1}). |
|
1116 \end{align*} |
|
1117 where here $a_{k+1}$ is just notation for $z$. |
|
1118 \todo{err... here I mean $z \perp z \tensor x = x$...} |
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1119 \todo{actually, if you let $q$ start from 1 you don't need the first term} |
|
1120 |
|
1121 \begin{align*} |
|
1122 \bdy(\tm_2(a \tensor b)) & = (\tm_2(\bdy a \tensor b) + \tm_2(a \tensor \bdy b)) + \\ |
|
1123 & \qquad + a \tensor b + \\ |
1141 & \qquad + a \tensor b + \\ |
1124 & \qquad + m_2(a \tensor b) \\ |
1142 & \qquad + m_2(a \tensor b) \\ |
1125 \bdy(\tm_3(a \tensor b \tensor c)) & = (\tm_3(\bdy a \tensor b \tensor c) + \tm_3(a \tensor \bdy b \tensor c) + \tm_3(a \tensor b \tensor \bdy c)) + \\ |
1143 \bdy(\tm_3(a \tensor b \tensor c)) & = \left(\tm_3(\bdy a \tensor b \tensor c) + \tm_3(a \tensor \bdy b \tensor c) + \tm_3(a \tensor b \tensor \bdy c)\right) + \\ |
1126 & + (\tm_2(a \tensor b) \tensor c + a \tensor \tm_2(b \tensor c)) + \\ |
1144 & \qquad + \left(\tm_2(a \tensor b) \tensor c + a \tensor \tm_2(b \tensor c)\right) + \\ |
1127 & + (\tm_2(m_2(a \tensor b) \tensor c) + \tm_2(a, m_2(b \tensor c)) + m_3(a \tensor b \tensor c)) \\ |
1145 & \qquad + \left(\tm_2(m_2(a \tensor b) \tensor c) + \tm_2(a, m_2(b \tensor c)) + m_3(a \tensor b \tensor c)\right) |
1128 \bdy(\tm_4(a \tensor b \tensor c \tensor d)) & = (\tm_4(\bdy a \tensor b \tensor c \tensor d) + \cdots + \tm_4(a \tensor b \tensor c \tensor \bdy d)) + \\ |
|
1129 & + (\tm_3(a \tensor b \tensor c) \tensor d + \tm_2(a \tensor b) \tensor \tm_2(c \tensor d) + a \tensor \tm_3(b \tensor c \tensor d)) + \\ |
|
1130 & + (\tm_3(m_2(a \tensor b) \tensor c \tensor d) + \tm_3(a \tensor m_2(b \tensor c) \tensor d) + \tm_3(a \tensor b \tensor m_2(c \tensor d)) + \\ |
|
1131 & + \tm_2(m_3(a \tensor b \tensor c) \tensor d) + \tm_2(a \tensor m_3(b \tensor c \tensor d)) + m_4(a \tensor b \tensor c \tensor d)) \\ |
|
1132 %d(\tm_k(x_1 \tensor \cdots \tensor x_k)) & = \sum_{i=1}^k (-1)^{\sum_{j=1}^{i-1} \deg(x_j)} \tm_k(x_1 \tensor \cdots \tensor d x_i \tensor \cdots \tensor x_k) + \\ |
|
1133 % & \qquad + + \\ |
|
1134 % & \qquad + |
|
1135 \end{align*} |
1146 \end{align*} |
|
1147 \begin{align*} |
|
1148 \bdy(& \tm_4(a \tensor b \tensor c \tensor d)) = \left(\tm_4(\bdy a \tensor b \tensor c \tensor d) + \cdots + \tm_4(a \tensor b \tensor c \tensor \bdy d)\right) + \\ |
|
1149 & + \left(\tm_3(a \tensor b \tensor c) \tensor d + \tm_2(a \tensor b) \tensor \tm_2(c \tensor d) + a \tensor \tm_3(b \tensor c \tensor d)\right) + \\ |
|
1150 & + \left(\tm_3(m_2(a \tensor b) \tensor c \tensor d) + \tm_3(a \tensor m_2(b \tensor c) \tensor d) + \tm_3(a \tensor b \tensor m_2(c \tensor d))\right. + \\ |
|
1151 & + \left.\tm_2(m_3(a \tensor b \tensor c) \tensor d) + \tm_2(a \tensor m_3(b \tensor c \tensor d)) + m_4(a \tensor b \tensor c \tensor d)\right) \\ |
|
1152 \end{align*} |
|
1153 See Figure \ref{fig:A4-terms}, comparing it against Figure \ref{fig:A4-faces}, to see this illustrated in the case $k=4$. There the $3$ faces closest |
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1154 to the top of the diagram have two rightmost vertices, while the other $6$ faces have only one. |
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1155 |
|
1156 \begin{figure}[!ht] |
|
1157 \begin{equation*} |
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1158 \mathfig{1.0}{associahedron/A4-terms} |
|
1159 \end{equation*} |
|
1160 \caption{The terms of $\bdy(\tm_k(a_1 \tensor \cdots \tensor a_k))$ correspond to the faces of the $k-1$ dimensional associahedron.} |
|
1161 \label{fig:A4-terms} |
|
1162 \end{figure} |
1136 |
1163 |
1137 \nn{Need to let the input $n$-category $C$ be a graded thing (e.g. DG |
1164 \nn{Need to let the input $n$-category $C$ be a graded thing (e.g. DG |
1138 $n$-category or $A_\infty$ $n$-category). DG $n$-category case is pretty |
1165 $n$-category or $A_\infty$ $n$-category). DG $n$-category case is pretty |
1139 easy, I think, so maybe it should be done earlier??} |
1166 easy, I think, so maybe it should be done earlier??} |
1140 |
1167 |