author | kevin@6e1638ff-ae45-0410-89bd-df963105f760 |
Tue, 18 Aug 2009 19:27:44 +0000 | |
changeset 114 | 1e50c1a5e8c0 |
parent 100 | c5a43be00ed4 |
child 117 | b62214646c4f |
permissions | -rw-r--r-- |
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%!TEX root = ../blob1.tex |
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\section{Definitions} |
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\label{sec:definitions} |
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\subsection{Systems of fields} |
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\label{sec:fields} |
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Let $\cM_k$ denote the category (groupoid, in fact) with objects |
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oriented PL manifolds of dimension |
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$k$ and morphisms homeomorphisms. |
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(We could equally well work with a different category of manifolds --- |
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unoriented, topological, smooth, spin, etc. --- but for definiteness we |
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will stick with oriented PL.) |
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Fix a top dimension $n$, and a symmetric monoidal category $\cS$ whose objects are sets. While reading the definition, you should just think about the case $\cS = \Set$ with cartesian product, until you reach the discussion of a \emph{linear system of fields} later in this section, where $\cS = \Vect$, and \S \ref{sec:homological-fields}, where $\cS = \Kom$. |
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A $n$-dimensional {\it system of fields} in $\cS$ |
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is a collection of functors $\cC_k : \cM_k \to \Set$ for $0 \leq k \leq n$ |
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together with some additional data and satisfying some additional conditions, all specified below. |
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\nn{refer somewhere to my TQFT notes \cite{kw:tqft}, and possibly also to paper with Chris} |
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Before finishing the definition of fields, we give two motivating examples |
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(actually, families of examples) of systems of fields. |
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The first examples: Fix a target space $B$, and let $\cC(X)$ be the set of continuous maps |
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from X to $B$. |
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The second examples: Fix an $n$-category $C$, and let $\cC(X)$ be |
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the set of sub-cell-complexes of $X$ with codimension-$j$ cells labeled by |
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$j$-morphisms of $C$. |
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One can think of such sub-cell-complexes as dual to pasting diagrams for $C$. |
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This is described in more detail below. |
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Now for the rest of the definition of system of fields. |
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\begin{enumerate} |
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\item There are boundary restriction maps $\cC_k(X) \to \cC_{k-1}(\bd X)$, |
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and these maps are a natural |
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transformation between the functors $\cC_k$ and $\cC_{k-1}\circ\bd$. |
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For $c \in \cC_{k-1}(\bd X)$, we will denote by $\cC_k(X; c)$ the subset of |
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$\cC(X)$ which restricts to $c$. |
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In this context, we will call $c$ a boundary condition. |
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\item The subset $\cC_n(X;c)$ of top fields with a given boundary condition is an object in our symmetric monoidal category $\cS$. (This condition is of course trivial when $\cS = \Set$.) If the objects are sets with extra structure (e.g. $\cS = \Vect$ or $\Kom$), then this extra structure is considered part of the definition of $\cC_n$. Any maps mentioned below between top level fields must be morphisms in $\cS$. |
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\item There are orientation reversal maps $\cC_k(X) \to \cC_k(-X)$, and these maps |
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again comprise a natural transformation of functors. |
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In addition, the orientation reversal maps are compatible with the boundary restriction maps. |
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\item $\cC_k$ is compatible with the symmetric monoidal |
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structures on $\cM_k$, $\Set$ and $\cS$: $\cC_k(X \du W) \cong \cC_k(X)\times \cC_k(W)$, |
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compatibly with homeomorphisms, restriction to boundary, and orientation reversal. |
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We will call the projections $\cC(X_1 \du X_2) \to \cC(X_i)$ |
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restriction maps. |
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\item Gluing without corners. |
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Let $\bd X = Y \du -Y \du W$, where $Y$ and $W$ are closed $k{-}1$-manifolds. |
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Let $X\sgl$ denote $X$ glued to itself along $\pm Y$. |
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Using the boundary restriction, disjoint union, and (in one case) orientation reversal |
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maps, we get two maps $\cC_k(X) \to \cC(Y)$, corresponding to the two |
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copies of $Y$ in $\bd X$. |
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Let $\Eq_Y(\cC_k(X))$ denote the equalizer of these two maps. |
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Then (here's the axiom/definition part) there is an injective ``gluing" map |
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\[ |
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\Eq_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl) , |
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\] |
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and this gluing map is compatible with all of the above structure (actions |
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of homeomorphisms, boundary restrictions, orientation reversal, disjoint union). |
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Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity, |
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the gluing map is surjective. |
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From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the |
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gluing surface, we say that fields in the image of the gluing map |
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are transverse to $Y$ or cuttable along $Y$. |
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\item Gluing with corners. |
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Let $\bd X = Y \cup -Y \cup W$, where $\pm Y$ and $W$ might intersect along their boundaries. |
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Let $X\sgl$ denote $X$ glued to itself along $\pm Y$. |
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Note that $\bd X\sgl = W\sgl$, where $W\sgl$ denotes $W$ glued to itself |
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(without corners) along two copies of $\bd Y$. |
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Let $c\sgl \in \cC_{k-1}(W\sgl)$ be a be a cuttable field on $W\sgl$ and let |
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$c \in \cC_{k-1}(W)$ be the cut open version of $c\sgl$. |
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Let $\cC^c_k(X)$ denote the subset of $\cC(X)$ which restricts to $c$ on $W$. |
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(This restriction map uses the gluing without corners map above.) |
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Using the boundary restriction, gluing without corners, and (in one case) orientation reversal |
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maps, we get two maps $\cC^c_k(X) \to \cC(Y)$, corresponding to the two |
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copies of $Y$ in $\bd X$. |
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Let $\Eq^c_Y(\cC_k(X))$ denote the equalizer of these two maps. |
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Then (here's the axiom/definition part) there is an injective ``gluing" map |
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\[ |
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\Eq^c_Y(\cC_k(X)) \hookrightarrow \cC_k(X\sgl, c\sgl) , |
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\] |
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and this gluing map is compatible with all of the above structure (actions |
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of homeomorphisms, boundary restrictions, orientation reversal, disjoint union). |
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Furthermore, up to homeomorphisms of $X\sgl$ isotopic to the identity, |
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91 |
the gluing map is surjective. |
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92 |
From the point of view of $X\sgl$ and the image $Y \subset X\sgl$ of the |
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93 |
gluing surface, we say that fields in the image of the gluing map |
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94 |
are transverse to $Y$ or cuttable along $Y$. |
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95 |
\item There are maps $\cC_{k-1}(Y) \to \cC_k(Y \times I)$, denoted |
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96 |
$c \mapsto c\times I$. |
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97 |
These maps comprise a natural transformation of functors, and commute appropriately |
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98 |
with all the structure maps above (disjoint union, boundary restriction, etc.). |
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99 |
Furthermore, if $f: Y\times I \to Y\times I$ is a fiber-preserving homeomorphism |
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100 |
covering $\bar{f}:Y\to Y$, then $f(c\times I) = \bar{f}(c)\times I$. |
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101 |
\end{enumerate} |
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102 |
|
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103 |
\nn{need to introduce two notations for glued fields --- $x\bullet y$ and $x\sgl$} |
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104 |
|
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105 |
\bigskip |
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106 |
Using the functoriality and $\bullet\times I$ properties above, together |
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107 |
with boundary collar homeomorphisms of manifolds, we can define the notion of |
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108 |
{\it extended isotopy}. |
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109 |
Let $M$ be an $n$-manifold and $Y \subset \bd M$ be a codimension zero submanifold |
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110 |
of $\bd M$. |
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111 |
Let $x \in \cC(M)$ be a field on $M$ and such that $\bd x$ is cuttable along $\bd Y$. |
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112 |
Let $c$ be $x$ restricted to $Y$. |
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113 |
Let $M \cup (Y\times I)$ denote $M$ glued to $Y\times I$ along $Y$. |
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114 |
Then we have the glued field $x \bullet (c\times I)$ on $M \cup (Y\times I)$. |
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115 |
Let $f: M \cup (Y\times I) \to M$ be a collaring homeomorphism. |
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116 |
Then we say that $x$ is {\it extended isotopic} to $f(x \bullet (c\times I))$. |
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117 |
More generally, we define extended isotopy to be the equivalence relation on fields |
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118 |
on $M$ generated by isotopy plus all instance of the above construction |
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119 |
(for all appropriate $Y$ and $x$). |
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120 |
|
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121 |
\nn{should also say something about pseudo-isotopy} |
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122 |
|
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123 |
%\bigskip |
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124 |
%\hrule |
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125 |
%\bigskip |
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126 |
% |
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127 |
%\input{text/fields.tex} |
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128 |
% |
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129 |
% |
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130 |
%\bigskip |
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131 |
%\hrule |
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132 |
%\bigskip |
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133 |
|
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134 |
\nn{note: probably will suppress from notation the distinction |
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135 |
between fields and their (orientation-reversal) duals} |
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136 |
|
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137 |
\nn{remark that if top dimensional fields are not already linear |
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138 |
then we will soon linearize them(?)} |
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139 |
|
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140 |
We now describe in more detail systems of fields coming from sub-cell-complexes labeled |
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141 |
by $n$-category morphisms. |
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142 |
|
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143 |
Given an $n$-category $C$ with the right sort of duality |
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144 |
(e.g. pivotal 2-category, 1-category with duals, star 1-category, disklike $n$-category), |
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145 |
we can construct a system of fields as follows. |
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146 |
Roughly speaking, $\cC(X)$ will the set of all embedded cell complexes in $X$ |
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147 |
with codimension $i$ cells labeled by $i$-morphisms of $C$. |
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148 |
We'll spell this out for $n=1,2$ and then describe the general case. |
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149 |
|
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150 |
If $X$ has boundary, we require that the cell decompositions are in general |
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151 |
position with respect to the boundary --- the boundary intersects each cell |
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152 |
transversely, so cells meeting the boundary are mere half-cells. |
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153 |
|
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154 |
Put another way, the cell decompositions we consider are dual to standard cell |
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155 |
decompositions of $X$. |
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156 |
|
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157 |
We will always assume that our $n$-categories have linear $n$-morphisms. |
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158 |
|
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159 |
For $n=1$, a field on a 0-manifold $P$ is a labeling of each point of $P$ with |
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160 |
an object (0-morphism) of the 1-category $C$. |
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161 |
A field on a 1-manifold $S$ consists of |
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162 |
\begin{itemize} |
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163 |
\item A cell decomposition of $S$ (equivalently, a finite collection |
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164 |
of points in the interior of $S$); |
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165 |
\item a labeling of each 1-cell (and each half 1-cell adjacent to $\bd S$) |
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166 |
by an object (0-morphism) of $C$; |
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167 |
\item a transverse orientation of each 0-cell, thought of as a choice of |
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168 |
``domain" and ``range" for the two adjacent 1-cells; and |
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169 |
\item a labeling of each 0-cell by a morphism (1-morphism) of $C$, with |
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170 |
domain and range determined by the transverse orientation and the labelings of the 1-cells. |
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171 |
\end{itemize} |
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172 |
|
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173 |
If $C$ is an algebra (i.e. if $C$ has only one 0-morphism) we can ignore the labels |
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174 |
of 1-cells, so a field on a 1-manifold $S$ is a finite collection of points in the |
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175 |
interior of $S$, each transversely oriented and each labeled by an element (1-morphism) |
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176 |
of the algebra. |
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177 |
|
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178 |
\medskip |
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179 |
|
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180 |
For $n=2$, fields are just the sort of pictures based on 2-categories (e.g.\ tensor categories) |
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181 |
that are common in the literature. |
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182 |
We describe these carefully here. |
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183 |
|
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184 |
A field on a 0-manifold $P$ is a labeling of each point of $P$ with |
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185 |
an object of the 2-category $C$. |
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186 |
A field of a 1-manifold is defined as in the $n=1$ case, using the 0- and 1-morphisms of $C$. |
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187 |
A field on a 2-manifold $Y$ consists of |
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188 |
\begin{itemize} |
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189 |
\item A cell decomposition of $Y$ (equivalently, a graph embedded in $Y$ such |
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190 |
that each component of the complement is homeomorphic to a disk); |
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191 |
\item a labeling of each 2-cell (and each partial 2-cell adjacent to $\bd Y$) |
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by a 0-morphism of $C$; |
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193 |
\item a transverse orientation of each 1-cell, thought of as a choice of |
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``domain" and ``range" for the two adjacent 2-cells; |
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195 |
\item a labeling of each 1-cell by a 1-morphism of $C$, with |
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domain and range determined by the transverse orientation of the 1-cell |
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and the labelings of the 2-cells; |
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\item for each 0-cell, a homeomorphism of the boundary $R$ of a small neighborhood |
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of the 0-cell to $S^1$ such that the intersections of the 1-cells with $R$ are not mapped |
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to $\pm 1 \in S^1$; and |
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\item a labeling of each 0-cell by a 2-morphism of $C$, with domain and range |
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determined by the labelings of the 1-cells and the parameterizations of the previous |
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203 |
bullet. |
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204 |
\end{itemize} |
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\nn{need to say this better; don't try to fit everything into the bulleted list} |
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206 |
|
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207 |
For general $n$, a field on a $k$-manifold $X^k$ consists of |
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208 |
\begin{itemize} |
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209 |
\item A cell decomposition of $X$; |
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\item an explicit general position homeomorphism from the link of each $j$-cell |
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to the boundary of the standard $(k-j)$-dimensional bihedron; and |
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\item a labeling of each $j$-cell by a $(k-j)$-dimensional morphism of $C$, with |
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domain and range determined by the labelings of the link of $j$-cell. |
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214 |
\end{itemize} |
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215 |
|
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216 |
%\nn{next definition might need some work; I think linearity relations should |
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%be treated differently (segregated) from other local relations, but I'm not sure |
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%the next definition is the best way to do it} |
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219 |
|
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220 |
\medskip |
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221 |
|
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For top dimensional ($n$-dimensional) manifolds, we're actually interested |
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223 |
in the linearized space of fields. |
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By default, define $\lf(X) = \c[\cC(X)]$; that is, $\lf(X)$ is |
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the vector space of finite |
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linear combinations of fields on $X$. |
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If $X$ has boundary, we of course fix a boundary condition: $\lf(X; a) = \c[\cC(X; a)]$. |
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Thus the restriction (to boundary) maps are well defined because we never |
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take linear combinations of fields with differing boundary conditions. |
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230 |
|
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231 |
In some cases we don't linearize the default way; instead we take the |
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spaces $\lf(X; a)$ to be part of the data for the system of fields. |
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233 |
In particular, for fields based on linear $n$-category pictures we linearize as follows. |
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234 |
Define $\lf(X; a) = \c[\cC(X; a)]/K$, where $K$ is the space generated by |
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235 |
obvious relations on 0-cell labels. |
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236 |
More specifically, let $L$ be a cell decomposition of $X$ |
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and let $p$ be a 0-cell of $L$. |
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Let $\alpha_c$ and $\alpha_d$ be two labelings of $L$ which are identical except that |
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$\alpha_c$ labels $p$ by $c$ and $\alpha_d$ labels $p$ by $d$. |
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240 |
Then the subspace $K$ is generated by things of the form |
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$\lambda \alpha_c + \alpha_d - \alpha_{\lambda c + d}$, where we leave it to the reader |
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242 |
to infer the meaning of $\alpha_{\lambda c + d}$. |
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243 |
Note that we are still assuming that $n$-categories have linear spaces of $n$-morphisms. |
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244 |
|
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\nn{Maybe comment further: if there's a natural basis of morphisms, then no need; |
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246 |
will do something similar below; in general, whenever a label lives in a linear |
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247 |
space we do something like this; ? say something about tensor |
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248 |
product of all the linear label spaces? Yes:} |
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249 |
|
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For top dimensional ($n$-dimensional) manifolds, we linearize as follows. |
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251 |
Define an ``almost-field" to be a field without labels on the 0-cells. |
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252 |
(Recall that 0-cells are labeled by $n$-morphisms.) |
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253 |
To each unlabeled 0-cell in an almost field there corresponds a (linear) $n$-morphism |
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254 |
space determined by the labeling of the link of the 0-cell. |
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255 |
(If the 0-cell were labeled, the label would live in this space.) |
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256 |
We associate to each almost-labeling the tensor product of these spaces (one for each 0-cell). |
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257 |
We now define $\lf(X; a)$ to be the direct sum over all almost labelings of the |
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258 |
above tensor products. |
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259 |
|
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260 |
|
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261 |
|
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262 |
\subsection{Local relations} |
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263 |
\label{sec:local-relations} |
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264 |
|
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265 |
|
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A {\it local relation} is a collection subspaces $U(B; c) \sub \lf(B; c)$, |
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267 |
for all $n$-manifolds $B$ which are |
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268 |
homeomorphic to the standard $n$-ball and all $c \in \cC(\bd B)$, |
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269 |
satisfying the following properties. |
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270 |
\begin{enumerate} |
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271 |
\item functoriality: |
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$f(U(B; c)) = U(B', f(c))$ for all homeomorphisms $f: B \to B'$ |
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273 |
\item local relations imply extended isotopy: |
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274 |
if $x, y \in \cC(B; c)$ and $x$ is extended isotopic |
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275 |
to $y$, then $x-y \in U(B; c)$. |
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276 |
\item ideal with respect to gluing: |
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if $B = B' \cup B''$, $x\in U(B')$, and $c\in \cC(B'')$, then $x\bullet r \in U(B)$ |
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278 |
\end{enumerate} |
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279 |
See \cite{kw:tqft} for details. |
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280 |
|
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281 |
|
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282 |
For maps into spaces, $U(B; c)$ is generated by things of the form $a-b \in \lf(B; c)$, |
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where $a$ and $b$ are maps (fields) which are homotopic rel boundary. |
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284 |
|
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285 |
For $n$-category pictures, $U(B; c)$ is equal to the kernel of the evaluation map |
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286 |
$\lf(B; c) \to \mor(c', c'')$, where $(c', c'')$ is some (any) division of $c$ into |
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287 |
domain and range. |
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288 |
|
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289 |
\nn{maybe examples of local relations before general def?} |
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290 |
|
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291 |
Given a system of fields and local relations, we define the skein space |
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292 |
$A(Y^n; c)$ to be the space of all finite linear combinations of fields on |
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293 |
the $n$-manifold $Y$ modulo local relations. |
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294 |
The Hilbert space $Z(Y; c)$ for the TQFT based on the fields and local relations |
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295 |
is defined to be the dual of $A(Y; c)$. |
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296 |
(See \cite{kw:tqft} or xxxx for details.) |
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297 |
|
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298 |
\nn{should expand above paragraph} |
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299 |
|
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300 |
The blob complex is in some sense the derived version of $A(Y; c)$. |
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301 |
|
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302 |
|
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303 |
|
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304 |
\subsection{The blob complex} |
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305 |
\label{sec:blob-definition} |
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306 |
|
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307 |
Let $X$ be an $n$-manifold. |
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308 |
Assume a fixed system of fields and local relations. |
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309 |
In this section we will usually suppress boundary conditions on $X$ from the notation |
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310 |
(e.g. write $\lf(X)$ instead of $\lf(X; c)$). |
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311 |
|
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312 |
We only consider compact manifolds, so if $Y \sub X$ is a closed codimension 0 |
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313 |
submanifold of $X$, then $X \setmin Y$ implicitly means the closure |
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314 |
$\overline{X \setmin Y}$. |
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315 |
|
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316 |
We will define $\bc_0(X)$, $\bc_1(X)$ and $\bc_2(X)$, then give the general case $\bc_k(X)$. |
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317 |
|
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318 |
Define $\bc_0(X) = \lf(X)$. |
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319 |
(If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \lf(X; c)$. |
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320 |
We'll omit this sort of detail in the rest of this section.) |
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321 |
In other words, $\bc_0(X)$ is just the space of all linearized fields on $X$. |
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322 |
|
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323 |
$\bc_1(X)$ is, roughly, the space of all local relations that can be imposed on $\bc_0(X)$. |
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324 |
Less roughly (but still not the official definition), $\bc_1(X)$ is finite linear |
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325 |
combinations of 1-blob diagrams, where a 1-blob diagram to consists of |
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326 |
\begin{itemize} |
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327 |
\item An embedded closed ball (``blob") $B \sub X$. |
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328 |
\item A field $r \in \cC(X \setmin B; c)$ |
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329 |
(for some $c \in \cC(\bd B) = \cC(\bd(X \setmin B))$). |
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330 |
\item A local relation field $u \in U(B; c)$ |
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331 |
(same $c$ as previous bullet). |
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332 |
\end{itemize} |
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333 |
In order to get the linear structure correct, we (officially) define |
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334 |
\[ |
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335 |
\bc_1(X) \deq \bigoplus_B \bigoplus_c U(B; c) \otimes \lf(X \setmin B; c) . |
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336 |
\] |
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337 |
The first direct sum is indexed by all blobs $B\subset X$, and the second |
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338 |
by all boundary conditions $c \in \cC(\bd B)$. |
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339 |
Note that $\bc_1(X)$ is spanned by 1-blob diagrams $(B, u, r)$. |
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340 |
|
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341 |
Define the boundary map $\bd : \bc_1(X) \to \bc_0(X)$ by |
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342 |
\[ |
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343 |
(B, u, r) \mapsto u\bullet r, |
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344 |
\] |
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345 |
where $u\bullet r$ denotes the linear |
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346 |
combination of fields on $X$ obtained by gluing $u$ to $r$. |
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347 |
In other words $\bd : \bc_1(X) \to \bc_0(X)$ is given by |
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348 |
just erasing the blob from the picture |
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349 |
(but keeping the blob label $u$). |
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350 |
|
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351 |
Note that the skein space $A(X)$ |
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352 |
is naturally isomorphic to $\bc_0(X)/\bd(\bc_1(X))) = H_0(\bc_*(X))$. |
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353 |
|
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354 |
$\bc_2(X)$ is, roughly, the space of all relations (redundancies) among the |
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355 |
local relations encoded in $\bc_1(X)$. |
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356 |
More specifically, $\bc_2(X)$ is the space of all finite linear combinations of |
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357 |
2-blob diagrams, of which there are two types, disjoint and nested. |
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358 |
|
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359 |
A disjoint 2-blob diagram consists of |
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360 |
\begin{itemize} |
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361 |
\item A pair of closed balls (blobs) $B_0, B_1 \sub X$ with disjoint interiors. |
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362 |
\item A field $r \in \cC(X \setmin (B_0 \cup B_1); c_0, c_1)$ |
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363 |
(where $c_i \in \cC(\bd B_i)$). |
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364 |
\item Local relation fields $u_i \in U(B_i; c_i)$, $i=1,2$. |
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365 |
\end{itemize} |
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366 |
We also identify $(B_0, B_1, u_0, u_1, r)$ with $-(B_1, B_0, u_1, u_0, r)$; |
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367 |
reversing the order of the blobs changes the sign. |
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368 |
Define $\bd(B_0, B_1, u_0, u_1, r) = |
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369 |
(B_1, u_1, u_0\bullet r) - (B_0, u_0, u_1\bullet r) \in \bc_1(X)$. |
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370 |
In other words, the boundary of a disjoint 2-blob diagram |
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371 |
is the sum (with alternating signs) |
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372 |
of the two ways of erasing one of the blobs. |
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373 |
It's easy to check that $\bd^2 = 0$. |
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374 |
|
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375 |
A nested 2-blob diagram consists of |
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376 |
\begin{itemize} |
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377 |
\item A pair of nested balls (blobs) $B_0 \sub B_1 \sub X$. |
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378 |
\item A field $r \in \cC(X \setmin B_0; c_0)$ |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
379 |
(for some $c_0 \in \cC(\bd B_0)$), which is cuttable along $\bd B_1$. |
c5a43be00ed4
No new content, just rearranging (and procrastinating)
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
380 |
\item A local relation field $u_0 \in U(B_0; c_0)$. |
c5a43be00ed4
No new content, just rearranging (and procrastinating)
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
381 |
\end{itemize} |
c5a43be00ed4
No new content, just rearranging (and procrastinating)
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
382 |
Let $r = r_1 \bullet r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$ |
c5a43be00ed4
No new content, just rearranging (and procrastinating)
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
383 |
(for some $c_1 \in \cC(B_1)$) and |
c5a43be00ed4
No new content, just rearranging (and procrastinating)
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
384 |
$r' \in \cC(X \setmin B_1; c_1)$. |
c5a43be00ed4
No new content, just rearranging (and procrastinating)
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
385 |
Define $\bd(B_0, B_1, u_0, r) = (B_1, u_0\bullet r_1, r') - (B_0, u_0, r)$. |
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No new content, just rearranging (and procrastinating)
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
386 |
Note that the requirement that |
c5a43be00ed4
No new content, just rearranging (and procrastinating)
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
387 |
local relations are an ideal with respect to gluing guarantees that $u_0\bullet r_1 \in U(B_1)$. |
c5a43be00ed4
No new content, just rearranging (and procrastinating)
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
388 |
As in the disjoint 2-blob case, the boundary of a nested 2-blob is the alternating |
c5a43be00ed4
No new content, just rearranging (and procrastinating)
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
389 |
sum of the two ways of erasing one of the blobs. |
c5a43be00ed4
No new content, just rearranging (and procrastinating)
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
390 |
If we erase the inner blob, the outer blob inherits the label $u_0\bullet r_1$. |
c5a43be00ed4
No new content, just rearranging (and procrastinating)
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
391 |
It is again easy to check that $\bd^2 = 0$. |
c5a43be00ed4
No new content, just rearranging (and procrastinating)
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
392 |
|
c5a43be00ed4
No new content, just rearranging (and procrastinating)
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
393 |
\nn{should draw figures for 1, 2 and $k$-blob diagrams} |
c5a43be00ed4
No new content, just rearranging (and procrastinating)
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
394 |
|
c5a43be00ed4
No new content, just rearranging (and procrastinating)
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
395 |
As with the 1-blob diagrams, in order to get the linear structure correct it is better to define |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
396 |
(officially) |
c5a43be00ed4
No new content, just rearranging (and procrastinating)
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
397 |
\begin{eqnarray*} |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
398 |
\bc_2(X) & \deq & |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
399 |
\left( |
c5a43be00ed4
No new content, just rearranging (and procrastinating)
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
400 |
\bigoplus_{B_0, B_1 \text{disjoint}} \bigoplus_{c_0, c_1} |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
401 |
U(B_0; c_0) \otimes U(B_1; c_1) \otimes \lf(X\setmin (B_0\cup B_1); c_0, c_1) |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
402 |
\right) \\ |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
403 |
&& \bigoplus \left( |
c5a43be00ed4
No new content, just rearranging (and procrastinating)
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
404 |
\bigoplus_{B_0 \subset B_1} \bigoplus_{c_0} |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
405 |
U(B_0; c_0) \otimes \lf(X\setmin B_0; c_0) |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
406 |
\right) . |
c5a43be00ed4
No new content, just rearranging (and procrastinating)
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
407 |
\end{eqnarray*} |
c5a43be00ed4
No new content, just rearranging (and procrastinating)
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
408 |
The final $\lf(X\setmin B_0; c_0)$ above really means fields cuttable along $\bd B_1$, |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
409 |
but we didn't feel like introducing a notation for that. |
c5a43be00ed4
No new content, just rearranging (and procrastinating)
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
410 |
For the disjoint blobs, reversing the ordering of $B_0$ and $B_1$ introduces a minus sign |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
411 |
(rather than a new, linearly independent 2-blob diagram). |
c5a43be00ed4
No new content, just rearranging (and procrastinating)
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
412 |
|
c5a43be00ed4
No new content, just rearranging (and procrastinating)
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
413 |
Now for the general case. |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
414 |
A $k$-blob diagram consists of |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
415 |
\begin{itemize} |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
416 |
\item A collection of blobs $B_i \sub X$, $i = 0, \ldots, k-1$. |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
417 |
For each $i$ and $j$, we require that either $B_i$ and $B_j$have disjoint interiors or |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
418 |
$B_i \sub B_j$ or $B_j \sub B_i$. |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
419 |
(The case $B_i = B_j$ is allowed. |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
420 |
If $B_i \sub B_j$ the boundaries of $B_i$ and $B_j$ are allowed to intersect.) |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
421 |
If a blob has no other blobs strictly contained in it, we call it a twig blob. |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
422 |
\item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$. |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
423 |
(These are implied by the data in the next bullets, so we usually |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
424 |
suppress them from the notation.) |
c5a43be00ed4
No new content, just rearranging (and procrastinating)
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
425 |
$c_i$ and $c_j$ must have identical restrictions to $\bd B_i \cap \bd B_j$ |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
426 |
if the latter space is not empty. |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
427 |
\item A field $r \in \cC(X \setmin B^t; c^t)$, |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
428 |
where $B^t$ is the union of all the twig blobs and $c^t \in \cC(\bd B^t)$ |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
429 |
is determined by the $c_i$'s. |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
430 |
$r$ is required to be cuttable along the boundaries of all blobs, twigs or not. |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
431 |
\item For each twig blob $B_j$ a local relation field $u_j \in U(B_j; c_j)$, |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
432 |
where $c_j$ is the restriction of $c^t$ to $\bd B_j$. |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
433 |
If $B_i = B_j$ then $u_i = u_j$. |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
434 |
\end{itemize} |
c5a43be00ed4
No new content, just rearranging (and procrastinating)
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
435 |
|
c5a43be00ed4
No new content, just rearranging (and procrastinating)
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
436 |
If two blob diagrams $D_1$ and $D_2$ |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
437 |
differ only by a reordering of the blobs, then we identify |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
438 |
$D_1 = \pm D_2$, where the sign is the sign of the permutation relating $D_1$ and $D_2$. |
c5a43be00ed4
No new content, just rearranging (and procrastinating)
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
439 |
|
c5a43be00ed4
No new content, just rearranging (and procrastinating)
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
440 |
$\bc_k(X)$ is, roughly, all finite linear combinations of $k$-blob diagrams. |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
441 |
As before, the official definition is in terms of direct sums |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
442 |
of tensor products: |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
443 |
\[ |
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No new content, just rearranging (and procrastinating)
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
444 |
\bc_k(X) \deq \bigoplus_{\overline{B}} \bigoplus_{\overline{c}} |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
445 |
\left( \otimes_j U(B_j; c_j)\right) \otimes \lf(X \setmin B^t; c^t) . |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
446 |
\] |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
447 |
Here $\overline{B}$ runs over all configurations of blobs, satisfying the conditions above. |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
448 |
$\overline{c}$ runs over all boundary conditions, again as described above. |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
449 |
$j$ runs over all indices of twig blobs. The final $\lf(X \setmin B^t; c^t)$ must be interpreted as fields which are cuttable along all of the blobs in $\overline{B}$. |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
450 |
|
c5a43be00ed4
No new content, just rearranging (and procrastinating)
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
451 |
The boundary map $\bd : \bc_k(X) \to \bc_{k-1}(X)$ is defined as follows. |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
452 |
Let $b = (\{B_i\}, \{u_j\}, r)$ be a $k$-blob diagram. |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
453 |
Let $E_j(b)$ denote the result of erasing the $j$-th blob. |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
454 |
If $B_j$ is not a twig blob, this involves only decrementing |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
455 |
the indices of blobs $B_{j+1},\ldots,B_{k-1}$. |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
456 |
If $B_j$ is a twig blob, we have to assign new local relation labels |
c5a43be00ed4
No new content, just rearranging (and procrastinating)
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
457 |
if removing $B_j$ creates new twig blobs. |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
458 |
If $B_l$ becomes a twig after removing $B_j$, then set $u_l = u_j\bullet r_l$, |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
459 |
where $r_l$ is the restriction of $r$ to $B_l \setmin B_j$. |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
460 |
Finally, define |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
461 |
\eq{ |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
462 |
\bd(b) = \sum_{j=0}^{k-1} (-1)^j E_j(b). |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
463 |
} |
c5a43be00ed4
No new content, just rearranging (and procrastinating)
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
464 |
The $(-1)^j$ factors imply that the terms of $\bd^2(b)$ all cancel. |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
465 |
Thus we have a chain complex. |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
466 |
|
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
467 |
\nn{?? say something about the ``shape" of tree? (incl = cone, disj = product)} |
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
468 |
|
c5a43be00ed4
No new content, just rearranging (and procrastinating)
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
469 |
\nn{?? remark about dendroidal sets} |
c5a43be00ed4
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kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
diff
changeset
|
470 |
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c5a43be00ed4
No new content, just rearranging (and procrastinating)
kevin@6e1638ff-ae45-0410-89bd-df963105f760
parents:
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changeset
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