1 %!TEX root = ../blob1.tex |
1 %!TEX root = ../blob1.tex |
2 |
2 |
3 \section{The blob complex for $A_\infty$ $n$-categories} |
3 \section{The blob complex for $A_\infty$ $n$-categories} |
4 \label{sec:ainfblob} |
4 \label{sec:ainfblob} |
5 Given an $A_\infty$ $n$-category $\cC$ and an $n$-manifold $M$, we make the anticlimactically tautological definition of the blob |
5 Given an $A_\infty$ $n$-category $\cC$ and an $n$-manifold $M$, we make the anticlimactically tautological definition of the blob |
6 complex $\bc_*(M;\cC)$ to be the homotopy colimit $\cl{\cC}(M)$ of Section \ref{ss:ncat_fields}. |
6 complex $\bc_*(M;\cC)$ to be the homotopy colimit $\cl{\cC}(M)$ of \S\ref{ss:ncat_fields}. |
7 |
7 |
8 We will show below |
8 We will show below |
9 in Corollary \ref{cor:new-old} |
9 in Corollary \ref{cor:new-old} |
10 that when $\cC$ is obtained from a topological $n$-category $\cD$ as the blob complex of a point, this agrees (up to homotopy) with our original definition of the blob complex |
10 that when $\cC$ is obtained from a topological $n$-category $\cD$ as the blob complex of a point, this agrees (up to homotopy) with our original definition of the blob complex |
11 for $\cD$. |
11 for $\cD$. |
85 Let $a$ be a generator of $G_*$. |
85 Let $a$ be a generator of $G_*$. |
86 Let $D(a)$ denote the subcomplex of $\cl{\bc_*(F; C)}(Y)$ generated by all $(b, \ol{K})$ |
86 Let $D(a)$ denote the subcomplex of $\cl{\bc_*(F; C)}(Y)$ generated by all $(b, \ol{K})$ |
87 such that $a$ splits along $K_0\times F$ and $b$ is a generator appearing |
87 such that $a$ splits along $K_0\times F$ and $b$ is a generator appearing |
88 in an iterated boundary of $a$ (this includes $a$ itself). |
88 in an iterated boundary of $a$ (this includes $a$ itself). |
89 (Recall that $\ol{K} = (K_0,\ldots,K_l)$ denotes a chain of decompositions; |
89 (Recall that $\ol{K} = (K_0,\ldots,K_l)$ denotes a chain of decompositions; |
90 see Subsection \ref{ss:ncat_fields}.) |
90 see \S\ref{ss:ncat_fields}.) |
91 By $(b, \ol{K})$ we really mean $(b^\sharp, \ol{K})$, where $b^\sharp$ is |
91 By $(b, \ol{K})$ we really mean $(b^\sharp, \ol{K})$, where $b^\sharp$ is |
92 $b$ split according to $K_0\times F$. |
92 $b$ split according to $K_0\times F$. |
93 To simplify notation we will just write plain $b$ instead of $b^\sharp$. |
93 To simplify notation we will just write plain $b$ instead of $b^\sharp$. |
94 Roughly speaking, $D(a)$ consists of filtration degree 0 stuff which glues up to give |
94 Roughly speaking, $D(a)$ consists of filtration degree 0 stuff which glues up to give |
95 $a$ (or one of its iterated boundaries), filtration degree 1 stuff which makes all of the filtration degree 0 stuff homologous, |
95 $a$ (or one of its iterated boundaries), filtration degree 1 stuff which makes all of the filtration degree 0 stuff homologous, |