
diff -r 0df8bde1c896 -r 638be64bd329 text/a_inf_blob.tex
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+++ b/text/a_inf_blob.tex	Mon Aug 17 22:51:08 2009 +0000
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+%!TEX root = ../blob1.tex
+
+\section{The blob complex for $A_\infty$ $n$-categories}
+\label{sec:ainfblob}
+
+Given an $A_\infty$ $n$-category $\cC$ and an $n$-manifold $M$, we define the blob
+complex $\bc_*(M)$ to the be the colimit $\cC(M)$ of Section \ref{sec:ncats}.
+\nn{say something about this being anticlimatically tautological?}
+We will show below 
+\nn{give ref}
+that this agrees (up to homotopy) with our original definition of the blob complex
+in the case of plain $n$-categories.
+When we need to distinguish between the new and old definitions, we will refer to the 
+new-fangled and old-fashioned blob complex.
+
+\medskip
+
+Let $M^n = Y^k\times F^{n-k}$.  
+Let $C$ be a plain $n$-category.
+Let $\cF$ be the $A_\infty$ $k$-category which assigns to a $k$-ball
+$X$ the old-fashioned blob complex $\bc_*(X\times F)$.
+
+\begin{thm}
+The old-fashioned blob complex $\bc_*^C(Y\times F)$ is homotopy equivalent to the
+new-fangled blob complex $\bc_*^\cF(Y)$.
+\end{thm}
+
+\begin{proof}
+We will use the concrete description of the colimit from Subsection \ref{ss:ncat_fields}.
+
+First we define a map from $\bc_*^\cF(Y)$ to $\bc_*^C(Y\times F)$.
+In filtration degree 0 we just glue together the various blob diagrams on $X\times F$
+(where $X$ is a component of a permissible decomposition of $Y$) to get a blob diagram on
+$Y\times F$.
+In filtration degrees 1 and higher we define the map to be zero.
+It is easy to check that this is a chain map.
+
+Next we define a map from $\bc_*^C(Y\times F)$ to $\bc_*^\cF(Y)$.
+Actually, we will define it on the homotopy equivalent subcomplex
+$\cS_* \sub \bc_*^C(Y\times F)$ generated by blob diagrams which are small with respect to some open cover
+of $Y\times F$.
+\nn{need reference to small blob lemma}
+We will have to show eventually that this is independent (up to homotopy) of the choice of cover.
+Also, for a fixed choice of cover we will only be able to define the map for blob degree less than
+some bound, but this bound goes to infinity as the cover become finer.
+
+\nn{....}
+\end{proof}
+
+\nn{need to say something about dim $< n$ above}
+
+
+
+\medskip
+\hrule
+\medskip
+
+\nn{to be continued...}
+\medskip
+




