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