1 %!TEX root = ../blob1.tex

     2 

     3 \section{The blob complex for $A_\infty$ $n$-categories}

     4 \label{sec:ainfblob}

     5 

     6 Given an $A_\infty$ $n$-category $\cC$ and an $n$-manifold $M$, we define the blob

     7 complex $\bc_*(M)$ to the be the colimit $\cC(M)$ of Section \ref{sec:ncats}.

     8 \nn{say something about this being anticlimatically tautological?}

     9 We will show below 

    10 \nn{give ref}

    11 that this agrees (up to homotopy) with our original definition of the blob complex

    12 in the case of plain $n$-categories.

    13 When we need to distinguish between the new and old definitions, we will refer to the 

    14 new-fangled and old-fashioned blob complex.

    15 

    16 \medskip

    17 

    18 Let $M^n = Y^k\times F^{n-k}$.  

    19 Let $C$ be a plain $n$-category.

    20 Let $\cF$ be the $A_\infty$ $k$-category which assigns to a $k$-ball

    21 $X$ the old-fashioned blob complex $\bc_*(X\times F)$.

    22 

    23 \begin{thm}

    24 The old-fashioned blob complex $\bc_*^C(Y\times F)$ is homotopy equivalent to the

    25 new-fangled blob complex $\bc_*^\cF(Y)$.

    26 \end{thm}

    27 

    28 \begin{proof}

    29 We will use the concrete description of the colimit from Subsection \ref{ss:ncat_fields}.

    30 

    31 First we define a map from $\bc_*^\cF(Y)$ to $\bc_*^C(Y\times F)$.

    32 In filtration degree 0 we just glue together the various blob diagrams on $X\times F$

    33 (where $X$ is a component of a permissible decomposition of $Y$) to get a blob diagram on

    34 $Y\times F$.

    35 In filtration degrees 1 and higher we define the map to be zero.

    36 It is easy to check that this is a chain map.

    37 

    38 Next we define a map from $\bc_*^C(Y\times F)$ to $\bc_*^\cF(Y)$.

    39 Actually, we will define it on the homotopy equivalent subcomplex

    40 $\cS_* \sub \bc_*^C(Y\times F)$ generated by blob diagrams which are small with respect to some open cover

    41 of $Y\times F$.

    42 \nn{need reference to small blob lemma}

    43 We will have to show eventually that this is independent (up to homotopy) of the choice of cover.

    44 Also, for a fixed choice of cover we will only be able to define the map for blob degree less than

    45 some bound, but this bound goes to infinity as the cover become finer.

    46 

    47 \nn{....}

    48 \end{proof}

    49 

    50 \nn{need to say something about dim $< n$ above}

    51 

    52 

    53 

    54 \medskip

    55 \hrule

    56 \medskip

    57 

    58 \nn{to be continued...}

    59 \medskip

    60