5 

     5 

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

     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 homotopy colimit $\cC(M)$ of Section \ref{sec:ncats}.

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

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

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

     9 We will show below 

     9 We will show below 

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

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

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

    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 

    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.

    14 new-fangled and old-fashioned blob complex.

    15 

    15 

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

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

   203 

   203 

   204 \medskip

   204 \medskip

   205 

   205 

   206 \begin{cor}

   206 \begin{cor}

   207 The new-fangled and old-fashioned blob complexes are homotopic.

   208 The new-fangled and old-fashioned blob complexes are homotopic.

   208 \end{cor}

   209 \end{cor}

   209 \begin{proof}

   210 \begin{proof}

   210 Apply Theorem \ref{product_thm} with the fiber $F$ equal to a point.

   211 Apply Theorem \ref{product_thm} with the fiber $F$ equal to a point.

   211 \end{proof}

   212 \end{proof}