1022 $a\in \psi_{\cC;W,c}(x)$ for some decomposition $x$, and $g: \psi_{\cC;W,c}(x)

  1022 $a\in \psi_{\cC;W,c}(x)$ for some decomposition $x$, and $g: \psi_{\cC;W,c}(x)

  1023 \to \psi_{\cC;W,c}(y)$ is value of $\psi_{\cC;W,c}$ on some antirefinement $x \leq y$.

  1023 \to \psi_{\cC;W,c}(y)$ is value of $\psi_{\cC;W,c}$ on some antirefinement $x \leq y$.

  1024 

  1024 

  1025 In the $A_\infty$ case, enriched over chain complexes, the concrete description of the homotopy colimit

  1025 In the $A_\infty$ case, enriched over chain complexes, the concrete description of the homotopy colimit

  1026 is more involved.

  1026 is more involved.

  1028 Define an $m$-sequence in $W$ to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions of $W$.

  1029 Define an $m$-sequence in $W$ to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions of $W$.

  1029 Such sequences (for all $m$) form a simplicial set in $\cell(W)$.

  1030 Such sequences (for all $m$) form a simplicial set in $\cell(W)$.

  1030 Define $\cl{\cC}(W)$ as a vector space via

  1031 Define $\cl{\cC}(W)$ as a vector space via

  1031 \[

  1032 \[

  1032 	\cl{\cC}(W) = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] ,

  1033 	\cl{\cC}(W) = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] ,

  2171 	arc (90:135:3) node[circle,fill=black,inner sep=2pt] {}

  2172 	arc (90:135:3) node[circle,fill=black,inner sep=2pt] {}

  2172 	arc (-135:-90:3) node[below] {$A$}

  2173 	arc (-135:-90:3) node[below] {$A$}

  2173 	arc (-90:-45:3);

  2174 	arc (-90:-45:3);

  2174 \draw[fill]  (150:1.5) circle (2pt) node[below=4pt] {$D'$};

  2175 \draw[fill]  (150:1.5) circle (2pt) node[below=4pt] {$D'$};

  2175 \node[green!50!brown] at (-2,0) {\scalebox{2.0}{$f'\uparrow $}};

  2176 \node[green!50!brown] at (-2,0) {\scalebox{2.0}{$f'\uparrow $}};

  2177 \end{tikzpicture}

  2178 \end{tikzpicture}

  2178 \end{equation*}

  2179 \end{equation*}

  2179 \caption{Moving $B$ from bottom to top}

  2180 \caption{Moving $B$ from bottom to top}

  2180 \label{jun23c}

  2181 \label{jun23c}

  2181 \end{figure}

  2182 \end{figure}