equal
deleted
inserted
replaced
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. |
1027 %\nn{should probably rewrite this to be compatible with some standard reference} |
1027 \nn{should change to less strange terminology: ``filtration" to ``simplex" |
|
1028 (search for all occurrences of ``filtration")} |
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 $}}; |
2176 \node[green!50!brown] at (0.2,0.8) {\scalebox{2.0}{$\psi^+\uparrow $}}; |
2177 \node[green!50!brown] at (0.2,0.8) {\scalebox{2.0}{$\psi^\dagger \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} |