1032 

  1032 

  1033 In the $A_\infty$ case, enriched over chain complexes, the concrete description of the homotopy colimit

  1033 In the $A_\infty$ case, enriched over chain complexes, the concrete description of the homotopy colimit

  1034 is more involved.

  1034 is more involved.

  1035 We will describe two different (but homotopy equivalent) versions of the homotopy colimit of $\psi_{\cC;W}$.

  1035 We will describe two different (but homotopy equivalent) versions of the homotopy colimit of $\psi_{\cC;W}$.

  1036 The first is the usual one, which works for any indexing category.

  1036 The first is the usual one, which works for any indexing category.

  1039 is more closely related to the blob complex

  1038 is more closely related to the blob complex

  1040 construction of \S \ref{sec:blob-definition} and takes advantage of local (gluing) properties

  1039 construction of \S \ref{sec:blob-definition} and takes advantage of local (gluing) properties

  1041 of the indexing category $\cell(W)$.

  1040 of the indexing category $\cell(W)$.

  1042 

  1041 

  1043 Define an $m$-sequence in $W$ to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions of $W$.

  1042 Define an $m$-sequence in $W$ to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions of $W$.

  1349 

  1348 

  1350 We can define marked pinched products $\pi:E\to M$ of marked balls analogously to the 

  1349 We can define marked pinched products $\pi:E\to M$ of marked balls analogously to the 

  1351 plain ball case.

  1350 plain ball case.

  1352 Note that a marked pinched product can be decomposed into either

  1351 Note that a marked pinched product can be decomposed into either

  1353 two marked pinched products or a plain pinched product and a marked pinched product.

  1352 two marked pinched products or a plain pinched product and a marked pinched product.

  1355 

  1354 

  1356 \begin{module-axiom}[Product (identity) morphisms]

  1355 \begin{module-axiom}[Product (identity) morphisms]

  1357 For each pinched product $\pi:E\to M$, with $M$ a marked $k$-ball and $E$ a marked

  1356 For each pinched product $\pi:E\to M$, with $M$ a marked $k$-ball and $E$ a marked

  1358 $k{+}m$-ball ($m\ge 1$),

  1357 $k{+}m$-ball ($m\ge 1$),

  1359 there is a map $\pi^*:\cM(M)\to \cM(E)$.

  1358 there is a map $\pi^*:\cM(M)\to \cM(E)$.

  1826 %For the time being, let's say they are.}

  1825 %For the time being, let's say they are.}

  1827 A 1-marked $k$-ball is anything homeomorphic to $B^j \times C(S)$, $0\le j\le n-2$, 

  1826 A 1-marked $k$-ball is anything homeomorphic to $B^j \times C(S)$, $0\le j\le n-2$, 

  1828 where $B^j$ is the standard $j$-ball.

  1827 where $B^j$ is the standard $j$-ball.

  1829 A 1-marked $k$-ball can be decomposed in various ways into smaller balls, which are either 

  1828 A 1-marked $k$-ball can be decomposed in various ways into smaller balls, which are either 

  1830 (a) smaller 1-marked $k$-balls, (b) 0-marked $k$-balls, or (c) plain $k$-balls.

  1829 (a) smaller 1-marked $k$-balls, (b) 0-marked $k$-balls, or (c) plain $k$-balls.

  1832 We now proceed as in the above module definitions.

  1831 We now proceed as in the above module definitions.

  1833 

  1832 

  1834 \begin{figure}[t] \centering

  1833 \begin{figure}[t] \centering

  1835 \begin{tikzpicture}[baseline,line width = 2pt]

  1834 \begin{tikzpicture}[baseline,line width = 2pt]

  1836 \draw[blue][fill=blue!15!white] (0,0) circle (2);

  1835 \draw[blue][fill=blue!15!white] (0,0) circle (2);

  2188 

  2187 

  2189 If $n\ge 2$, these two movie move suffice:

  2188 If $n\ge 2$, these two movie move suffice:

  2190 

  2189 

  2191 \begin{lem}

  2190 \begin{lem}

  2192 Assume $n\ge 2$ and fix $E$ and $E'$ as above.

  2191 Assume $n\ge 2$ and fix $E$ and $E'$ as above.

  2194 are related by a sequence of the two movie moves defined above.

  2193 are related by a sequence of the two movie moves defined above.

  2195 \end{lem}

  2194 \end{lem}

  2196 

  2195 

  2197 \begin{proof}

  2196 \begin{proof}

  2198 (Sketch)

  2197 (Sketch)

  2209 

  2208 

  2210 For $n=1$ we have to check an additional ``global" relations corresponding to 

  2209 For $n=1$ we have to check an additional ``global" relations corresponding to 

  2211 rotating the 0-sphere $E$ around the 1-sphere $\bd X$.

  2210 rotating the 0-sphere $E$ around the 1-sphere $\bd X$.

  2212 But if $n=1$, then we are in the case of ordinary algebroids and bimodules,

  2211 But if $n=1$, then we are in the case of ordinary algebroids and bimodules,

  2213 and this is just the well-known ``Frobenius reciprocity" result for bimodules.

  2212 and this is just the well-known ``Frobenius reciprocity" result for bimodules.

  2215 

  2214 

  2216 \medskip

  2215 \medskip

  2217 

  2216 

  2218 We have now defined $\cS(X; c)$ for any $n{+}1$-ball $X$ with boundary decoration $c$.

  2217 We have now defined $\cS(X; c)$ for any $n{+}1$-ball $X$ with boundary decoration $c$.

  2219 We must also define, for any homeomorphism $X\to X'$, an action $f: \cS(X; c) \to \cS(X', f(c))$.

  2218 We must also define, for any homeomorphism $X\to X'$, an action $f: \cS(X; c) \to \cS(X', f(c))$.

  2238 

  2237 

  2239 \nn{still to do: associativity}

  2238 \nn{still to do: associativity}

  2240 

  2239 

  2241 \medskip

  2240 \medskip

  2242 

  2241