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. |
1037 The second construction, we we call the {\it local} homotopy colimit, |
1037 The second construction, which we call the {\it local} homotopy colimit, |
1038 \nn{give it a different name?} |
|
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. |
1354 \nn{should give figure} |
1353 \nn{should maybe give figure} |
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. |
1831 (See Figure \nn{need figure}.) |
1830 (See Figure \nn{need figure, and improve caption on other figure}.) |
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); |
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. |
2214 \nn{find citation for this. Evans and Kawahigashi?} |
2213 \nn{find citation for this. Evans and Kawahigashi? Bisch!} |
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 $f: \cS(X; c) \to \cS(X', f(c))$. |