1250 \nn{surely $\cbar'$ can't be empy: we don't allow $D_1$ to be empty.} |
1250 \nn{surely $\cbar'$ can't be empy: we don't allow $D_1$ to be empty.} |
1251 Translating from the boundary map for $(\cM_\cC\ot {_\cC\cN})^*$ appearing in Equation \eqref{eq:tensor-product-boundary}, |
1251 Translating from the boundary map for $(\cM_\cC\ot {_\cC\cN})^*$ appearing in Equation \eqref{eq:tensor-product-boundary}, |
1252 we have |
1252 we have |
1253 \begin{eqnarray*} |
1253 \begin{eqnarray*} |
1254 (\bd g)(\olD\ot x \ot \cbar) &=& \bd(g(\olD\ot x \ot \cbar)) + g(\olD\ot\bd(x\ot\cbar)) + \\ |
1254 (\bd g)(\olD\ot x \ot \cbar) &=& \bd(g(\olD\ot x \ot \cbar)) + g(\olD\ot\bd(x\ot\cbar)) + \\ |
1255 & & \;\; g((\bd_+\olD)\ot x\ot\cbar) + \gl(g((\bd_0\olD)\ot x\ot\cbar')\ot\cbar'') . |
1255 & & \;\; g((\bd_+\olD)\ot x\ot\cbar) + \gl''(g((\bd_0\olD)\ot \gl'(x\ot\cbar'))\ot\cbar'') . |
1256 \end{eqnarray*} |
1256 \end{eqnarray*} |
1257 \nn{put in signs, rearrange terms to match order in previous formulas} |
1257 \nn{put in signs, rearrange terms to match order in previous formulas} |
1258 Here $\gl$ denotes the module action in $\cY_\cC$. |
1258 Here $\gl''$ denotes the module action in $\cY_\cC$ |
|
1259 and $\gl'$ denotes the module action in $\cX_\cC$. |
1259 This completes the definition of $\hom_\cC(\cX_\cC \to \cY_\cC)$. |
1260 This completes the definition of $\hom_\cC(\cX_\cC \to \cY_\cC)$. |
1260 |
1261 |
1261 Note that if $\bd g = 0$, then each |
1262 Note that if $\bd g = 0$, then each |
1262 \[ |
1263 \[ |
1263 g(\olD\ot -) : \cX(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{p-1}) \to \cY(I_1\cup\cdots\cup I_{p-1}) |
1264 g(\olD\ot -) : \cX(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{p-1}) \to \cY(I_1\cup\cdots\cup I_{p-1}) |
1290 Trivially, we have $(\bd g)(\olD\ot x \ot \cbar) = 0$ if $\deg(\olD) > 1$. |
1291 Trivially, we have $(\bd g)(\olD\ot x \ot \cbar) = 0$ if $\deg(\olD) > 1$. |
1291 If $\deg(\olD) = 1$, $(\bd g) = 0$ is equivalent to the fact that $h$ commutes with gluing. |
1292 If $\deg(\olD) = 1$, $(\bd g) = 0$ is equivalent to the fact that $h$ commutes with gluing. |
1292 If $\deg(\olD) = 0$, $(\bd g) = 0$ is equivalent to the fact |
1293 If $\deg(\olD) = 0$, $(\bd g) = 0$ is equivalent to the fact |
1293 that each $h_K$ is a chain map. |
1294 that each $h_K$ is a chain map. |
1294 |
1295 |
|
1296 We can think of a general closed element $g\in \hom_\cC(\cX_\cC \to \cY_\cC)$ |
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1297 as a collection of chain maps which commute with the module action (gluing) up to coherent homotopy. |
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1298 \nn{ideally should give explicit examples of this in low degrees, |
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1299 but skip that for now.} |
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1300 \nn{should also say something about composition of morphisms; well-defined up to homotopy, or maybe |
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1301 should make some arbitrary choice} |
1295 \medskip |
1302 \medskip |
1296 |
1303 |
1297 Given $_\cC\cZ$ and $g: \cX_\cC \to \cY_\cC$ with $\bd g = 0$ as above, we next define a chain map |
1304 Given $_\cC\cZ$ and $g: \cX_\cC \to \cY_\cC$ with $\bd g = 0$ as above, we next define a chain map |
1298 \[ |
1305 \[ |
1299 g\ot\id : \cX_\cC \ot {}_\cC\cZ \to \cY_\cC \ot {}_\cC\cZ . |
1306 g\ot\id : \cX_\cC \ot {}_\cC\cZ \to \cY_\cC \ot {}_\cC\cZ . |
1300 \] |
1307 \] |
1301 \nn{this is fairly straightforward, but the details are messy enough that I'm inclined |
1308 |
1302 to postpone writing it up, in the hopes that I'll think of a better way to organize things.} |
1309 \nn{not sure whether to do low degree examples or try to state the general case; ideally both, |
1303 |
1310 but maybe just low degrees for now.} |
1304 |
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1305 |
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1306 |
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1307 \medskip |
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1308 |
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1309 |
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1310 \nn{do we need to say anything about composing morphisms of modules?} |
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1311 |
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1312 \nn{should we define functors between $n$-cats in a similar way?} |
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1313 |
1311 |
1314 |
1312 |
1315 \nn{...} |
1313 \nn{...} |
1316 |
1314 |
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1315 |
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1316 |
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1317 |
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1318 \medskip |
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1319 |
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1320 |
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1321 \nn{should we define functors between $n$-cats in a similar way? i.e.\ natural transformations |
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1322 of the $\cC$ functors which commute with gluing only up to higher morphisms? |
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1323 perhaps worth having both definitions available. |
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1324 certainly the simple kind (strictly commute with gluing) arise in nature.} |
1317 |
1325 |
1318 |
1326 |
1319 |
1327 |
1320 |
1328 |
1321 |
1329 |