1267 \nn{surely $\cbar'$ can't be empy: we don't allow $D_1$ to be empty.} |
1267 \nn{surely $\cbar'$ can't be empy: we don't allow $D_1$ to be empty.} |
1268 Translating from the boundary map for $(\cM_\cC\ot {_\cC\cN})^*$ appearing in Equation \eqref{eq:tensor-product-boundary}, |
1268 Translating from the boundary map for $(\cM_\cC\ot {_\cC\cN})^*$ appearing in Equation \eqref{eq:tensor-product-boundary}, |
1269 we have |
1269 we have |
1270 \begin{eqnarray*} |
1270 \begin{eqnarray*} |
1271 (\bd g)(\olD\ot x \ot \cbar) &=& \bd(g(\olD\ot x \ot \cbar)) + g(\olD\ot\bd(x\ot\cbar)) + \\ |
1271 (\bd g)(\olD\ot x \ot \cbar) &=& \bd(g(\olD\ot x \ot \cbar)) + g(\olD\ot\bd(x\ot\cbar)) + \\ |
1272 & & \;\; g((\bd_+\olD)\ot x\ot\cbar) + \gl(g((\bd_0\olD)\ot x\ot\cbar')\ot\cbar'') . |
1272 & & \;\; g((\bd_+\olD)\ot x\ot\cbar) + \gl''(g((\bd_0\olD)\ot \gl'(x\ot\cbar'))\ot\cbar'') . |
1273 \end{eqnarray*} |
1273 \end{eqnarray*} |
1274 \nn{put in signs, rearrange terms to match order in previous formulas} |
1274 \nn{put in signs, rearrange terms to match order in previous formulas} |
1275 Here $\gl$ denotes the module action in $\cY_\cC$. |
1275 Here $\gl''$ denotes the module action in $\cY_\cC$ |
|
1276 and $\gl'$ denotes the module action in $\cX_\cC$. |
1276 This completes the definition of $\hom_\cC(\cX_\cC \to \cY_\cC)$. |
1277 This completes the definition of $\hom_\cC(\cX_\cC \to \cY_\cC)$. |
1277 |
1278 |
1278 Note that if $\bd g = 0$, then each |
1279 Note that if $\bd g = 0$, then each |
1279 \[ |
1280 \[ |
1280 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}) |
1281 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}) |
1307 Trivially, we have $(\bd g)(\olD\ot x \ot \cbar) = 0$ if $\deg(\olD) > 1$. |
1308 Trivially, we have $(\bd g)(\olD\ot x \ot \cbar) = 0$ if $\deg(\olD) > 1$. |
1308 If $\deg(\olD) = 1$, $(\bd g) = 0$ is equivalent to the fact that $h$ commutes with gluing. |
1309 If $\deg(\olD) = 1$, $(\bd g) = 0$ is equivalent to the fact that $h$ commutes with gluing. |
1309 If $\deg(\olD) = 0$, $(\bd g) = 0$ is equivalent to the fact |
1310 If $\deg(\olD) = 0$, $(\bd g) = 0$ is equivalent to the fact |
1310 that each $h_K$ is a chain map. |
1311 that each $h_K$ is a chain map. |
1311 |
1312 |
|
1313 We can think of a general closed element $g\in \hom_\cC(\cX_\cC \to \cY_\cC)$ |
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1314 as a collection of chain maps which commute with the module action (gluing) up to coherent homotopy. |
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1315 \nn{ideally should give explicit examples of this in low degrees, |
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1316 but skip that for now.} |
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1317 \nn{should also say something about composition of morphisms; well-defined up to homotopy, or maybe |
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1318 should make some arbitrary choice} |
1312 \medskip |
1319 \medskip |
1313 |
1320 |
1314 Given $_\cC\cZ$ and $g: \cX_\cC \to \cY_\cC$ with $\bd g = 0$ as above, we next define a chain map |
1321 Given $_\cC\cZ$ and $g: \cX_\cC \to \cY_\cC$ with $\bd g = 0$ as above, we next define a chain map |
1315 \[ |
1322 \[ |
1316 g\ot\id : \cX_\cC \ot {}_\cC\cZ \to \cY_\cC \ot {}_\cC\cZ . |
1323 g\ot\id : \cX_\cC \ot {}_\cC\cZ \to \cY_\cC \ot {}_\cC\cZ . |
1317 \] |
1324 \] |
1318 \nn{this is fairly straightforward, but the details are messy enough that I'm inclined |
1325 |
1319 to postpone writing it up, in the hopes that I'll think of a better way to organize things.} |
1326 \nn{not sure whether to do low degree examples or try to state the general case; ideally both, |
1320 |
1327 but maybe just low degrees for now.} |
1321 |
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1322 |
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1323 |
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1324 \medskip |
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1325 |
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1326 |
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1327 \nn{do we need to say anything about composing morphisms of modules?} |
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1328 |
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1329 \nn{should we define functors between $n$-cats in a similar way?} |
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1330 |
1328 |
1331 |
1329 |
1332 \nn{...} |
1330 \nn{...} |
1333 |
1331 |
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1332 |
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1333 |
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1334 |
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1335 \medskip |
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1336 |
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1337 |
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1338 \nn{should we define functors between $n$-cats in a similar way? i.e.\ natural transformations |
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1339 of the $\cC$ functors which commute with gluing only up to higher morphisms? |
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1340 perhaps worth having both definitions available. |
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1341 certainly the simple kind (strictly commute with gluing) arise in nature.} |
1334 |
1342 |
1335 |
1343 |
1336 |
1344 |
1337 |
1345 |
1338 |
1346 |