1182 It's easy to verify the remaining module axioms. |
1182 It's easy to verify the remaining module axioms. |
1183 |
1183 |
1184 Now we reinterpret $(\cM_\cC\ot {_\cC\cN})^*$ |
1184 Now we reinterpret $(\cM_\cC\ot {_\cC\cN})^*$ |
1185 as some sort of morphism $\cM_\cC \to (_\cC\cN)^*$. |
1185 as some sort of morphism $\cM_\cC \to (_\cC\cN)^*$. |
1186 Let $f\in (\cM_\cC\ot {_\cC\cN})^*$. |
1186 Let $f\in (\cM_\cC\ot {_\cC\cN})^*$. |
1187 Let $\olD$ be a chain of subdivisions with $D_0 = [J = I_1\cup\cdots\cup I_m]$, and let |
1187 Let $\olD = (D_0\cdots D_l)$ be a chain of subdivisions with $D_0 = [J = I_1\cup\cdots\cup I_m]$. |
1188 $m\ot \cbar \in \cM(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{m-1})$. |
1188 Recall that $(_\cC\cN)^*(I_1\cup\cdots\cup I_{p-1}) = (_\cC\cN(I_p))^*$. |
1189 |
1189 Then for each such $\olD$ we have a degree $l$ map |
1190 |
1190 \begin{eqnarray*} |
1191 |
1191 \cM(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{p-1}) &\to& (_\cC\cN)^*(I_1\cup\cdots\cup I_{p-1}) \\ |
1192 |
1192 m\ot \cbar &\mapsto& [n\mapsto f(\olD\ot m\ot \cbar\ot n)] |
1193 |
1193 \end{eqnarray*} |
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1194 |
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1195 We are almost ready to give the definition of morphisms between arbitrary modules |
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1196 $\cX_\cC$ and $\cY_\cC$. |
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1197 Note that the rightmost interval $I_m$ does not appear above, except implicitly in $\olD$. |
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1198 To fix this, we define subdivisions are antirefinements of left-marked intervals. |
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1199 Subdivisions are just the obvious thing, but antirefinements are defined to mimic |
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1200 the above antirefinements of the fixed interval $J$, but with the rightmost subinterval $I_m$ always |
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1201 omitted. |
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1202 More specifically, $D\to D'$ is an antirefinement if $D'$ is obtained from $D$ by |
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1203 gluing subintervals together and/or omitting some of the rightmost subintervals. |
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1204 (See Figure xxxx.) |
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1205 |
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1206 Now we define the chain complex $\hom_\cC(\cX_\cC \to \cY_\cC)$. |
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1207 The underlying vector space is |
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1208 \[ |
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1209 \prod_l \prod_{\olD} \hom[l]\left( |
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1210 \cX(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{p-1}) \to |
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1211 \cY(I_1\cup\cdots\cup I_{p-1}) \rule{0pt}{1.1em}\right) , |
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1212 \] |
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1213 where, as usual $\olD = (D_0\cdots D_l)$ is a chain of antirefinements |
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1214 (but now of left-marked intervals) and $D_0$ is the subdivision $I_1\cup\cdots\cup I_{p-1}$. |
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1215 $\hom[l](- \to -)$ means graded linear maps of degree $l$. |
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1216 |
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1217 \nn{small issue (pun intended): |
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1218 the above is a vector space only if the class of subdivisions is a set, e.g. only if |
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1219 all of our left-marked intervals are contained in some universal interval (like $J$ above). |
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1220 perhaps we should give another version of the definition in terms of natural transformations of functors.} |
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1221 |
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1222 Abusing notation slightly, we will denote elements of the above space by $g$, with |
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1223 \[ |
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1224 \olD\ot x \ot \cbar \mapsto g(\olD\ot x \ot \cbar) \in \cY(I_1\cup\cdots\cup I_{p-1}) . |
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1225 \] |
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1226 For fixed $D_0$ and $D_1$, let $\cbar = \cbar'\ot\cbar''$, where $\cbar'$ corresponds to the subintervals of $D_0$ which map to $D_1$ and $\cbar''$ corresponds to the subintervals |
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1227 which are dropped off the right side. |
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1228 (Either $\cbar'$ or $\cbar''$ might be empty.) |
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1229 Translating from the boundary map for $(\cM_\cC\ot {_\cC\cN})^*$ \nn{give ref?}, |
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1230 we have |
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1231 \begin{eqnarray*} |
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1232 (\bd g)(\olD\ot x \ot \cbar) &=& \bd(g(\olD\ot x \ot \cbar)) + g(\olD\ot\bd(x\ot\cbar)) + \\ |
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1233 & & \;\; g((\bd_+\olD)\ot x\ot\cbar) + \gl(g((\bd_0\olD)\ot x\ot\cbar')\ot\cbar'') . |
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1234 \end{eqnarray*} |
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1235 Here $\gl$ denotes the module action in $\cY_\cC$. |
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1236 This completes the definition of $\hom_\cC(\cX_\cC \to \cY_\cC)$. |
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1237 |
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1238 Note that if $\bd g = 0$, then each |
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1239 \[ |
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1240 g(\olD\ot -) : \cX(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{p-1}) \to \cY(I_1\cup\cdots\cup I_{p-1}) |
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1241 \] |
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1242 constitutes a null homotopy of |
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1243 $g((\bd \olD)\ot -)$ (where the $g((\bd_0 \olD)\ot -)$ part of $g((\bd \olD)\ot -)$ |
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1244 should be interpreted as above). |
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1245 |
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1246 \nn{do we need to say anything about composing morphisms of modules?} |
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1247 |
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1248 \nn{should we define functors between $n$-cats in a similar way?} |
1194 |
1249 |
1195 |
1250 |
1196 \nn{...} |
1251 \nn{...} |
1197 |
1252 |
1198 |
1253 |